Lecture 4. QUANTUM MECHANICS FOR MULTIPLE QUBIT SYSTEMS 4.1 Multiple Qubits Next we consider a system of two qubits. If these were two classical bits, then there would be four possible states,, 1, 1, and 11. Correspondingly, to describe a two-qubit system, we may introduce four basis states denoted >, 1 >, 1 >, and 11 >. More formally, we can treat these basis states as constructing from the single-qubit basis states >, 1 > according to lm >= l > m > l > m >, (1) where l, m =, 1. In mathematics, this construction is known as the direct or tensor product of vectors l > and m >. Note the following useful rules for this product: ( x > + y >) u >= x > u > + y > u > xu > + yu >; u > ( x > + y >) = u > x > + u > y > ux > + uy >; a x > b y >= ab x > y > ab xy >, () where a, b are arbitrary constants. In general, if l > and m > are basis vectors for two vector spaces X 1 and X, their tensor product, X = X 1 X, is defined as the space built on the basis given by Eq. (1). The dimensions of these spaces obey an obvious relation where dimx = dimx 1 dimx, (3) n 1 = dimx 1 and n = dimx (4) are the dimensions of the factor spaces X 1 and X, respectively. To show the validity of the relation (3) one may consider lm as a single vector index whose values are listed as l 1 m 1, l 1 m,..., l 1 m n, l m 1, l m,..., l m n,..., l n1 m 1, l n1 m,..., l n1 m n. It is instructive to compare this with the Cartesian product of two spaces, X = X 1 X, which has as its basis the union of the bases of component spaces, (l 1,..., l n1, m 1,..., m n ), and hence dimx = dimx 1 + dimx. (5) We notice that the tensor product of a n 1 -dimensional state vector Ψ 1 > and n - dimensional state vector Ψ > defines a n 1 n -dimensional state vector of combined system according to Ψ 1 > Ψ >= l l >< l Ψ 1 > m m >< m Ψ >= lm lm >< l Ψ 1 >< m Ψ >. On the other hand, we can expand an arbitrary state of the composite system as (6) Ψ >= lm >< lm Ψ >, (7) l,m Comparing (6) and (7), we see that the state Ψ > can be factored to Ψ 1 > Ψ > only if < lm Ψ >=< l Ψ 1 >< m Ψ > (8) Otherwise the composite state Ψ > is called correlated or entangled. Naturally, the entanglement of states is a phemomena which has no counterpart in classical physics. A little bit later we will discuss this phenomena in more detail.
Finally, we notice that the generalization of the formalism to the case of the composition of more than two subsystems is obvious: the state of such composite system is given by a vector in the space X 1 X... X n with the dimension dim (X 1 X... X n ) = dimx 1 dimx... dimx n = n 1 n... n n. (9) After these general remarks on the theory of composite systems, let return to the twoqubit system. In matrix representation the basis states for such system are as follows: 1 1 >=, 1 >=, 1 >=, 11 >= 1, (1) 1 and a general state of a two-qubit system (or a two-qubit memory register) is therefore Ψ >= c > +c 1 1 > +c 1 1 > +c 11 11 >. (11) Similar to the case of a single qubit, the complex amplitudes c x, x (, 1, 1, 11) {, 1} (1) determine the probabilities, c x, that the state of the qubits after the measurement in the state x > is x. Here we used the notation {, 1} for the set of strings of length two with each letter being either zero ore one. Naturally, x {,1} c x = 1. (13) Of course, for a two-qubit system we could measure not only the total set of two qubits but a subset of the qubits, say the first qubit. You can probably guess how this works: measuring the first qubit alone gives with probability c + c 1, leaving the post-measurement state ψ > out = c > +c 1 1 > c + c 1. (14) Note how the post-measurement state is renormalized by the factor c + c 1 so that it still satisfies the normalization condition, just as we might expect for any legitimate quantum state. Note also that the final state can be treated as the state obtained from an initial state (11) by applying the projection operator We have P = >< + 1 >< 1. (15) P ψ >= ( >< + 1 >< 1 )(c > +c 1 1 > +c 1 1 > +c 11 11 >) = c > +c 1 1 >, (16) which after renormalization gives just the state ψ > out. An important two-qubit state is the Bell state or EPR pair, > + 11 >. (17) This innocuous-looking state is responsible for many surprises in quantum computation and quantum information. In particular, it is the key object in quantum teleportation
which we shall consider later in more detail. Here we notice only that the Bell state has the property that upon measuring the first qubit, one obtains two possible results (both with probability 1/):, leaving the post-measurement state ψ >= >, and 1, leaving ψ >= 11 >. But if the second qubit had been previously measured, the probability that the state of the first qubit is > is 1 or depending on whether the second qubit was measured as > or 1 >, respectively. This observation shows that the measurement outcomes are strongly correlated. This type of correlation have been the subject of intense interest ever since a famous paper by Einstein, Podolsky and Rosen, who first pointed out the strange property of the state under consideration. Later EPR s insights were taken up and greatly improved by John Bell. For a deeper penetration into the unusual properties of the EPR pair, let us apply the relations () to the two one-qubit states (a 1 > +b 1 1 >) and (a > +b 1 >). We obtain (a 1 > +b 1 1 >) (a > +b 1 >) = (a 1 > a >) + (a 1 > b 1 >) + (b 1 1 >) a >) + (b 1 1 > b 1 >) = a 1 a ( > >) + a 1 b ( > 1 >) + b 1 a ( 1 > >) + (b 1 b ( 1 > 1 >) = a 1 a > +a 1 b 1 > +b 1 a 1 > +b 1 b 11 > (18) This equality expresses the fact that the combined state of a two-qubit system given by the right-hand side of Eq. (18) can be represented by the composition of the separate states of two qubits, i.e. the states (a 1 > +b 1 1 >) and (a > +b 1 >). However, it is easy to convince that there are the states which don t obey Eq. (18), i.e. cannot be expressed, mathematically, as the direct product of states of the individual qubits that comprise the register. We already know that these are the entangled states, and a striking example of such state is bestowed by nothing but the Bell state! Indeed, attempting to obtain such state from the product (18) one must put a 1 b = but this implies that either a 1 a = or b 1 b =. Thus for this state the equality given by Eq. (18) doesn t exist, and therefore, it can be never decomposed in the separate states of two qubits. For comparison, let take the state > + 1 >. (19) This state is evidently not entangled since it can be expressed in terms of tensor product of the single-qubit states like Eq. (18). Namely, we have > + 1 > = > ( > + 1 >). () In other words, an entangled state can be never described in terms of the state of its component pieces in full contradictory with a classical object breaking apart which can be described completely by describing the state of each of its pieces separately. Note that in entangled states like EPR pair the quantum memory register as a whole still has a well-defined quantum states even though its component qubits do not! In general, we may consider a system of n qubits. The computational basis states of this system are of the form x 1...x n >, x k =, 1, and so a quantum state of such a system can be written as 1 1 Ψ >=... c x1...x n x 1...x n >, (1) x 1 = x n= In view of the measurement anzatz of quantum mechanics the square modulus of the coefficients c x1...x n determine the probabilities of the basis states x 1...x n >, P ( x 1...x n >) = c x1...x n, ()
so that 1 x 1 =... 1 x n= c x1...x n = 1. (3) Realise that () is the probability of finding the whole memory in a particular configuration (a particular sequence of bits) when it is finally observed. The measurement of a subset of the qubits in the register projects out the state of the whole register into a subset of eigenstates consistent with the answers obtained for the measured qubits analogous to an example which we have already considered in the case of a two qubit system. We see that the general state of a system of n qubits is specified by n amplitudes - in full accordance with Eq. (9) because in the case of qubits n 1 = n =... = n n =. This leads to the most important property of quantum memory registers which is their ability to store an exponential amount of classical information in only a polynomial number of qubits by exploiting the principle of superposition. For an illustration, let take n = 5. Then the number n is larger than the estimated number of atoms in the Universe! That is trying to store all these complex numbers would not be possible on any conceivable classical computer - Hilbert space is indeed a big space! However, quantum Nature can manipulates such enormous quantities of data, using even systems containing only a few hundred atoms. It looks as if Nature were keeping 5 hidden pieces of scratch paper on the side, on which she performs calculations as the system evolves. This enormous computational power is something we would very much like to take advantage of. 4. EPR Paradox Teleportation, as it is meant here, relies upon a quintessential quantum phenomenon known as Einstein-Podolsky-Rosen paradox or EPR paradox named after its discoverers Albert Einstein, Boris Podolsky and Nathan Rosen. In their classic article, published in 1935, a composite quantum system consisting of two distant particles being in an entangled state ψ(x 1, p 1 ; x, p ) < x 1, p 1 ; x, p Ψ >= δ(x 1 x L)δ(p 1 + p ), (4) where the symbol δ does not represent a true δ-function, but a normalized function with a high and narrow peak, and L is a large distance - much larger than the range of mutual interaction between particles 1 and. The state represented by the function (4) corresponds to the two particles which have been prepared in such a way that their relative distance is arbitrarily close to L, and their total momentum is arbitrarily close to zero. In this state we know nothing of positions of the individual particles (we only know their distance from each other); and we know nothing of positions of the individual momenta of particles (we only know the total momentum). Once a measurement shows particle to be located at the point x, then particle 1 is certain to be found at position x 1 = x + L, and nowhere else. On the other hand, if not position but momentum of particle is measured and found to have the value p, then particle 1 is certain to be found to have a definite momentum value p 1 = p. Therefore, we must conclude that depending on where the coordinate or the momentum of particle is measured, the particle 1 after such measurement is represented by a one-particle state of scharp position or a momentum, for the distance L between particles as large as we please! Such unequivocal conclusions of quantum mechanics have been perceived by many physicists as paradoxical. The essence of this situation is as follows. If we assume (as we everywhere did) that the quantum mechanical amplitude gives a complete statistical description of a single system, it appears that, even when two particles are arbitrarily far apart, what can be known about the state of particle 1, after a measurement on particle
is performed, depends on the choice of measurement made on particle. Of course, for the people who belive, like Einstein, that the world of physics is governed by local realism, this strange dependence of the expected behavior of particle 1 on the subjective caprice of a distant human experimenter, seemed nothing but misticism and paradox which relate to the incomplete description of physical reality by quantum mechanics. The EPR claim that the description of physical reality by means of quantum mechanics is not complete suggests the existence of a more detailed description of Nature dealt with the hypothetical additional data which have been given the name hidden variables. The tentative goal of a hidden variable theory is the following. In the absence of a detailed knowledge of the hidden variables, calculation could be based on an ensemble average over their purported statistical distribution, and would then yield the statistical predictions of quantum theory. It was just a line along which Einstein and his associates tried to resolve their paradox by suggesting that quantum mechanical amplitudes pertain only to ensembles of systems, rather than single systems. But this development has been broken by the discovery of the so-called Bell s Inequalities and the experimental evidence of their violation. 4.3 Bell s Inequalities In principle, there is nothing unacceptable in the assumption that deterministic hidden variables underlie the statistical features of quantum theory. When Boltzmann created classical statistical mechanics, he assumes the existence of atoms, well before anyone was able to observe - let alone manipulate - individual atoms. Boltzmann s work was attacked by the school of energeticists who did not believe in atoms but later discoveries, relying on new experimental techniques fully vindicated Boltzmann s work. Why then we could not likewise speculate that future discoveries will some day give us access to a subquantum world, described by the hypothetical hidden variables? Let assume that we have at hand the complete set of the deterministic hidden variables which obeys the principle of local causes which is also called the Einstein locality though it has been conjectured well before Einstein. Then the Bell s theorem stands: there is an upper limit to the correlation of distant events which is established by the Bell s inequalities. Note that Bell s theorem is not a specific property of a quantum theory, rather it applies to any physical system with dichotomic variables, whose values are arbitrarily called +1 and 1. Consider then the system which is most easily amenable to experimental verification, i.e. a pair of photons emitted in opposite directions and observed by two distant observers by means of polarizers with different orientations as shown in Figure 9.1 [Williams, p. 19]. Suppose that the first observer has a choice between two different orientation of his polarization analyzer, fixed by the angles α and γ with an arbitrary axis. For each orientation, an experiment has two possible (and unpredictable) outcomes. The hypothesis that we want to test is that any outcome is causally determined by the local hidden variables, of unknown nature, but due to the locality pertaining only to the photon and to the apparatus of the first observer. If we choose α and γ we obtain the outcomes a and c, respectively, both equal ±1. Einstein locality asserts that these outcomes cannot depend on the orientation of the analyzers used by the second observer. Suppose, that the latter also has a choice of two alternative directions β or γ (γ being the same as may be chosen by the first observer). The outcomes of the second observer are b = ±1 or c = ±1, respectively, and are determined by the hidden variables inherent in the second photon and apparatus.
We notice that in any case the results a, b and c identically satisfy a(b c) ±(1 bc), (5) since both sides of this equations vanish if b=c and are equal to ± if b c. Now, let repeat the same joint experiment many times, with many consecutive photon pairs. Then, for j-th photon we have a j b j a j c j ±(1 b j c j ). (6) Naturally, the hidden variables, which we cannot control, are different for each j. The serial number j can thus be understood as a shorthand notation for the unknown values of these hidden variables. In particular, taking an average over the hidden variables is the same as taking an average over j, and we thus obtain after average < ab > < ac > 1 < bc >, (7) where < ab > is the sum of all products a j b j, divided by the total number of photon pairs or, in other words, < ab > is the correlation of the outcomes a and b (the same for < ac > and < bc >). Now comes the crux of this consideration: Although quantum theory is unable to predict individual values a j, b j, c j, it can very well predict average values, and, in particular, correlations like those which appear in an inequality (7). Moreover, these correlations can be measured experimentally, regardless of any theory. The result of quantum calculations for an entangled state of polarized photons is so, inequality (4) becomes < ab >= cos (α β), (8) cos (α β) cos (α γ) + cos (β γ) 1. (9) If the principles underlying in the derivation of the enequality (6) are valid, then this inequality must be valid for any set of the angles α, β, γ. But it is not the case! For instance, if the three directions α, β, γ are separated by angles of 3, as in Figure 6.7 (a) [Peres, p. 164], then the three cosines are 1, 1 and 1, respectively, and the left hand side of Eq. (6) is 3. Therefore the Bell s inequality is violated by quantum theory. For experimental verification of this remarkable conclusion, it is sufficient to show the validity of the cosine-law (8), and it has been done! It is here that Bell s theorem comes to put a cap on science fiction: there are no local hidden variables theory for quantum mechanics! Therefore, the Einstein program cannot be carried out, as long as the theory is required to be local from the very beginning. Any theory built on such strict local realism fails to reproduce some strange predictions of multiparticle quantum mechanics - predictions which have been verified experimentally to a high degree of acuracy. Therefore, the quest for return to local realism in physics must remain unfulfilled, and we have to accept the existence of the mysterious long-range correlations between widely separated subsystems. The entangled EPR pair is real - but in the nonlocal world! Such is our world, at least, at the level of the individual phenomena which are fundamentally unpredictable. However, the Einstein locality retains for quantum averages, which can be predicted and causally controlled. It is instructive to exectify what we mean saying nonlocal interaction. The best starting point for this is to recall that the well-familiar local interactions are characterized by three criteria: (i) they are mediated by another entity, such as a particle or field, (ii)
they propagate no faster than the speed of light, and (iii) their strength drops off with distance. By flipping each of these criteria, we must say that the nonlocal interactions are (i) not mediated by anything, (ii) not limited to acting at the speed of light, and (iii) do not drop off in strength with distance. Too much to satisfy the majority of physicists, isn t it? Nevertheless, the scientific evidence that arose in the mid-198s and that we just demonstrated, proved, theoretically and empirically, that the reality is nonlocal, and hence the EPR phenomena is worth to be called not EPR paradox but EPR effect concerning the behavior of entangled quantum systems. And it is namely this remarkable type of interaction that is exploited in quantum teleportation.