EPR Paradox and Bell s Theorem

Transcription

1 Università degli Studi di Perugia Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Triennale in Fisica Tesi di Laurea Triennale EPR Paradox and Bell s Theorem Candidato: Danny Laghi Relatore: Prof. Gianluca Grignani Anno Accademico Sessione di Laurea 30 Settembre 2013

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3 Alla mia famiglia, senza della quale questo (piccolo) traguardo non sarebbe stato raggiunto

4 Entanglement is not one, but rather the characteristic trait of Quantum Mechanics. Erwin Schrödinger

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6 Contents 1 Introduction 7 2 The EPR Paradox Definition of entanglement Reality, Locality and Completeness EPR Paradox Bohm-Aharonov s formulation of the paradox Bell s Theorem Bell s Inequalities CHSH Inequalities Optical version of Bell s Gedankenexperiment A. Aspect s experiments Experiment with single-channel polarizers Experiment with two-channel polarizers Experiment with time-varying polarizers Towards the impossibility of FTL communications No-Communication Theorem Some outlines about the FLASH Project Conclusions on Quantum Mechanics Interpretations of Quantum Mechanics: Copenhagen, Bohm Copenhagen interpretation Bohm s interpretation Open questions on Quantum Mechanics A Some expectation values of Quantum Mechanics 71 Bibliography 77 6

7 Chapter 1 Introduction Quantum Mechanics, as its own peculiarity, introduces some ideas that depart from everyday experience and common sense. Until the end of the 19th century, the interpretation of macroscopic physical phenomena was based on Newton s Laws, which explained mechanical, acoustic and thermal phenomena, and on Maxwell s Equations, which governed the electric, magnetic and optical ones. These two large classes of physical phenomena yielded a distinction between the wave-like behavior (or the electromagnetic field radiation) and the corpuscolar character of matter, but from the beginning of the last century, physicists realized that there were some phenomena that could not be organized within the system of classic laws, and some of them voided the distinction between particle and field: for instance, the atom structure and the mechanism of emission and absorption of matter radiation. Gradually, there was the necessity of introducing a theory, without vanishing the excellent results obtained with classical mechanics until then, which was capable to extend the latter in order to describe even those phenomena that were remaining not understood. There were too facts for whom there were not any explanations, according to classical laws and statistics which were incredibly losing their own descriptive and predictive power for certain physical phenomena, like the black body radiation spectrum and the spectral lines of hydrogen atom. It is in this scenary that the first quantum theory was born, laying its foundations with the introduction of the quantum of energy, introduced in 1900 by Max Planck in order to give a plausible explanation for the spectrum of black body radiation. Planck postulated (and then showed) that the energy exchanged between matter and radiation does not show itself in a continuous manner, but by discrete and indivisible quantities, or quanta of energy, that must be proportional to the frequency of the radiation; he obtained an expression for the spectrum, in accordance with the experimental distribution, by properly adjusting the constant of proportionality. So the energy quanti- 7

8 zation was the first step toward a quantum theory able to give reason for the experimental data, incompatible with the classical theory. The second important step was taken in 1913 by Niels Bohr, who, basing his works on the emission spectrum of hydrogenoid atoms, succeeded in generalizing the well-known Balmer s empirical formula for the spectrum of hydrogen atom, obtaining a first, fundamental, model for hydrogenoid atoms. This model was based on some postulates, the first of them imposing a new discretization, this time for another physical quantity that was always been considered like a continuous variable: the angular momentum. When the quantization of the angular momentum is considered, some orbits were imposed to the electron (second postulate): in these orbits (called stationary levels) the electron could not emit energy quanta. It will be only with the introduction of two formalisms, the matrix mechanics and the wave mechanics, introduced by Werner Heisenberg (1925) and Erwin Schrödinger (1926) respectively, that began to emerge a real quantum theory, which was able to clear up in a more complete way the experiments, one over all, the double-slit experiment that Richard Feynman was used to say it holds the essential mystery, rather, the only mystery of Quantum Mechanics. This theory shows a double nature wave-like and corpuscular of elementary particles, confirms the presence of discrete energy levels as assumed by Planck, and gives an inferior limit about how much precisely it is possible to investigate on physical world, because of the well-known Uncertainty Principle. But undoubtedly the most peculiar feature of Quantum Mechanics is the introduction of a wave function ψ, solution of the Schrödinger equation and representation of the physical state of a quantum system, for which it is the complete description. And it is the completeness of the wave function, that is, the quantum theory behind it, that was called into question in 1935 by Albert Einstein, Boris Podolsky and Nathan Rosen (EPR in the following), in their famous paper Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?, where they showed through a Gedankenexperiment their so famous argument, later baptized as EPR paradox, pointing out how the quantum theory couldn t simultaneously satisfy three principles: reality, locality and completeness. Being assumed the first and the second as obvious and incontrovertible (it was indeed the impossibility to accept an istantaneous spooky action at a distance one of their key point), they inferred that it was the third principle to be abandoned, that is, the wave function could not contain all of the possible information about quantum system. Many controversies arose from this paper, mostly of them contesting the hypotheses on the basis of their demonstration, so that the paradox became one of the principal 8

9 subject of discussion. Two main tendencies of opposite thought standed from that: on one hand, those belonging to the Copenhagen school, which supported a probabilistic and undetermined vision of the reality and thought that within this interpretation the paradox was based on inconsistent hypotheses; on the other hand, there were those which, according to the EPR paradox conclusions, were convinced of the incompleteness of Quantum Mechanics, so that the latter had to be completed with the introduction of additional parameters, that could explain the queer correlations that, like in EPR demonstration, seemed to emerge between remote systems. In this way came up the hypothesis of the introduction of additional degrees of freedom with respect to those considered in Quantum Mechanics, which allowed to make it a deterministic and complete theory. Nevertheless in 1964 an Irish physicist, John Stewart Bell, demonstrated that considering local theories with additional hidden variables, there would arise some inequalities which generally are violated by Quantum Mechanics. The importance of this work is that the experiments have the possibility to determine if such a theory is possible or not, according to the results stemming from laboratory; thus the possibility of choosing what interpretation to assume for Quantum Mechanics, either a local deterministic theory, or a nonlocal probabilistic one, is no longer arbitrary, but it is told by the physics itself; as we shall see, by means of the experimental results obtained and comparated with Bell s predictions, Nature has showed an exclusion about the possibility of being local. Since 1972 several experiments, more refined as they were executed, have been conceived and conducted to verify the effectiveness of Bell s inequalities (in particular, as we shall see, a series of decisive experiments has been performed in by Alain Aspect et al.). The experiments, within an acceptable range of experimental errors, proved that: (a) nonlocal correlations predicted by Quantum Mechanics exist; (b) Bell s inequalities are violated. Generally the arguments we are going to deal with are still not treated in a complete uniform manner by Quantum Mechanics books, mainly because of the subtlety of the questions and, of course, the difficulty to verify them experimentally that arise out of these subjects and because these arguments, though some sure points have been reached, 9

10 maybe have not been completely understood in their many facets yet 1. The following work just wants to run through the fundamental stages in the development of EPR problem, pointing out and describing as well the several possible solutions for this problematic argument, and analyzing the main significative experiments that have been made in order to verify some possible theoretic models, based on alternative principles instead of orthodox Quantum Mechanics. It is a sure thing that the concept that Einstein involved for questioning about the completeness of quantum theory, i.e. the entanglement of two particles far apart from each other, is a real physical phenomenon, whatever the meaning one could assume for the latter terms. The life and work of Einstein, hard critic of quantum theory, just because he was not able to accept that Nature expresses itself in a probabilistic way ( God doesn t play dice with the world ), showed that curiously the German physicist was right even when he thought he was wrong (e.g. about the cosmological constant). And as for the quantum world, Einstein s paper of 1935, was actually the seed for one of the most important discoveries in physics in the twentieth century: the actual discovery of entanglement through physical experiments. Bell s contribution, whose works have been of fundamental importance in the development of this subject, as we shall see, set out to prove that the Einstein-Podolsky-Rosen thought experiment was far from being an absurd idea to be used just to invalidate the completeness of the quantum theory, but rather the description of a real phenomenon. The existence of this phenomenon had to provide a new proof in favour of Quantum Mechanics, while against a limiting view of reality. We shall see how the experiments have actually shown that. 1 For instance, see [10]. 10

11 Chapter 2 The EPR Paradox Quantum Mechanics has been founded on radical revisions of many classical concepts. For example, in order to take into account the wave-particle duality, it had to give up the concept of classical trajectory, because of Heisenberg s Uncertainty Principle: choosing as canonically conjugate observables the position and momentum of a particle, it describes quantitatively the impossibility of defining precisely and simultaneously its position and its velocity. According to that, the thing which Einstein thought was that quantum formalism was incomplete. The paper of 1935 by A. Einstein, B. Podolsky and N. Rosen [2] wants to demonstrate the non-completeness of the wave function, thought as description of reality. What EPR show is substantially a thought experiment (Gedankenexperiment is the Deutch term) that proofs how, assuming two principles indicated as reality and locality principle, Quantum Mechanics formalism leads to a contradiction, unless one introduces the existence of hidden variables. EPR show that quantum formalism can allow for the existence of states describing two particles, for which one is able to expect strong correlations either for the velocity or the position, even if the particles are widely separated and no longer interacting to each other. The position measurements would always give symmetric values with respect to the origin, so that a measure of one particle would allow one to know with certainty the value of the position of the other. Similarly, measurements of the momentum of the two particles would lead to opposite values, so that the measurement on one of them would be sufficient for having knowledge of the value of the other with certainty. Undoubtedly, one has to choose between a precise measurement of position or momentum for one particle, because of Heisenberg s Uncertainty Principle. But the measurement on the first particle cannot disturb the second (distant) particle, and it is here that the 11

12 principle of locality is introduced, by means of which EPR conclude that the second particle must have had, long before the measurement, predetermined values of position and momentum. Since for Quantum Mechanics it is not possible to know precisely and simultaneously the values of these quantitities, Einstein and his co-authors inferred that the theory is incomplete. Following up the publication of this paper, E. Schrödinger coined the term entanglement, just to describe the impossibility of factorizing a quantum state like the one assumed by EPR. But before looking in detail the EPR paper and the paradox that arises from it, it is best to define what we intend for entangled states and, in general, for completeness, reality and locality. 2.1 Definition of entanglement Entanglement is a purely quantum phenomenon, i.e. it is not supplied with a classical counterpart; it is in this way that the quantum state of two or more physical systems depends on the states of everyone of the systems that compose it. A first definition of entanglement that we propose here is just the first absolute definition of the term, due to E. Schrödinger, who conied it himself following up EPR paper. But in order to get the Schrödinger s definition, it is primarily necessary to define the important concepts of pure states and mixed states. Definition 1. Given a vector space X, a finite linear combination i α ix i is called convex if α i [0, 1] and i α i = 1. Moreover, C X is called convex if for any pair x, y C, λx + (1 λ)y C, λ [0, 1] (and thus any convex combination of elements in C belongs to C). Definition 2. If C is convex, an element e C is called extreme if it cannot be written as e = λx + (1 λ)y, with λ [0, 1], x, y C \{e}. Definition 3. Let H be a separable Hilbert space ( H is separable H has finite dimension). Let S(H) be a convex subset of H. Then the extreme elements in S(H) are called pure states. Non extreme-states are mixed states, or nonpure states. 12

13 Schrödinger defined as entangled those quantum pure states Ψ, from an ensamble of systems, that cannot be represented in a form of simple tensorial products of eigenstates of the systems, that is, Ψ ψ 1 ψ 2 ψ n, where denotes the tensorial product and ψ i are some vectors that represent the states of some Hilbert spaces. The states of composed systems, that can be instead represented as tensorial products of the subset states, constitute the complement set of pure states, known as product states. It is easy to determine whether a pure state of a system, composed of only two subsystems, is an entangled state making use of Schmidt s decomposition theorem [1], that is always valid for these systems 1. Indeed any bifactorized pure state Ψ can be written as a sum of bi-orthogonal terms: it is always possible to write Ψ in a form Ψ = i c i u i v i, with c i C, where the sets of vectors { u i } and { v i } consist of orthonormal unit vectors spanning the space of possible-state vectors for the system, the index i running up to the smaller of the dimensions of the two subsystem Hilbert spaces. On the basis of our previous considerations, it is easy to get a second more formal definition: Definition 4. Let A and B be two non-interacting systems with their respective Hilbert spaces H A and H B, and let ψ A and φ B be two states in which there are the first and the second system respectively. Then the state of the composed system, belonging to the Hilbert space given by the tensorial product H A H B, is ψ A φ B. The states of the composed system that can be written in this way are called product states or separable states. Generally, not all of the states of a composed system are product states; let { u i A } and { v i B } be some bases in their respective Hilbert spaces H A and H B. According to 1 In the simple case of finite dimensional spaces, Schmidt s decomposition theorem asserts that any double sum Ψ = kl X klx my n, can be converted into a single sum Ψ = α aαξαηα, by means of unitary transformations ξ α = k U αkx k and η α = l V αly l. If {x k } and {y l } are two orthonormal bases for two distinct vector spaces, then {ξ α} and {η α} are also two, possibly complete, orthonormal vector bases for these spaces. The absolute values a α are called the singular values of the matrix X, and are easily calculated by noting that a α 2 are the nonvanishing eigenvalues of the Hermitian matrices XX and X X. The corresponding sets of eigenvectors are {ξ α} and {η α}, respectively. 13

14 Schmidt s decomposition, any state in the composed system can readily be written as ψ AB = i c i u i A v i B. (2.1) Then the state (2.1) is a product state, or a non-entangled state, if there is one and only one i so that c i 0, and all of the other c j = 0, j i; that is, a state is non-entangled if and only if ψ AB = c i u i A v i B, i fixed. (2.2) If instead there are more than one c i 0 for the state 2.1, then such a state is called entangled state. If i = 1,..., N and if c i = 1 N, i, the state (2.1) is called to be maximally entangled. In other words, a state is entangled if and only if it cannot be factorized. As an explanatory example of this definition, let s consider two bases { 0 A, 1 A } H A and { 0 B, 1 B } H B. Then an entangled state of the composed system H A H B is represented by 1 2 ( 0 A 1 B 1 A 0 B ), that is, omitting tensorial product symbols, 1 2 ( 0 A 1 B 1 A 0 B ), where it is evident the impossibility of attributing separately a quantum state either to the system A or to the system B without involving the other. A similar state to the one above is a singlet state for a pair of electrons. Thus if we knew, by experimental measurements, that for example the system A would be in the state 1 A, then, unless of coefficients that we can neglect, we will able to know automatically that the system B will be in the state 0 B. How to obtain entangled states There are some different ways for achieving entangled states of two systems. 14

15 For instance, in the decay of a spin 0 particle, the final products are two spin 1/2 particles in a singlet state (like π 0 γ + e + + e ). Another example is in the e + e annihilation, that can produce a φ meson, that can decay, for instance, in a couple of K 0 K0 in a spin singlet state. Similarly it is possible to create photons entangled in polarization as result of positronium annihilation (e + + e γ + γ). One more example of entangled state is the above mentioned singlet spin of two electrons near an atomic nucleus. Entanglement has numerous applications beyond its use in the paradox topic as well; for instance an application of such states is in Quantum Information Theory, in Quantum Cryptography, where in the one time pad transmission, a couple of entangled particles is used to exchange a random coding key between sender and receiver, in a completely secure way from eavesdropping attacks. 2.2 Reality, Locality and Completeness As we already said, EPR paper relies substantially upon the assumption of some principles, i.e. the completeness, the reality and the locality. So it is good to find for these concepts some definitions applicable to our contest [2]. Principle of reality The elements of physical reality cannot be determined by a priori philosophical considerations, but they are to be found using measurements and experimental results. Far from being an exhaustive definition, EPR consider as satisfactory the following definition for an element of physical reality. If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exist an element of physical reality corresponding to this physical quantity, that is, an objective property of the system, independent of any eventual external observer. Thus, all of the possible physical observables must have some preexistent values for any possible measurement, before of carrying out that measurement. 15

16 Principle of locality Accordingly to the existence of an upper limit for the velocity of propagation of a signal, coincident with the speed light c in vacuum, an object is affected only directly by his immediate neighbourhood. This lead us to a definition of locality. Given two physical systems, let s suppose that during a certain time interval they remain isolated from each other. Then the temporal evolution of the physical properties of one of them during that time interval cannot be influenced by operations executed on the other (Einstein s locality). Condition of completeness of a theory According to the definition adopted by EPR in their paper (called by them condition of completeness ): In order for a theory to be complete, it is necessary that any elements of physical reality must have a counterpart in a physical theory. Whether an element of physical reality has no counterpart in the theory, then we shall say that the theory is not complete. 2.3 EPR Paradox Given two noncommuting physical quantities, because of the Uncertainty Principle, the exact knowledge of one of them precludes the exact knowledge of the other. In fact, any attempt of determining a quantity by means of measurement changes the system state, so that the value of the other quantity is undetermined. In other words, two noncommuting physical quantities cannot have both expected values. Starting from this assumption, EPR state that: 1. either the Quantum Mechanics description of reality given by the wave function is not complete; 2. or the physical quantities associated with noncommuting operators cannot have simultaneous reality. 16

17 It can be easily shown that one of the above conclusions has to be necessarily true, for there are no other possible situations. Actually, if we supposed ab absurdum that both the assumptions were wrong, we could consider two noncommuting physical quantities A, B, that have simultaneous reality (negation of 2.). Then, for the principle of reality, both the solutions should have some defined values that, by means of the completeness condition, would be part of the complete description of reality. But then, if the wave function was able to provide such complete description of the reality (negation of 1.), it would include those values, which would be therefore predictable. But this is an absurd, for it would contradict the Uncertainty Principle. In this way, we have showed that 1. and 2. are complementary: whether we assume the first as false, then the second must be necessarily true, and vice versa. EPR create the paradox assuming the completeness of the wave function and, at the same time, deducing the physical reality for two quantities, whose associated operators do not commute, that is, inferring that both 1. and 2. are false, thus obtaining an absurd, as we have seen above. So they conclude that, in order to solve the paradox, one of the three assumptions has to be dropped out, that is one amongst: the principle of reality the principle of locality the completeness of Quantum Mechanics. Since it seemed that the first two hypotheses could not be debated, unless of having a senseless theory (but actually it is just the debate on these assumptions to open to other possible solutions of the paradox), EPR drop out the validity of the remaining assumption, that is, Quantum Mechanics is an incomplete theory. We now shall see what are the arguments of EPR paper and how the paradox arises from it, i.e. how to get both the negation of 1. and 2.. Paradox formulation It is helpful to run through the sheer introductory passages of the EPR paper [2]. Let s consider a particle having a single degree of freedom, i.e. in a unidimensional space. The fundamental concept of the theory is the concept of state, which is supposed 17

18 to be completely characterized by the wave function ψ, which is a function of the variables chosen to describe the particle s behaviour. Corresponding to each physically observable quantity A there is an operator, which may be designated by the same letter (with a hat ). If ψ is an eigenfunction of the operator Â, that is, if: Âψ = aψ, (2.3) where a R, then the physical quantity A has with certainty the value a whenever the particle is in the state described by ψ. In accordance with the principle of reality, for this particle there is an element of physical reality corresponding to the physical quantity A. For instance, let s consider, ψ = e i p 0x, (2.4) where p 0 is a numerical constant and x the independent variable. Taking the operator corresponding to the linear momentum of a particle, we obtain ˆp = i x, (2.5) ˆpψ = i ψ x = p 0ψ, (2.6) and therefore, in the state (2.4), the momentum assume with certainty the value p 0. The momentum of the particle therefore has physical reality. Now let s consider the case in which the eigenvalue equation (2.3) is not valid. In this case we can speak no longer of the quantity A as if it had a definite value. Such a case is that, for example, of the position of the particle. Its corresponding operator, ˆx, is the operator of multiplication by the independent variable x. Hence, ˆxψ = xψ aψ. (2.7) In accordance to Quantum Mechanics, it is just possible to say that the relative probability of a coordinate measurement giving a result lying between a and b is P (a, b) = b a ψ ψdx = b a dx = b a. (2.8) Since this probability is independent of a, but depends only upon the difference b a, 18

19 we see that all of the values for the coordinate are equally probable. From another point of view, we could have obtained such a conclusion without any computation, noting that the state (2.4) is representative of a unidimensional free wave of definite momentum p 0, so it is not possible to localize exactly a free wave, which extends as far as infinity, for it has an equivalent probability of being in any point of the (unidimensional) space. So a defined value of the position of a particle in a state given by (2.4) is unpredictable, but it can be reached only by direct measurement. In accordance with Quantum Mechanics, this action alters the particle state, that becomes a position eigentstate (and no longer of momentum). Therefore, after the determination of the position, the particle will be no longer in the state given by Eq. (2.4). We can conclude that whenever the momentum of a particle is known, its position has no physical reality. Now, supposing that the wave function of a certain physical system is complete (negation of 1.), yet Einstein, Podolsky and Rosen found that it is possible to have two simultaneous real physical quantities in spite of the fact that their respective operators do not commute (negation of 2.), that is an absurd. In order to deduce such an absurd, EPR built the following reasoning. Let s consider two systems I and II (which could be punctiform or hard rigid bodies, or whatever other generic physical systems), described by the variables x 1 and x 2 respectively. We suppose that they interact with each other from the time t = 0 to the time t = T. We suppose further that the states of the two systems before t = 0 are known. Then, by means of the wave function Ψ, it is possible to describe the state of the entangled system I + II for any successive time t > T using the Schrödinger s equation, even if, after the interaction, it is no longer possible to work out the states of I or II, for they are no longer factorable states. Now let s consider the physical quantity A, relative to the system I, which has some eigenvalues a 1, a 2, a 3,... and eigenfunctions u 1 (x 1 ), u 2 (x 1 ), u 3 (x 1 ),.... The wave function Ψ, considered as function of x 1, at a certain fixed time t > T, can then be expressed as: Ψ(x 1, x 2 ) = ψ n (x 2 )u n (x 1 ), (2.9) n=1 where x 2 stands for the variables used to describe the second system. Here ψ n (x 2 ) are to be regarded merely as the coefficients of the expansion of Ψ into a series of orthogonal functions u n (x 1 ), i.e. as Fourier s coefficients (u n (x 1 ), Ψ). 19

20 Now we want to measure A; suppose that it is found that it has the value a k. According to the precipitation postulate, after a measurement, the state of the system is given by Ψ = P Ψ, where P is a projection operator on the eigenspace relative to the found eigenvalue. Since the system I has fallen in the state described by the wave function u k (x 1 ), we get: Ψ(x 1, x 2 ) = ψ k (x 2 )u k (x 1 ). (2.10) Since for systems formed by independent subsystems the wave function is the product of the wave functions of the single subsystems, from Eq. (2.10) we deduce that, after the measurement, the second system will collapse into a definite state too, which is described by the wave function ψ k (x 2 ). So the wave packet given by the infinite series (2.9) is reduced to a single term given by (2.10). We now propose to consider, instead of the physical quantity A, another physical quantity, say B, relative to the system I as well, with eigenvalues b 1, b 2, b 3,... and eigenfunctions v 1 (x 1 ), v 2 (x 1 ), v 3 (x 1 ),.... In this way, analogously to Eq. (2.9), we can write Ψ(x 1, x 2 ) = φ m (x 2 )v m (x 1 ), (2.11) m=1 where φ m (x 2 ) are the new coefficients. In analogy with the previous passages, we measure B and suppose we find it in a value b j ; then the system collapses into the state described by the wave function v j (x 1 ); consequently we obtain Ψ(x 1, x 2 ) = φ j (x 2 )v j (x 1 ), (2.12) from which we deduce that system II has collapsed into the state described by the wave function φ j (x 2 ). We see therefore that, as a consequence of the two different measurements performed upon the first system, the second system may be left in states with two different wave functions: ψ k (x 2 ) and φ j (x 2 ), and, since at the time of measurement the two systems I, II no longer interact, no real change can take place in the second system in consequence of anything that may be done in the first system. 20

21 Thus, it is possible to assign two different wave functions, ψ k and φ j, to the same reality (the second system after the interaction with the first). Therefore, if we were able to choose two functions for the eigenfunctions belonging to two noncommuting operators, relative to two, non compatible, physical quantities P and Q, which assume the values p k and q j respectively, then we would automatically obtain the proof that two noncommuting quantities are simultaneously real (negation of 2.), that is exactly our aim. To show this, let s consider two generic systems, for example two point-like particles. We suppose that the wave function of the total system is: where x 0 is a constant. Ψ(x 1, x 2 ) = + e i (x 1 x 2 +x 0 )p dp, (2.13) Choosing as physical quantity A the observable linear momentum P, to which is naturally associated the operator ˆP of the form ˆP = i x, and measuring that quantity for the particle I, it will assume a value p; consequently the particle will fall into a state described by the eigenfunction u p (x 1 ) = e i x1p. (2.14) From now on, we consider the case of a continuous spectrum; hence Eq.(2.9), representative of the wave function of the total system, is Ψ(x 1, x 2 ) = + the projection on u p (x 1 ) will naturally lead to ψ p (x 2 )u p (x 1 )dp; (2.15) ψ p (x 2 ) = e i (x 2 x 0 )p, (2.16) that is nothing but the eigenfunction of the operator ˆP = i x 2, relative to the eigenvalue p for the momentum of the second particle. On the other hand, if we choose as physical quantity B the observable position Q, whose operator is the operator ˆQ of multiplication by x, and performing a measurement on the particle I, the latter will assume a value x, and consequently it will fall into the 21

22 state described by the eigenfunction where δ(x 1 x) is the well-known Dirac delta-function. v x (x 1 ) = δ(x 1 x), (2.17) Eq. (2.11), representative of the wave function of the total system, now becomes so we clearly have Ψ(x 1, x 2 ) = + + v x (x 1 )dx 1 e i (x x 2+x 0 )p dp; (2.18) φ x (x 2 ) = + e i (x x 2+x 0 )p dp = hδ(x x 2 + x 0 ), (2.19) that is the eigenfunction of the operator ˆQ = x 2, relative to the eigenvalue x + x 0 for the position of the second particle. At this point the paradox is obtained. In fact it is immediate to verify that ˆP and ˆQ are a pair of quantum operators associated with a couple of canonically conjugate variables, which do not commute: [ ˆP, ˆQ] = i. (2.20) Finally, we have obtained two wave functions, ψ k and φ j, associated with two non compatible quantities, that is, two noncommuting quantities P and Q, belonging to the second system, that became simultaneously real: thus it follows the paradox. Starting from the assumption that the wave function gave a real complete description of physical reality, EPR came to the conclusion that two physical quantities, with noncommuting operators, could have simultaneous reality. Consequently the negation of 1. leads to the negation of the only alternative 2.. Thus we conclude that the quantum mechanical description of physical reality given by the wave function is not complete. EPR conclude their paper saying that one could object them that the criterion of reality they have chosen is not sufficiently restrictive. Indeed, we would not arrive to a paradox if we just insisted on the point that two or more physical quantities could be viewed as simultaneous elements of reality only whether they could be measured or predicted simultaneously. From this point of view, since either one or the other, but not both simultaneously of the quantitites P and Q could be expected, they are not simultaneously real and the paradox therefore would not arise. Nevertheless, this procedure is such as to make the reality of P and Q depending on 22

23 the measuring process that is executed on the first system, which does not interfere in any way with the second one. And it is here that the principle of locality makes its part, preventing from taking into account that possibility. With EPR words: No reasonable definition of reality could be expected to permit this [2]. The paper ends opening to the search of new possible methods capable to describe completely the physical reality, i.e. seeking for a new theory comprehensive of the lacking information, so that, with the addition of additional parameters it would be able to completely describe the physical reality. 2.4 Bohm-Aharonov s formulation of the paradox Now we shall examine another formulation of the EPR paradox, following the example ideated by David Bohm and Yakir Aharonov [4]. In this model, a quantum system is considered from the point of view of its own spin variables. This formulation of the paradox is more handy and comfortable from a mathematical point of view, since, working with spins, the operators and the eigenfunctions will become matrices and vectors respectively, reducing all of the problem to a simple matricial calculus; moreover, the theoretical formulation of the problem is nearer to the experimental situations where it has been really examinated. Let s consider a diatomic molecule of spin 0, whose atoms (I and II) are our two systems taken into account, having spin 1/2. It is straightforward that the wave function of the composed system will be given by a singlet state of spin. We suppose that the couple of atoms interact for a certain time and then become entangled; after that time they are separated. From the time of separation on, there is no longer any interaction between them. We can certainly define the wave function of the composed system, even if we do not know the functions that describe the state of each single atom separately; omitting the orbital part, which is quite inessential for the description of the problem, the state describing the composed system is a singlet state of spin: Ψ = 1 2 [Ψ + (I)Ψ (II) Ψ (I)Ψ + (II)], (2.21) where Ψ ± (I) e Ψ ± (II) are the vectors that represent the state of the first and the second particle respectively, with spin ± 2 on the direction of the axis z. In that state, for the conservation of the angular momentum, the total spin always must vanish; whether one 23

24 of the two atoms assumes a positive value, the other will be certainly negative. Given the description of Bohm-Aharonov s model, it is easy to see that the quantities to be measured are two of the three components of the spin between one of the two atoms, that we know from Quantum Mechanics to be incompatible; that is, the relative operators generally obey to the relation: [S i, S j ] = i ε ijk S k. In order to see this, we maintain an analogy with EPR steps, choosing as physical quantity the S x component of the spin for I; measuring it, we shall obtain two possible values: + 2 and 2. In the first case, as we know from the knowledge of the eigenvectors of Ŝx, the atom I collapses into the state given by ( Ψ + (I) = 1 2 eigenvector of the operator Ŝx, relative to the assumed eigenvalue + 2. Therefore, for the conservation of angular momentum, it is necessarily 1 1 ), Ψ = Ψ + (I)Ψ (II), (2.22) from which we also deduce the state of the second atom, which will have a wave function given by ( Ψ (II) = 1 2 eigenvector of the operator S x, relative to the eigenvalue 2 of the spin for atom II. In the second case all is analogous, but with inverted eigenvalues and eigenvectors for the two atoms. 1 1 ), Choosing as physical quantity B the component S z of the spin for I, and measuring it, we will get the same two possible values of the previous case. So we see that whether the observable assumes a value + 2, then the atom I collapses into the state given by Ψ + (II) = ( 1 0 ), 24

25 eigenvector of the operator Ŝz, relative to the assumed value + 2. Then the state of the total system will be Ψ = Ψ + (I)Ψ (II), from which we can work out a state for the second atom, that will be equal to ( ) 0 Ψ (II) =, 1 eigenvector of the operator Ŝz, relative to the eigenvalue 2 of spin for atom II. In accordance with what we saw for Ŝx, when the measure of the observable B assumes the second of the two possible values, it is all analogous, but with inverted eigenvalues and eigenvectors for the atoms. In this way Bohm and Aharonov demonstrate that the generic quantities A and B are reduced to a pair of observables (in our example: S x and S z ), whose operators do not commute. Then the two wave functions, called by Einstein ψ k and φ j, represent two states of simultaneous reality for these operators, related to the second atom (in the example: Ψ +/ (II) for the S x -spin and ψ +/ (II) for the S z -spin). The result obtained is the same as that obtained by Einstein, Podolsky and Rosen: the steps followed are equivalent. Indeed both start considering a pair of incompatible quantities, and they successively measure these observables on one of the two considered systems. Both of them finally demonstrate that, measuring these quantities on one of the two systems, they are able to determine these values on the other system too, but without altering its state; that is, both demonstrate that the wave functions for the quantitites of the second systems, could correspond with the eigenfunctions of two noncommuting operators, related to two incompatible physical quantities. In order to solve the contradiction of both the examined formulation, one can drop one of the three hypotheses assumed, that we list here again: i) Principle of reality, ii) Principle of locality, iii) Completeness of Quantum Mechanics. 25

26 According to EPR, the hypothesis to be abandoned is nothing but the completeness of Quantum Mechanics, since, as they have demonstrated, the quantum description of reality, provided by the wave function, comes out to be inconsistent with the assumed principles. As well, since completeness of Quantum Mechanics means indeterminism, they are sure of the need to complete the latter by means of a more fundamental theory, in which the incompleteness is overwhelmed introducing additional dynamical parameters, the so-called hidden variables : in this way, in addition to the solution of the paradox, we would get back an essentially deterministic vision of the world, in which it is possible to associate with certainty a definite value for every quantity, i.e. a definite element of reality. As a matter of fact, the weak point of their proof is precisely to take for granted the assumptions i) and ii), as we already said, but this is not to be considered as a gross mistake of the authors, but rather a historical limit of their work. It is better to stress again the importance of one of the critical point of the argumentation of Einstein and collaborators. Expressing the principle of reality for the Gedankenexperiment of Bohm-Aharonov, it states that, if one can execute an operation, that allows him to predict with certainty the value of a measurement for a quantity without disturbing the measured spin, then the measurement has a definite value, apart from the fact that this operation is actually executed or not. In accordance with the Copenhagen interpretation that is the same of Bohr who, by the way, replied with an article [3] to that of EPR as follows the concept of reality cannot be legitimately applied to a property unless there is a device able to measure it, whereas Einstein intended this vision as anthropocentric, assering that instead physical systems have intrinsic properties, apart from the fact that they are observed or not. So it is here that arises the most large difference between the two currents of thought, each of them defending their own position with their arguments. But it is important to point out at once the concept of simultaneity, used more than once by EPR, since this word was a surprising expression for people who knew very well that this term was undefined in the teory of relativity. Let us examine this issue with Bohm s singlet model. [5] One observer, conventionally called Alice, measures the z-component of the spin for her particle and find + /2. Then she immediately knows that if another distant observer, Bob, measures (or has measured, or will measure) the z-component of the spin for his particle, the result is certainly /2. One could then ask when Bob s particle acquires the state with s z = /2. This question is meaningless, but it has a definite answer: Bob s particle acquires this state instantaneously in the Lorentz 26

27 frame that we arbitrarily choose to perform our calculations. Of course, Lorentz frames are not material objects: they exist only in our imagination. When Alice measures her spin, the information she gets is localized at her position, and will remain so until she decides to broadcast it. Absolutely nothing happens at Bob s location. From Bob s point of view, any spin directions are equally probable, as can be verified experimentally by repeating the experiment many times with a large number of singlets without taking in consideration Alice s results. Thus, after each one of her measurements, Alice assigns a definite pure state to Bob s particle, while from Bob s point of view the state is completely random. It is only if and when Alice informs Bob of the result she got (by mail, telephone, radio, or by means of any other material carrier, which is naturally restricted to the speed of light) that Bob realizes that his particle has a definite pure state. Until then, the two observers can legitimately ascribe different quantum states to the same system. For Bob, the state of his particle suddenly changes, not because anything happens to that particle, but because Bob receives information about a distant event. So reality might differ for different observers. Anyway, the conclusions reached by EPR raised a discussion that nowadays is still in other ways object of heated debates. Is it really possible to complete Quantum Mechanics? In other terms: is it possible to consider quantum states as averages over those states for which the results of any possible measures are, in principle, completely determined? 27

28 Chapter 3 Bell s Theorem There were two principal currents of thought that wanted to give an explanation to the question opened with the paper of Einstein, Podolsky and Rosen: on one hand there were the determinists, i.e. those people who believed that Quantum Mechanics should be extended in a theory with local hidden variables 1 ; on the other hand, there were the indeterminists, that instead did not consider as right the formulation of the paradox because of its wrong assumptions. In 1964 the Irish physicist John Stewart Bell published the paper On the Einstein- Podolsky-Rosen Paradox [6], where he provided an elegant solution that could be verified experimentally, in order to consolidate the more or less validity of EPR argumentations. He proved, through his inequalities, that any possible theory with hidden variables, with the requirement of locality, is in contradiction with the statistical previsions of Quantum Mechanics, e.g. the former cannot be considered as the completion of the latter, but they are to be set on different planes. As we shall emphasize several times, it is best to notice from now on that a possible interpretation with nonlocal hidden variables is not forbidden; in such a case, the conflict does not arise. We can illustrate and summarize the Bell s Theorem as follows: (Bell s) Theorem. There is no local hidden variable theory that can reproduce (all) the predictions of Quantum Mechanics. Bell demonstrated that for certain phenomena local reality implies some constraints i.e. the inequalities, which are not required but violated by Quantum Mechanics. The experiments that have been carried so far have shown an evident violation of Bell s 1 Here we assume local in the sense of Einstein, that is, remote systems do not interact with each others and they behave as if they were independent. 28

29 inequalities; so we get an empiric evidence against local reality, showing that some of the spooky actions at a distance expected by EPR, actually occur. Nevertheless, the principles of special relativity are not violated. Indeed, it has been proved theoretically and experimentally that, because of the No-Communication Theorem, it is impossible for one experimenter to use these special quantum effects to communicate information to another with a velocity faster than light. As a first point, Bell was interested in formulating Einstein s theory within a mathematical frame. In order to do that, he based his work on Bohm s formulation of the paradox, considering a couple of spin-1/2 particles in a singlet spin state. The EPR additional variables should have had the aim of making the theory causal and local. What Bell was trying to (and finally) show is that such an operation is incompatible with the statistical previsions of Quantum Mechanics. 3.1 Bell s Inequalities We consider a couple of 1 2-spin particles, moving freely in opposite directions and forming a system in a singlet spin state, of the form (2.21). It is possible to make measurements, e.g. by Stern-Gerlach magnets, on selected components of the spins σ 1 and σ 2. From Quantum Mechanics we know that, whether a measurement of the component σ 1 â, where â is a unit vector, yields value +1, then measurement of σ 2 â will give with certainty value 1 and vice versa. We now introduce the principle of locality seen in the last chapter, that establishes that whether the measurements are made at places remote from one another, the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of σ 2 after a measurement of the same component of σ 1, it follows that the result of any such measurements must actually be predetermined. Since the wave function describing the initial state does not determine the result of an individual measurement, this predetermination implies the possibility of adding some new variables that would represent some properties intrinsic to any considered pair of particles, and that are not considered by the wave function just because of their difference from couple to couple. These additional parameters are the so-called hidden variables, that Bell labels with the letter λ. It is the same argument whether λ stands for a single variable or a set of variables, or even a set of functions, or if the variables are discrete or continuous. We consider for simplicity λ as a single continuous parameter. 29

30 With such addition, the result A of the measurement of σ 1 â is determined by the vector â and by the parameter λ, and the result B of the measurement of σ 2 ˆb is in a similar manner determined by ˆb and λ, that is A(â, λ) = ±1; B(ˆb, λ) = ±1. (3.1) The principle of locality, the crucial point in the argumentation of Einstein, Podolsky and Rosen, implies that the result B for particle 2 does not depend from the arrangement â of the magnet for particle 1, neither A from ˆb. Defining by ρ(λ) the probability distribution of λ, then the expectation value of the product of the two components σ 1 â and σ 2 ˆb is P (â, ˆb) = dλρ(λ)a(â, λ)b(ˆb, λ). (3.2) This value should be equal to the quantum mechanical expectation value, which for a singlet spin state is 2 σ 1 â σ 2 ˆb = â ˆb. (3.3) Actually, as will be shown below, these two expressions are not equal and they are mutually exclusive. Since ρ(λ) is a probability distribution, it is normalized, dλρ(λ) = 1, (3.4) and because the relations (3.1) are valid, (3.2) cannot be less than 1. It could reach the value 1 only for â = ˆb if A(â, λ) = B(ˆb, λ), (3.5) except for a set of points λ of zero probability. Thus (3.2) becomes P (â, ˆb) = dλρ(λ)a(â, λ)b(ˆb, λ). (3.6) Now define ĉ, another unit vector; therefore, with the help of (3.1), 2 For an explicit evaluation, see Appendix A. 30

31 P (â, ˆb) P (â, ĉ) = = [ ] dλρ(λ) A(â, λ)a(ˆb, λ) A(â, λ)a(ĉ, λ) [ ] dλρ(λ)a(â, λ)a(ˆb, λ) A(ˆb, λ)a(ĉ, λ) 1, (3.7) from which P (â, ˆb) P (â, ĉ) [ ] dλρ(λ) 1 A(ˆb, λ)a(ĉ, λ). (3.8) For (3.6), the second term of the RHS of the last Eq. is equal to P (ˆb, ĉ), then 1 + P (ˆb, ĉ) P (â, ˆb) P (â, ĉ). (3.9) The inequality (3.9) is the first of a family of inequalities that are collectively called Bell s Inequalities. This inequality sets a constraint on the expectation values with hidden variables, valuated along the three considered directions. Indeed, generally the RHS is of order ˆb ĉ for small ˆb ĉ. Consequently, the value P (ˆb, ĉ) cannot be stationary at the minimum value (-1 for ˆb = ĉ) and therefore cannot be equal to the quantum mechanical result (3.3) in these conditions. Moreover neither quantum mechanical correlation (3.3) can be approximated arbitrarily by (3.2). Just to prove it, we now suppose to make more repeated experiments. The directions of unit vectors â, ˆb, ĉ will not be strictly determined, but will lie into right circular cones of small width since physically the directions are always affected by certain errors. Accordingly, instead of (3.2) and (3.3) we consider their averaged functions P (â, ˆb) â ˆb, (3.10) where the bar denotes independent average of P (â, ˆb ) and â ˆb over vectors â and ˆb within specified small angles of â and ˆb. We suppose that for any unit vector â and ˆb, P (â, ˆb) approximates quantum mechanical results well, that is, the difference between the averages is bounded on the upper side by a small quantity ε: P (â, ˆb) + â ˆb ε. (3.11) 31

32 Our aim is to show that ε cannot be made arbitrarily small, making the impossibility of (3.2) to be equal to (3.3). then Let s consider that for all â and ˆb â ˆb â ˆb δ; (3.12) P (â, ˆb) + â ˆb P (â, ˆb) + â ˆb + â ˆb â ˆb ε + δ. (3.13) From (3.2) we can define P (â, ˆb) = dλρ(λ)a(â, λ)b(ˆb, λ) 1, (3.14) where From (3.13) and (3.14), taking â = ˆb, we get Now, from definition (3.14) it turns out that A(â, λ) 1 e B(ˆb, λ) 1. (3.15) [ ] dλρ(λ) A(ˆb, λ)b(ˆb, λ) + 1 ε + δ. (3.16) P (â, ˆb) P (â, ĉ) = = [ ] dλρ(λ) A(â, λ)b(ˆb, λ) A(â, λ)b(ĉ, λ) [ ] dλρ(λ)a(â, λ)b(ˆb, λ) 1 + A(ˆb, λ)b(ĉ, λ) [ ] dλρ(λ)a(â, λ)b(ĉ, λ) 1 + A(ˆb, λ)b(ˆb, λ). With the help of the two conditions (3.15), P (â, ˆb) P (â, ĉ) + [ ] dλρ(λ) 1 + A(ˆb, λ)b(ĉ, λ) [ ] dλρ(λ) 1 + A(ˆb, λ)b(ˆb, λ). (3.17) Making use of (3.14) and (3.16), 32

33 P (â, ˆb) P (â, ĉ) 1 + P (ˆb, ĉ) + ε + δ. (3.18) Finally, using inequalities (3.13) we have for the RHS, 1 + P (ˆb, ĉ) + ε + δ = 1 + P (ˆb, ĉ) + ˆb ĉ ˆb ĉ + ε + δ 1 + P (ˆb, ĉ) + ˆb ĉ ˆb ĉ + ε + δ 1 ˆb ĉ + 2(ε + δ), (3.19) while for the LHS, P (â, ˆb) P (â, ĉ) = P (â, ˆb) + â ˆb â ˆb P (â, ĉ) + â ĉ â ĉ â ĉ â ˆb P (â, ˆb) + â ˆb P (â, ĉ) + â ĉ â ĉ â ˆb 2(ε + δ), (3.20) from which, for (3.19) and (3.20), (3.18) becomes that is â ĉ â ˆb 2(ε + δ) 1 ˆb ĉ + 2(ε + δ), (3.21) 4(ε + δ) â ĉ â ˆb + ˆb ĉ 1. (3.22) The interesting result here is that this inequality is not always verified for small values of ε (as on the contrary it should be in order to make equal the expectation values (3.2) and (3.3)). Actually, if we consider, for instance, â ĉ = 0 and â ˆb = ˆb ĉ = 1 2, then from (3.22) we obtain 4(ε + δ) 2 1, (3.23) from which we conclude that, for little δ, ε cannot be made arbitrarily small. This is the proof that the difference between the expectation values (3.2) and (3.3) is necessarily finite, so that their corresponding theories are not compatible with each other. Therefore, it could be reasonable to debate the principle of locality, or even to state that if a theory with additional parameters wants to explain in a right manner quantum mechanical rules, it necessarily has to be nonlocal, e.g. it should be in such a way that it 33

34 could allow for any measurement made on a system to influence any further measurement made on other systems at any distance. Finally it is now justified the statement of Bell s Theorem given at the beginning of this chapter. It is important to emphasize that Bell does not exclude local realistic theories, but simply demonstrates that these latter would be in contradiction with quantum previsions, so these cannot be considered as a complement of a quantum mechanical theory. Admitting the existence of hidden variables automatically cancels the hypothesis of locality, but this does not mean that there would be absolutely no possibility of making a different model of additional parameters which could resolve the paradox and, at the same time, observe quantum mechanical predictions. For the moment, the problem raised by Bell is only the need to choose one or the other, otherwise we have to renounce the concept so intuitive and taken for granted of locality, the price to pay for allowing the introduction of a new theory with hidden variables that would be consistent with Quantum Mechanics. 3.2 CHSH Inequalities John Clauser, Michael Horne, Abner Shimony and Richard Holt (in the following CHSH), in a paper of 1969 [7] generalized Bell s inequalities in a way so that these could be applied to realizable experiments. Successively (1971), a generalization of Bell inequalities would be given by Bell himself as well [6]. The CHSH inequalities are based upon a combination of four correlation coefficients measured on four different orientations of the polarizers a definition for coefficient of polarization will be given in the next section as well. Let s consider a set of couple of entangled particles that move one in the opposite direction of the other, so that one enters apparatus I a and the other apparatus II b, where a and b are adjustable apparatus parameters. In each apparatus, a particle must select one of the two channels labeled +1 and 1 respectively. Let the results of selections be represented by A(a) and B(b), where a and b represent the adjustable parameters of the two measurement apparatus. So A(a) and B(b) can assume values ±1 according as the first or second channel is selected. Suppose now that there is a statistical correlation of A(a) and B(b), due to information carried by and localized within each particle, and that at some time in the past the particles constituting one pair were in contact and communication regarding this information. The 34

35 information, which emphatically is not quantum mechanical, is part of the content of a set of hidden variables, denoted collectively by λ. The results of the two selections must be deterministic functions, denoted by A(a, λ) and B(b, λ). A reasonable request of locality requires A(a, λ) to be independent of the parameter b, and B(b, λ) to be likewise independent of a, since the two selections may occur at an arbitrarily great distance from each other. Finally, since the pair of particles is generally emitted by a source in a manner physically independent of the adjustable parameters a and b, we assume that the normalized probability distribution, ρ(λ), characterizing the ensamble is independent of a and b. We can now define a correlation function as P (a, b) = dλρ(λ)a(a, λ)b(b, λ), (3.24) where Γ is the total space of the hidden variables λ. Using the definition (3.24) it follows that Γ P (a, b) P (a, c) dλρ(λ) A(a, λ)b(b, λ) A(a, λ)b(c, λ) Γ = dλρ(λ) A(a, λ)b(b, λ) [1 B(b, λ)b(c, λ)] Γ = dλρ(λ) [1 B(b, λ)b(c, λ)] Γ = 1 dλρ(λ)b(b, λ)b(c, λ). (3.25) Γ Let s suppose that for some parameters b and b we have P (b, b) = 1 δ, where 0 δ 1. Experimentally interesting cases will have δ close to but not equal to zero. Note as well that in this point of the argumentation we are avoiding the experimentally unrealistic restriction of Bell that for some couple of b and b there is perfect correlation (i.e., δ = 0). Dividing the region Γ into two regions Γ + and Γ such that Γ ± = {λ A(b, λ) = ±B(b, λ)}, from (3.24) we have 35

36 P (b, b) = dλρ(λ)a(b, λ)b(b, λ) Γ = dλρ(λ)a(b, λ)b(b, λ) + Γ + dλρ(λ)a(b, λ)b(b, λ) Γ = dλρ(λ) [B(b, λ)] 2 Γ + dλρ(λ) [B(b, λ)] 2 Γ = dλρ(λ) dλρ(λ) Γ + Γ = 1 δ = dλρ(λ) δ = Γ dλρ(λ) + Γ + dλρ(λ) δ, Γ from which dλρ(λ) = δ Γ 2. We can therefore write the integral of RHS of (3.25) as dλρ(λ)b(b, λ)b(c, λ) = dλρ(λ)b(b, λ)b(c, λ) + dλρ(λ)b(b, λ)b(c, λ) Γ Γ + Γ = dλρ(λ)a(b, λ)b(c, λ) dλρ(λ)a(b, λ)b(c, λ) Γ + Γ = dλρ(λ)a(b, λ)b(c, λ) + Γ + dλρ(λ)a(b, λ)b(c, λ) + Γ 2 dλρ(λ)a(b, λ)b(c, λ) Γ = dλρ(λ)a(b, λ)b(c, λ) 2 dλρ(λ)a(b, λ)b(c, λ) Γ Γ P (b, c) 2 dλρ(λ) A(b, λ)b(c, λ) Γ = P (b, c) δ = P (b, c) + P (b, b) 1, (3.26) therefore from (3.25) and (3.26), this is an expression for CHSH inequalities. P (a, b) P (a, c) 2 P (b, b) P (b, c); (3.27) In principle entire measuring devices, each consisting of a filter followed by a detector, 36

37 could be used for I a and II b, and the values ±1 of A(a) and B(b) would denote detection or nondetection of the particles. Inequalities (3.27) would then apply directly to experimental counting rates. Nevertheless, if photons are used, this manner could not make to a very check of (3.27), for the photomultipliers have a little efficiency; so from now on we shall interpret A(a) = ±1 and B(b) = ±1 as the coming out or less of photons from their respective filters, that for example could be linear polarizers having orientations defined by a and b. At this point it is worthwhile to introduce an exceptional value,, taken from parameters a and b for representing polarizer remotion; clearly, we will necessarily have A( ) = B( ) = +1. Since P (a, b) is a correlation function of emission (of photons), it has to be made a further assumption in order to derive an experimental prediction: that if a pair of photons emerges from I a and II b, the probability of their joint detection is independent of a and b. Then if the flux into I a and II b is a constant independent of a and b, the rate of coincidence detection R(a, b) will be proportional to w [A(a) +, B(b) + ], where w [A(a) ±, B(b) ± ] is the probability that A(a) = ±1 and B(b) = ±1. Letting R 0 = R(, ), R 1 (a) = R(a, ), R 2 (b) = R(, b), and making use of the evident formulas P (a, b) = w [A(a) +, B(b) + ] w [A(a) +, B(b) ] w [A(a), B(b) + ] + w [A(a), B(b) ], and w [A(a) +, B( ) + ] = w [A(a) +, B(b) + ] + w [A(a) +, B(b) ], w [A( ) +, B(+) + ] = w [A(a) +, B(b) + ] + w [A(a), B(b) + ], w [A( ) +, B( ) + ] = P (a, b) + 2w [A(a) +, B(b) ] + 2w [A(a), B(b) + ], we get P (a, b) = 4R(a, b) R 0 2R 1(a) R 0 2R 2(b) R (3.28) 37

38 Finally, we can now express (3.27) in terms of experimental quantities, namely coincidence rates with both polarizers in, and with one and then the other removed. Supposing that R 1 (a) and R 2 (b) are constants equal to R 1 and R 2 experimentally determined, CHSH inequalities are R(a, b) R(a, c) + R(b, b) + R(b, c) R 1 R 2 0. (3.29) This is the generalized formulation of Bell s inequalities by CHSH, in a favourable manner to be applied to experiments with photons. In the next section we shall see how to rewrite these in a more explicit manner, thus applying them on the optical version of Bell s Gedankenexperiment. 3.3 Optical version of Bell s Gedankenexperiment As far as we have seen until now, Bell s Theorem states that local realistic theories are in disagreement with Quantum Mechanics, but there is nothing which tells us that these theories are to be rejected. On the contrary, the disagreement between the theories illustrated by Bell arises from rather unusual situations. It is just for this reason that an empirical testing by means of appropriately designed experiments is required to test those critical regions in which conflicts have origin. The more significative experiments, as well as complete, are those by Alain Aspect, who was interested in the individuation of critical regions and in the analysis of photon behavior in situations built up ad hoc. Section 3.4 will deal with the experimental aspect and the results obtained by Aspect. What we wish to study now is a reformulation of Bell s Theorem [9], adopted by Aspect himself as starting point for his whole experimental examinations: a rewriting of the inequalities in terms of correlation coefficients between the directions of the photons, making more clear the importance of the hypothesis of locality within the theorem. In the optical version of Gedankenexperiment of Einstein, Podolsky and Rosen, due to Bohm (see Fig. 3.1), a source S emits pairs of photons with different frequencies, ν 1 and ν 2, which propagate along opposite directions ±O z, in a way similar to that of a pair of 1 2-spin particles in a singlet spin state. Let s suppose that the state entangled which describes polarization of the two photons is given by the ket 38

39 Figure 3.1: The Einstein, Podolsky, Rosen and Bohm Gedankenexperiment. Ψ(ν 1, ν 2 ) = 1 2 ( x, x + y, y ), (3.30) where x and y represent states describing two different orthogonal directions of linear polarization. This state is not a product state, therefore it is not possible to fix any single photons with arbitrary polarization, instead we have to see the system in its whole entirety. Two linear polarizers I and II, placed at the sizes of the source and oriented in the directions given by unit vectors â and ˆb, have the aim to analyse photons and see their tracks. So it is possible to make measurements on linear polarization of two photons through the analysis of two detectors (represented by + and in the Figure) placed next to each of the polarizers. Both will give result +1 or 1 depending on whether the photon polarization occurs in a direction parallel or perpendicular to the one of the polarizer itself. Labeled P ± (â) the probability of obtaining the result ±1 for ν 1, and P ± (ˆb) that one of obtaining ±1 for ν 2, for these measurements quantum mechanical predictions for single detection are 3 P + (â) = P (â) = 1 2, P + (ˆb) = P (ˆb) = 1 2, (3.31) according to the fact that no polarization cannot to be established for single photons; therefore any measurement will give a random result. Defining now P ±± (â, ˆb) to be the probability of combined detection of ν 1 in channel ± of I (oriented along â), and ν 2 in channel ± of II (ˆb), quantum mechanical previsions for combined detection are 4 3 For an explicit evaluation, see Appendix A. 4 See note 3. 39

40 P ++ (â, ˆb) = P (â, ˆb) = 1 2 cos2 ϑ ab, P + (â, ˆb) = P + (â, ˆb) = 1 2 sin2 ϑ ab, (3.32) where ϑ ab is the angle between â and ˆb. In particular, we can see that, in the case for ϑ ab = 0, we get P ++ (â, ˆb) = P (â, ˆb) = 1 2, P + (â, ˆb) = P + (â, ˆb) = 0, (3.33) that is, whether the photon ν 1 gives result +1 (whose probability is 50%), then photon ν 2 as well will give certainly result +1 (analogously for -1), that it means total correlation. This prevision is perfectly in accordance with all we have discussed so far, since measuring the polarization along the same direction of two photons emicted by the same source at the same time, we obtain two results equal and opposite, as it happens with the spin of two atoms belonging to the same molecule. In this contest it is very useful to define the following quantity. Definition (Correlation coefficient of polarization). It is called correlation coefficient of polarization the quantity E(â, ˆb) = P ++ (â, ˆb) + P (â, ˆb) P + (â, ˆb) P + (â, ˆb). (3.34) Hence, substituting (3.32), in the quantum mechanical case this coefficient becomes: For ϑ ab = 0 we have instead E MQ (â, ˆb) = cos (2ϑ ab ). (3.35) E MQ (0) = 1, i.e. total correlation. It is evident that the value obtained effectively explains strong correlations that relate measurement results of ν 1 and ν 2. Generally, the correlation coefficient provides a quantitative criterion in order to quantify the correlation between random results obtained from any individual measurement. 40

41 So we have to admit that there are some properties typical for any pair of photons Einstein called them elements of physical reality that explain correlations, according to the type of correlated results one obtains. These properties, different from pair to pair, are not taken in consideration by the vector state Ψ(ν 1, ν 2 ), that instead is the same for any pair of photons. Here it is evident again the reason that guided Einstein to conclude that Quantum Mechanics was not complete. And here there is the need of giving account of these properties introducing some additional parameters, or hidden variables. We can finally explain such interations through a classical description, and we can hope of finding again quantum previsions by taking the average of the expectation values over the hidden variables (procedure that brought Bell to his inequalities) Now we shall try to work out Bell s inequalities using the model described above. Labelling λ the set of additional parameters and A(λ, â) and B(λ, ˆb) the results respectively obtained from analyser I oriented respectively along â, and analyser II oriented along ˆb, these quantities can only assume values ±1, hence the quantity 1 2 [1 + A(λ, â)] could assume only values +1 (in case of result +) and 0 (otherwise), and analogously, the quantity 1 2 [1 A(λ, â)] could assume only values +1 (in case of result ) and 0 (otherwise). Hence, given the probability distribution of λ, that is ρ(λ), the expectation values for single detection are found to be: P ± (â) = 1 2 P ± (ˆb) = 1 2 dλρ(λ) [1 ± A(λ, â)], [ dλρ(λ) 1 ± B(λ, ˆb) ], whereas for combined detection: P ++ (â, ˆb) = 1 4 P (â, ˆb) = 1 4 P + (â, ˆb) = 1 4 P + (â, ˆb) = 1 4 [ dλρ(λ) [1 + A(λ, â)] 1 + B(λ, ˆb) ], [ dλρ(λ) [1 A(λ, â)] 1 B(λ, ˆb) ], [ dλρ(λ) [1 + A(λ, â)] 1 B(λ, ˆb) ], [ dλρ(λ) [1 A(λ, â)] 1 + B(λ, ˆb) ]. Substituting these quantities in definition (3.34), after some straightforward passages, one finds that the correlation coefficient, averaged over the distribution of λ, is given by: 41

42 E(â, ˆb) = dλρ(λ)a(λ, â)b(λ, ˆb). (3.36) Now we define a new quantity, that allows us to write the inequalities representing in explicit form the correlation coefficient, quantity which can be denoted by s (λ, â, â, ˆb, ˆb ) : ( s λ, â, â, ˆb, ˆb ) def = A(λ, â)b(λ, ˆb) A(λ, â)b(λ, ˆb ) + A(λ, â )B(λ, ˆb) + A(λ, â )B(λ, ˆb ) [ = A(λ, â) B(λ, ˆb) B(λ, ˆb ] [ ) + A(λ, â ) B(λ, ˆb) + B(λ, ˆb ] ). (3.37) Since A and B can assume only the values ±1, then ( s λ, â, â, ˆb, ˆb ) = ±2, from which, averaging over the distribution of λ one gets that is, defining 2 ( dλρ(λ) s λ, â, â, ˆb, ˆb ) +2, (3.38) we obtain the inequalities S := E(â, ˆb) E(â, ˆb ) + E(â, ˆb) + E(â, ˆb ), (3.39) 2 S(â, â, ˆb, ˆb ) +2. (3.40) (3.40) are well-known as BCHSH inequalities, i.e. inequalities of Bell generalized by Clauser, Horne, Shimony, Holt. These inequalities are based on a combination of four correlation coefficients of polarization, measured along four orientations of the polarizers. Then S is a measurable quantity. However, BCHSH inequalities in some particular situations (which will be specified shortly) are in conflict with Quantum Mechanics. Indeed if we put the system in the configuration illustrated by Figure 3.2, with ϑ ab = ϑ ba = ϑ a b = π 8, ϑ ab = ϑ ab + ϑ ba + ϑ a b, 42

43 substituting the quantities E in (3.39) with their quantum mechanical values (3.35), we get (3.40). S MQ = 2 2. This quantum mechanical prevision deeply violates the upper limit of inequalities Figure 3.2: Orientations with ϑ ab = ϑ ba = ϑ a b = π 8. In this particular situation it turns out that quantum previsions cannot be obtained from theories with hidden variables, therefore we can conclude that a local deterministic theory does not exist, according to the very general model showed in this Section, which reproduces all quantum mechanical previsions. It is reasonable to ask what are precisely the critical regions and for which angles one could have the maximum conflict. Deriving S with respect to the three independent angles ϑ ab, ϑ ba, ϑ a b, we have that S MQ is extreme in correspondence with and for (3.35), ϑ ab = ϑ ba = ϑ a b = ϑ, (3.41) S MQ (ϑ) = 3 cos(2ϑ) cos(6ϑ). (3.42) Finally, deriving this quantity with respect to ϑ and letting equal to zero, ds MQ dϑ = 6 sin(6ϑ) 6 sin(2ϑ) = 0, we obtain the angles for which S MQ has its maximum and minimum values, that are respectively 43

44 S max MQ = 2 2 per ϑ = π 8, S min MQ = 2 2 per ϑ = 3π 8, (situations illustrated in Figure 3.2 and 3.3 respectively). Figure 3.3: Orientations with ϑ ab = ϑ ba = ϑ a b = 3π 8. At last, with a brief study of function, it is possible to plot a graphic representing the behaviour of S varying with the angle ϑ, as shown in Figure 3.4. In conclusion, Bell s Theorem brings out a conflict between theories with hidden variables and certain quantum mechanical previsions (according to the model expounded in this section) and provides a quantitative criterion to clarify this conflict. Discussing the hypotheses, the fundamental assumptions on the bases of the described model are three: the existence of hidden variables, determinism and the locality condition. These hypotheses, if simultaneously assumed, give rise to an incoherence between the created theory and Quantum Mechanics. The first two of them are not to be under discussion since they are parts of the theory itself. The third instead is a handy and obvious assumption that it seems absurd not to accept. And yet, whether in the case that the first two hypotheses do not want to be dropped out, if one wish to complete 44

45 Figure 3.4: S(ϑ) = 3 cos(2ϑ) cos(6ϑ) as expected by Quantum Mechanics for pairs in the state (3.30). The conflict arises in the hatched zone. Quantum Mechanics by a theory with hidden variables, then this theory will have to be necessarily nonlocal. Indeed if the condition of locality is abandoned, it can be easily shown that with quantities like A(λ, â, ˆb) or ρ(λ, â, ˆb), the demonstration that brings to BCHSH inequalities drops, thus voiding their result. 3.4 A. Aspect s experiments Between 1980 and 1982 Alain Aspect and his équipe tried to create more complex experimental apparatuses, compared with those used until that time, to verify Bell s inequalities, and they made three experiments to study the validity of these constraints using atomic cascades [9]. The first experiment was been organized as a direct check for inequalities. Nevertheless, it had some limits, e.g. single-channel polarizers were used, so that only photons polarized parallel to â (or ˆb) could pass, while those polarized orthogonally were blocked. Therefore only the results + could be revealed, and measurements of coincidence could only provide an evaluation for P ++ (â, ˆb). So one could not know whether the missed measure of correlation between a pair of photons was due to a real absence of correlation (i.e. to the action of the polarizer), or to a scarce efficiency of the detector. For this reason a second experiment was performed, in this case preserving a configuration like that of the Bohm-Aharonov s thought experiment. This experiment implemented 45

46 two-channel polarizers that made possible to detect the correlation of every photons that reached the polarizer, thus even allowing for obtain a valuation for P (â, ˆb). Finally, the third experiment, which made use of polarizers with orientation variable in time, can be considered as the more accurate and complete experiment, to which the others can be reconducted. Moreover this experiment proves experimentally the impossibility of faster-than-light communications. As common source for each of the three equipments, it has been used an atomic cascade (J = 0) (J = 1) (J = 0). The main feature of these experiments with respect to the previous was the usage of a very powerful and stable source, that allowed to cut the data recording times from several hours to a few minutes. The source exploited the cascade 4p 2 1 S 0 4s4p 1 P 1 4s 2 1 S 0 of 40 Ca. Figure 3.5: Relevant energy levels for a 40 Ca cascade. The atoms, excited by absorption of two photons ν K and ν D, emit visible photons ν 1 and ν 2 correlated in polarization. This cascade, represented in Fig. 3.5, produces two visible photons ν 1 and ν 2 correlated in polarization. The atoms of 40 Ca are thus excited from the ground state to the upper energetic level by absorption of two photons, ν K and ν D, generated by two laser beams. The first one (λ K = nm) is generated by ions of Kryptons, as the second was a dye laser, brought to resonance for the two-photon process (λ D = 581 nm); lasers have parallel polarizations. First of all, there is a feedback cycle that checks on the dye laser wavelenght, in order to have the maximum signal of fluorescence by the cascade. Then a second feedback cycle checks on the power emitted by the Krypton laser in order to guarantee a constant emission from the cascade. Strictly speaking, as a result of the atom excitation, one electron of each atoms is led to jump up to two energy levels beyond its ground state. When the electron falls from two energy levels, it sometimes emits a pair of entangled photons. In this way, using a power of 40 mw for each laser, the typical 46

47 emission efficiency of the cascade is of photons per second Experiment with single-channel polarizers In their first experiment, Aspect and collegues have mounted a pile of two polarizers made up of ten glass plates to the Brewster s angle; in front of each of these polarizer, a linear polarizer has been placed, in order to transmit photons polarized in a way parallel with the axes of the polarizer, and to stop those polarized perpendicularly. In this type of experiment, one talks about single-channel polarizers because only the value +1 is measured for each photon belonging to the couples of photons emitted by the source. Therefore, the use of single-channel polarizers allows one to determine only the result R(â+, ˆb+) = R(â, ˆb) 5, since it cannot be established if the result 1 for a photon was due just to an orthogonal polarization with respect to the polarizer axes, or to a scarce efficiency of the counting system. Accordingly, auxiliary recordings with one or both the polarizers removed were needed, thus obtaining the following quantities: R(, ) = R 0 R(â+, ) = R 1 (â) R(, ˆb+) = R 2 (ˆb), from which new BCHSH inequalities are obtained: with 1 S 0, (3.43) S = 1 [R(â, R ˆb) + R(â, ˆb) + R(â, ˆb ) R(â, ˆb ] ) R 1 (â ) R 2 (ˆb). (3.44) 0 Experimental test of Bell s inequalities has provided S exp = ± 0.014, (3.45) that violates inequalities (3.43) by 9 standard deviations, and is on good agreement with quantum mechanical previsions S MQ = ± 0.005, (3.46) 5 With analogy to the notation used in Section 3.2, by R(â+, ˆb+) we intend the counting result of the detection of coincidences parallel to the single-channel polarizers (+) oriented with respect to â and ˆb. 47

48 (in this case the error accounts for the uncertainty in the measurements of the polarizer efficiences) Experiment with two-channel polarizers When a photon is stopped by the polarizer in the single-channel experiment, it is lost and there is no way to establish whether and how it was correlated to another photon. This is the reason why, in the second experiment made by Aspect, two channels have been used. Indeed, if a photon is stopped by the polarizer, then the photon is reflected by it and so could be still detected. In this way the coincidence rate was much greater and leaded to a more precise experiment. With two-channel polarizers, the experiment implemented is more similar to that of the scheme in Figure 3.1. The polarizers used were polarizing cubes that transmit a polarization (parallel to â, or to ˆb respectively) and reflect the orthogonal one. This polarization separator and its corresponding photomultipliers were mounted on a rotating device. The latter (i.e. a polarimeter) supplies the results + and for linear polarization measurements along â (ˆb respectively). Strictly speaking, it is an optical analogous of a Stern-Gerlach filter for 1/2 spin particles. Setting the polarimeters I and II oriented along â and ˆb, and with a four-coincidence counting system, it has been possible to measure, in a single data recording period, the four coincidence rates R ±± (â, ˆb). The correlation coefficient for measurement along â and ˆb is directly E(â, ˆb) = R ++ + R R + R + R ++ + R + R + + R +. (3.47) So it has been sufficient to repeat the same measurements for the other three orientations: in that way, BCHSH inequalities (3.40) could be directly verified. This procedure is correct if the measured values (3.47) are like the correlation coefficients of polarization previously defined by (3.34), i.e. if one assumes that a set of pair actually revealed is a significant sample of all of the emitted pairs. But, because of the simmetry of the experimental apparatus, this assumption is revealed very reasonable, since the measurements +1 and 1 are treated in equal manner, i.e. the efficiencies of revelation for both channels of a polarimeter are equal. Thus, after verification of the fact that, nevertheless each of the four rates R ±± (â, ˆb) changed drastically, their total sum was the same varying the orientations, it has been made the counting experiment in agreement with the orientations of Fig. 3.2 and 3.3, i.e. 48

49 the values for which the maximum conflict is expected. The value obtained is S exp = ± 0.015, (3.48) that violates inequalities (3.40) by more than 40 standard deviations, a result in a very excellent agreement with Quantum Mechanics predictions (with the adopted polarizers and solid angles): S MQ = 2.70 ± 0.05, whose uncertainty accounts for a small lack of symmetry for both channels of each polarizers (±1%). This asymmetric effect has been calculated and cannot however cause a variation more than 2% for S MQ Experiment with time-varying polarizers Maybe the most interesting experiment conducted by Aspect is the one in which timevarying polarizers have been mounted. This because the assumption of the principle of locality is reasonable, but it is not prescribed by any physical law. So, accordingly to the results obtained so far, we could imagine that maybe some fixed analysers can be placed along their respective directions sufficiently in advance, in order to allow them to communicate one each other by means of an exchange of signals faster than or equal to c 6. If such interactions existed, Bell s inequalities would no longer be valid, since they have as their hypothesis the independence of the measure made on a system with respect to any disposition of the other apparatus. In the experimental apparatus (see Fig. 3.6), each polarizer is substituted by a system composed of a commutation device (C 1 and C 2 ) followed by two polarizers in two different orientations: â and â by side I, ˆb and ˆb by side II. The optical switch is able to lead rapidly the incident flux from a polarizer to the other one. Every system is thus equal to a variable polarizer switched between two orientations. The distance between the two commutators is L = 12 m. The characteristic point of this Aspect s experiment that will be considered the very new feature that yields a definitive evidence of nonlocality is the light switching, 6 From now on, by c we mean the speed of light in the vacuum: c = ms 1. 49

50 Figure 3.6: Time-varying experiment with optical commutators C 1 and C 2. Time of commutation: 10 ns. which is made through acousto-optic interaction of light rays by an ultrasonic stationary wave in the water. Thus, when the wave changes in the transparent water container, the light beam, hitting the liquid, is deflected modifying the regulation of the polarizer. The incident angle (Bragg angle) and the acoustic power are regulated to obtain a complete commutation. At an acoustic frequency of 25 MHz, the switching frequency is 50 MHz. Accordingly, every 10 ns there is a change in orientation of the equivalent variable polarizer. Since this time interval (10 ns) and the mean life of the intermediate level of the atomic cascade (5 ns) are small with respect to L c (40 ns), the detection of an event on one hand, and the corresponding change of orientation on the other, are separated by a space-like interval 7. Nevertheless, by means of the large beams used, the commutation was not complete since the incidence angle was not exactly the Bragg angle. So, in order to obtain a better commutation, the divergence of the beams was reduced: such a thing led to a reduction of the coincidence rates detected with respect to those ones of previous experiments. According to this, the only consequence was a longer data-recording time. With a data-recording of 8000 s, the polarizers configured in the extreme configurations of Fig. 3.2 and 3.3, plus additional 8000 s, necessary for ausiliary calibration measurements with the removal of half of all the polarizers, and taking into account for systematic errors, the test for inequalities (3.43) led to S exp = ± 0.020, (3.49) 7 Two events are said to have a space-like separation if they do not influence between each other by signals that propagate at a speed less than or equal to the speed light. For a definition of space-like interval, see note

51 that violates inequality (3.43) by 5 standard deviations, and is in good agreement with quantum mechanical previsions: S MQ = ± (3.50) Therefore, even this experiment shows the reasonableness of Quantum Mechanics to disadvantage of a theory of hidden variables which obey to Einstein s locality. At this point, accordingly to the strong evidence of correlations, between the acceptation of local additional parameters that transmit themselves information faster-thanlight, or the rejection of an explanation that uses these additional parameters, we have to choose the second position. An exam of experimental results let us conclude that Quantum Mechanics describes correctly the reality, whereas local hidden variable theories give previsions proved wrong by the experiments Towards the impossibility of FTL communications Actually, the results obtained by Aspect s experiment with time-varying polarizers leads to a further outcome, as we anticipated: that is, the impossibility of faster-than-light communication, or FTL. To better understand how this is not possible, suppose that the observer I wants to transmit some information to the observer II using spin correlation measurements for two particles spatially separated and in opposite directions. Suppose that I and II decide to measure the S z component. Then, without asking for anything, the observer II will know exactly the result of the measure for the first observer. But this does not mean that the two observers are communicating one to each other as observed in Section 2.4, since the observer II obtains from his measure just a sequence of positive or negative signs without having any useful information. In order to get any kind of information, it would be necessary that the observers had a decoding key, by means of that one is able to establish what meaning (i.e. information) to give to the sequence of symbols + and, according to the order and the kind of spin that are obtained from measurements, in order to let the observers obtain useful information from a certain sequence of S z relevations. But this necessarily involves an exchange of a decoding key between the observers, that can occur only by classical channels, at a velocity less than or equal to c. The reason of this is that entanglement, being a markedly quantum (i.e. probabilistic) phenomenon, does not allow to the observer (that is, to the sender of the possible communication) to choose how the spin revelations should be made 51

52 by who measures, making impossible any kind of key exchange. In other words: only after matching their results (using a conventional method of communication, that cannot broadcast signals faster or equal-than-light), the observers I and II could notice the coincidence of their results. Thus we have seen at a quick first analysis how it is not possible to utilize entangled states for a direct exchange of information through FTL communications. Quantum Cryptography Instead, it is possible to exploit entangled states (for instance, entangled photons) for an absolutely secure exchange of a cryptographic key clearly at velocity less than c, by means of which to communicate successively by classical channels. The safety is guaranteed by the fact that if a third observer, say III, measured a certain value of the polarization component of one or more photons making part of the series that I and II are mutually exchanging i.e. of the cryptographic key, then, after that measurement, the polarization state of the photon would alter and, because of the Uncertainty Principle, some errors would be introduced into the measurement of the observer II even for measurements that would be correct; when, at the end of the key exchange (that is, the series of single polarized photons), after some procedures, observers I and II match their keys, they will find inevitably that these are different for some values, which proves that an external observer (III) has cut off some photons and hence that the key is not secure. This is what one does in Quantum Cryptography, where one of the fundamental problem is that of generating a sequence of random numbers, as long as the message to send, in order to use the One Time Pad communication 8 well-known as Vernam cipher, the unique cryptographic system whose security is mathematically demonstrated [11] 9 (for this reason it is also called perfect cipher ). Once obtained a key as described, 100% safe, it is possible then to establish a secure communication between sender and receiver. This is an example of how the entanglement is utilized practically, in this case in a communication contest. On June of 2004 in Cambridge this has been implemented for Quantum Net, the first net with more than two quantum cryptographic nodes. In Qnet data flow by usual cables in optic fibre, but the peculiarity is that they are codified using coding keys obtained by the exchange of an entangled series of single polarized photons. 8 Cryptographic method in which the key lenght, composed of random characters, is equal to the lenght of the message to code. 9 Firstly showed in 1949 by Claude Shannon. 52

53 3.5 No-Communication Theorem In the light of what we said, one question could arise: How does system I communicate to system II the outcome of the measurement of a certain observable A, in time to produce the correlations we know of, and without violating the special relativity limits? First note that the question is ill posed, because it understates that the outcome of measuring A causes the outcome of B. Relativistically speaking, in spacetime regions where the two measurements are taken (or can be taken) the latter are, in relativistic language, causally disjoint: there is no future-directed space-like or time-like path 10 in the cone of spacetime joining them. This means that the cronological order of the events is conventional, and depends on the choise of inertial frame. Indeed, special relativity states that it is possible to choose an inertial frame in which A s measurement precedes in time B s, or another frame in which the situation is opposed: that is, B s measurement precedes in time A s. So it makes no sense to say that the outcome of the experiment on B is the consequence, or the cause, of the outcome on A 11. Nevertheless, one could resort to the partial conventionality of Einstein s synchronisation procedure in order to dismiss that problem. But despite the conventional choises underpinning special relativity, it is known that the correlations between causally disjoint events are dangerous in relativistic theories, for they can produce causal paradoxes: with a chain of causally disjoint events we can put events in the history of a given system in any chronological order whatsoever. If it were possible to use the correlations of causally disjoint events to transfer information either way, we would be able to communicate with the past (inside the light cone) and thereby obtain causal paradoxes. It can be proved, by accepting the standard formulation of Quantum Mechanics for systems made by entangled states, that no piece of information can be transmitted from event X, where part of the system is measured, to event Y, where the other part is measured, by measuring any arbitrary pairs of quantities and exploiting the quantum correlations between the readings. Not only that, but observing the outcomes on one part of the system we cannot establish whether on the other part some measurements have been 10 Given two distinct simultaneous events, their interval is space-like : t = 0 in fact implies s 2 < 0, being s 2 = c 2 t 2 x 2. Instead if two events occur in the same space point, their interval is time-like : x = 0, thereby s 2 > We already noted that in Section 2.4, where we have talked about the concept of simultaneity used by EPR. 53

54 taken, if they are being taken as we speak, nor if they will be taken in the future. We shall consider in the following two ways of transferring information from X to Y via EPR correlations. Given a quantum system S composed of two subsystem A and B, the Hilbert space of S will be given by H S = H A H B with obvious notation. Let s consider also a pair of observables G A and G B, with discrete spectrum, defined in A and B respectively. Such a pair of observables with discrete spectrum can be relative to internal degrees of freedom of their respective systems in which they are defined. First possibility Consider the single pairs of measurements on A and B of the observables G A and G B respectively, which we know have correlated outcome. We cannot pass information from X to Y using the correlation, because the outcome, albeit correlated, is completely accidental. Making a practical example, it is like having two coins, A, B with the remarkable property that each time one shows heads, the other one gives tails, independent of the fact they are tossed far away, rapidly, and that A is tossed before or after B in some frame. The coins, though, have a quantum character and it is physically impossible to force one to give a certain result: the outcome of the toss is determined in a probabilistic way and whatever our wish is. Thus the two coins, i.e. our quantum system made by parts A and B, cannot be used as a sort of Morse telegraph to transmit information between X and Y. Second possibility Consider not the single measurements of G A and G B, but a large number thereof, and study the statistical features of the outcome distributions. The statistics of the measurements of G A might be different according to whether we measure G B as well, or whether we measure a new quantity G B. In this way, by measuring or not measuring G B (and measuring G B or measuring nothing at all) in Y, we can send an elementary signal to X, of the type yes or no, that we recover by checking esperimentally the statistics of A. It is possible to demonstrate that, even with such procedure, it is not possible to transmit any information, since the statistics relative to G A is exactly the same in case we also measure G B (or any other G B ) or we do not measure G B. But before this study, we recall some properties of the density matrix ρ. 54

55 A few notions about the density matrix ρ We recall that, given a dynamical quantum system of physical quantities, it is possible to associate with any dynamical variable of the system a definite statistical distribution of the possible values assumed by it. The dynamical state of a quantum system can be completely known if it is possible to determine precisely the variables of one of the possible complete sets of compatible variables associated with the system. When the knowledge of the system is not complete, one says that the system has a certain probability p 1, p 2,..., p m,..., of being in dynamical states represented by kets 1, 2,..., m,..., respectively. In other words, the dynamical state of a quantum system cannot be more represented by a unique vector, instead by a statistical mixture of vectors. A statistical mixture can be described by means of the density operator or statistical operator ρ = m p m m, m where the vectors m are normalized to unity (but not necessarily orthogonal) and the quantities p m have the characteristic properties of statistical weights, namely p m 0, p m = 1. The average value of the observable A is equal to m A = T r ρa. Indeed it can be easily shown that T r ρa = m p m T r ( m m A). So, having knowledge of ρ, it is possible to derive a statistical distribution of the results for measurements of A. Very generally, if P D is the projector upon the subspace spanned by the eigenvectors of A belonging to the eigenvalues located in a certain domain D of the spectrum of A, whether an observation made on the system shows that it is an eigenstate of A belonging to D, the density operator after the measurement is, without a normalization constant, the projection P D ρp D of the operator ρ, which represents the statistical mixture before 55

56 the measurement. The constant is determined by the constraint that this operator must have trace equal to unity. We therefore obtain that T r P D ρp D = T r ρp D ; then the evolution in time of density operator ρ during the act of measuring leads to ρ P DρP D T r ρp D. For a more complete study of the properties of the density operator, we refer the reader to [21], [26]. We now can begin by discussing the second possibility. We take the state ρ I(H A H B ) for the system composed of A and B. Suppose that G A = G (A) I B, where G (A) is a self-adjoint operator in H A having a discrete and finite spectrum: {g (A) 1, g (A) 2,..., g n (A) }, associated with its respective eigenspaces H (A) g H A H B, targets of orthogonal projectors P (G A) k := Pk G(A) I B. Analogously G B = I A G (B), where G (B) is a self-adjoint operator in H B with discrete and finite spectrum: {g (B) 1, g (B) 2,..., g m (B) }, associated with its correspondent eigenspaces H g (B) k H A H B, targets of orthogonal projections P (G B) k Now, if we measure G B on state ρ reading g (B) k 1 ( )P (G B) T r P (G B) k ρp (G B) k ρp (G B) k. k := I A P G(B) k., the post-measurement state is: Considering all of the possible readings of B, if we measure in some frame first B and successively A, the system we are intending to test on A is the quantum mixture: ρ = m k=1 ( ) where p k = T r P (G B) k ρp (G B) k Thus: p k ( )P (G B) T r P (G B) k ρp (G B) k ρp (G B) k, k is the probability of reading g (B) k for the measure of B. ρ = m k=1 P (G B) k ρp (G B) k. k 56

57 Hence the probability of getting g (A) h for A, when B has been measured (irrespective of the latter s outcome) is: ( ( ) m ) P(g (A) h B) = T r ρ P (G A) h = T r P (G B) k ρp (G B) k P (G A) h. At this point, with the help of the property of linearity and ciclicity of trace, we have: k=1 P(g (A) h B) = m = = k=1 m k=1 m k=1 ( ) T r P (G B) k ρp (G B) k P (G A) h ( ) T r ρp (G B) k P (G A) h P (G B) k ( ) T r ρp (G B) k P (G B) k P (G A) h, where in the last step we have used P (G B) k P (G A) h = P (G A) h P (G B) k projectors. On the other hand P (G B) k P (G B) k = P (G B) k e theorem 12. Therefore: k P (G B) k from the structure of the = I by the spectral P(g (A) h B) = m k=1 = T r ( ) T r ρp (G B) k P (G A) h ( ( = T r ( = P ρ m k=1 ρp (G A) h g (A) h ). P (G B) k P (G A) h ) ) Thus we have obtained that: the probability of obtaining g (A) h from A when the quantity B has been measured ( with any possible outcome), coincides with the probability of obtaining g (A) h from A without measuring B. 12 For finite-dimensional Hilbert spaces, the spectral theorem states that each Hermitean operator O can be written O = i oip ( oi ), where {oi} is the set of eigenvalues comprising the eigenvalue spectrum of O and P ( o i ) is the projector onto the finite Hilbert subspace spanned by o i. This provides the spectral decomposition (also called eigenvalue expansion) of the operator O. When the state of the system is instead mixed, by necessity being described by a statistical operator ρ that is not a projector, the expectation value is O ρ = T r (ρo). 57

58 So, measuring part B of the system, the presence or the absence of the correlation is completely irrelevant if we observe only part A of the system. From this follows that, in conclusion, even if we consider the statistics of the results for the measures of A, there is no way to transmit information using EPR correlations. There are several generalizations of the result obtained above. For instance, we refer the reader to a paper of G.C. Ghirardi, A. Rimini and T. Weber [12], entitled A General Argument Against Superluminal Transmission Through The Quantum Mechanical Measurement Process, where it is proved, by more general characters than ours, the impossibility of transmission of faster-than-light signals Some outlines about the FLASH Project In 1981 the American physicist N. Herbert proposed a Gedankenexperiment in order to use quantum nonlocality to obtain a FTL communication 13 [13]. Herbert s apparatus was an idealized laser gain tube which would have macroscopically distinguishable outputs when the input was a single arbitrarily polarized photon. Indeed, the word LASER is an acronym for Light Amplification by Stimulated Emission of Radiation. However, besides the stimulated emission, there is also spontaneous emission, which results in noise. Herbert s claim was that the noise would not prevent identifying the polarization of the incoming photon, at least statistically. For this, he used the notion of quantum compounds and many properties of laser physics. This apparatus has been realized for the first time in 2007 by a group of Italian research workers (T. De Angelis, F. De Martini, E. Nagali, F. Sciarrino) [15], using the optical parametric amplification 14 of a single photon forming an EPR-entangled pair into an output field involving photons. The experimental apparatus is represented in Figure 3.7. Two space-like distant observers, say Alice and Bob, share two polarization entangled photons generated by a common EPR source, represented in the figure by Crystal 1. Alice detects by two phototubes a certain polarization of her photon in a two orthogonal measurement basis, that with no lack of generalization are chosen as linear polarization and circular polarization with respect to a classical Cartesian (tridimensional) frame. 13 Actually, Nick Herbert s FLASH paper was wrong, but anyway it was published. For more details about this story and its contribution to the no-cloning theorem, see [14]. 14 An optical parametric amplifier (OPA) is a particular kind of amplifier that allows to directly amplify a weak optical signal through a highly non-linear medium, without converting it into an electric signal. 58

59 Figure 3.7: Configuration of the quantum injected optical parametric amplifier. The polarization entangled photon is generated by Crystal 1. The choise of the basis is the only coding method accessible to Alice in order to establish a meaningful communication with Bob. Well, if Bob could guess the coding basis chosen by Alice, then a FTL signaling process would be established; nevertheless, since the detection of an unknown single particle cannot carry any information on the coding basis (as the No-Communication Theorem states), the purpose by Herbert is been that of letting Bob make a new kind of measurement on the photon through the amplification by a polarization independent amplifier, in order to split the amplified beam by symmetric beam-splitter (BS), so that Bob could perform a measurement on half of the amplified particles in the linearly polarized basis, and on the other half in the circularly polarized basis. Actually this proposal cannot be carried out, because of the quantum no-cloning theorem, that asserts that for the Quantum Mechanics postulates, no quantum amplifier can duplicate accurately two or more nonorthogonal quantum states. Herbert, aware of the impossibility to produce perfect clones of any input qubit where a qubit is a quantum information unity, which in our specific case is represented by the photon, because of the noisy contributions from spontaneous emission, proposed to measure the mean values of two physical quantities, depending on the number of pair of electronic signals detected by two pairs of photodetectors, that should have had a dependance on the coding bases chosen by Alice. Omitting some technical and numerical details of the experiment (for more details we refer to the original paper of the experiment listed in Bibliography), the final result of the experiment, based on an accurate theoretical and experimental analysis, is that the mean 59