QUANTUM MYSTERIES AND THE LOGIC OF COMPUTATION QUANTUM COMPUTING AND ITS APPLICATIONS IN HEALTHCARE WORKSHOP MARCH 16, 2017 ALI ESKANDARIAN VIRGINIA SCIENCE AND TECHNOLOGY CAMPUS COLLEGE OF PROFESSIONAL STUDIES THE GEORGE WASHINGTON UNIVERSITY
Abstract 2 The notions of probability and statistics play important but fundamentally different roles in classical and modern theories of physics. What appears mysterious according to classical laws of nature, has important consequences in how we control and guide physical processes in tasks related to computation and communication. Here we attempt to elucidate the way quantum concepts are understood and employed by scientists, and the way they reshape our thinking about related fields. The emphasis will be on foundational issues.
19 th Century Physics! 3 The true logic of this world is the calculus of probabilities. James Clerk Maxwell In James Clerk Maxwell and Peter Michael Harman (ed.), The Scientific Letters and Papers of James Clerk Maxwell, Vol. 1, 1846-1862- (1990), 197.
James Clerk Maxwell 4 From a long view of the history of mankind the most significant event of the nineteenth century will be judged as Maxwell's discovery of the laws of electrodynamics. Richard P. Feynman Quoted in Robert J. Scully, The Demon and the Quantum (2007), 3.
James Clerk Maxwell 5 This change in the conception of reality is the most profound and the most fruitful that physics has experienced since the time of Newton. Referring to James Clerk Maxwell's contributions to physics. Albert Einstein 'Maxwell's Influence on the Development of the Conception of Physical Reality', James Clerk Maxwell: A Commemorative Volume 1831-1931 (1931), 71.
Outline 6 Motivation and prologue A brief review of principles of quantum mechanics, based on the double-slit experiment Classical and Quantum worlds Mermin s construction of a gedanken experiment based on Hardy s two-particle quantum states [time permitting] Concluding remarks and epilogue Appendix: More details of the double-slit experiment
Motivation & Prologue 7 I have been tasked by the organizers of this workshop, especially Dr. Sergei Leonov, to provide a clear and simple exposition on the role of probabilities in Quantum Theory for an audience of scientists, mostly non-physicists. I shall use the results of the double-slit experiment (a la Feynman) to motivate the discussions. It is a daunting task given the limitations of time and context. I hope I may be able to, at least, acquaint your imagination with some of the concepts and dilemmas that the non-intuitive aspects of quantum theory pose, and show how physics, probability theory, and logic are intricately intertwined in this most sophisticated of our theories of nature.
Motivation & Prologue 8 Nature demands that we walk a logical tightrope if we wish to describe her. - Richard Feynman (Quantum Mechanics and Path Integrals) Logical Tightrope is what I wish to focus on, for the purposes of this presentation! It is not necessary to know all the machinery or consequences of quantum mechanics in order to contribute, but one must know the foundations and principles well.
Motivation & Prologue 9 Most physicists spend their time building and extending knowledge of specific fields (nuclear, particle, solid-state, cosmology) based on the consequences of the known laws. (in modern times, laws of nature through quantum theory!) It just so happens that quantum theory poses not only scientific, but also philosophical questions (some misguided!) and difficulties of enormous importance to us as human beings that cast doubt on what we have taken for granted to be rational.
Motivation & Prologue 10 Einstein, Podolsky, and Rosen (EPR) pointed out one such difficulty in their celebrated paper of 1935, based on rational arguments with clear logical inferences, and based on what they considered to be clear logical premises. Unlike what many in the following generation of physicists think, Einstein was not resistant to the ideas of quantum theory, having played an important role in advancing the theory himself. Rather, Einstein was keenly interested in consistent and complete theories. The EPR work was to demonstrate the incompleteness of quantum theory.
Motivation & Prologue 11 I shall try to demonstrate that to build a coherent theory incorporating our experimental observations of the microscopic world, we need to extend and amend our mathematical and logical tools in appropriate instances/circumstances, including the rules of probability theory that apply successfully to all the phenomena in the classical world. I will also argue that practical applications of quantum theory may require the advent of a different logic. We start with the double-slit experiment following Feynman s approach.*
An Interference Experiment with Bullets 12
Interference Experiment with Water Waves 13
Interference Experiment with Electrons Wave Particle Duality 14
Modified Electron Experiment 15
Difficulty with the Concept of Probability Note that we are immediately confronted with a difficulty in the combination law of probability theory (a la Laplace): P(A B) P(A) + P(B) 16 It appears that the combination law on what we might rationally (logically) consider exclusive alternatives does NOT apply to the case of electrons in the double-slit experiment! One may be tempted to argue that there is no other alternative but for a particle to go through either one or the other hole in the course of the experiment! [Herein lies the main difficulty: The intuitive classical understanding and the marriage between logic and probabilities seems to be violated!]
Difficulty with the Concept of Probability 17 The experiments indicate the prospect for Two Alternatives: exclusive alternatives, interfering alternatives. In exclusive alternatives, the usual combination law for probabilities works. In interfering alternatives, the usual combination law of probabilities fails, instead, the concept of interfering probability amplitudes holds! In quantum mechanical calculations we use both. For example, to know what the probability of finding the particle in the range [-d, +d] is, we add the contributions of all detectors in that range using the old combination law (exclusive alternatives), but to find the probability at each detector we use the interfering alternatives.
General Principles of Quantum Mechanics The double-slit experiment suggests a notation for the non-intuitive process leading to the interference of electrons: 18 Φ 1 = x s through hole1 is called the probability amplitude, a complex number. 2 With P 1 = Φ 1 indicating the probability of the electron starting at the source and ending at the position x on the screen. Therefore, probabilities are calculated by computing the square modulus of probability amplitudes (complex numbers).
General Principles of Quantum Mechanics 19 Also, Φ 1 = x s through hole1 the probability amplitude for going from s to x through hole 1 is the amplitude to get from s to 1 multiplied by the amplitude to go from 1 to x. Φ 1 = x 1 1 s (assuming the hole itself does not introduce any complications) This is the multiplication rule for the probability amplitudes
More on Notation 20 One can think of the state of a system being designated by any set of labels that are needed to describe it, or a name, inserted in the mathematical notation as follows: s,+1/ 2 the spin state of an electron in spin up configuration s,+1/ 2 +1/ 2 + 1 ( for example in quantum computing) s, 1/ 2 the spin state of an electron in spin down configuration x x s,+1/ 2 s,+1/ 2 ( for example) =1 ( finding the system in its original state) x y s,+1/ 2 s, 1/ 2 ( for example) = 0 ( for mutually exclusive states of a system)
The Mathematics of Quantum Mechanics 21 The concise notation of bracket, (bra) and (ket), for quantum states was introduced by P.A.M. Dirac [see Principles of Quantum Mechanics], and the relationships are reminiscent of linear vector spaces over complex numbers. The complex conjugate of x s is designated by s x = x s * Through the machinery of linear spaces one can establish a dual relationship between bras and kets, and represent the inner products with brackets.
The Mathematics of Quantum Mechanics 22 The probability interpretation of the square of the absolute value of the amplitudes requires that the discrete sum or the continuous integral of the square modulus of the amplitude function over all space to be finite and properly normalized, which establishes a special metric on the vector space, turning it into a Hilbert space (more strictly, a Rigged Hilbert Space). Therefore, the geometry of quantum Mechanics is that of a Hilbert Space.
The Mathematics of Quantum Mechanics 23 A Physical Observable in this formalism is represented by an operator in the Hilbert space. The results of physical measurements (i.e., real numbers) are eigenvalues of Hermitian operators in the Hilbert space. Q ψ = q ψ Q = Q + with Q as a Hermitian operator, and q as one of its eigenvalues.
The Mathematics of Quantum Mechanics 24 A measurement on the system yields one of the eigenvalues of the system, and leaves the system in the corresponding eigenstate. Before the measurement is made the system can exist in the superposition of all possible states. This postulate is tied to the collapse of the wave packet and the Copenhagen Interpretation of quantum mechanics.
The Principle of Superposition of Amplitudes 25 When there are several alternatives (pathways) to get from one state of the system to another state, without knowing which path is actually taken, the total probability amplitude is the sum of the probability amplitudes for each path: Φ 12 = Φ 1 + Φ 2 = x s through hole1 + x s through hole 2 P 12 = Φ 12 2 Φ 12 = Φ 1 + Φ 2 = x s through both holes = x 1 1 s + x 2 2 s Note: So far we have been discussing the general structure of quantum mechanics without telling you how the probability amplitudes are actually calculated (and no dynamics).
The Superposition Principle 26 What about an experiment with two screens with different number of holes in each? Φ = x s = x a a 1 1 s + x b b 1 1 s + x c c 1 1 s + x c c 2 2 s Φ = i=1,2 α =a,b,c x α α i i s Feynman generalized this set up for his path integral formulation of quantum mechanics.
The Superposition Principle 27 Feynman generalized this set up for his path integral formulation of quantum mechanics. Φ = x s = x a a 1 1 s + x b b 1 1 s + x c c 1 1 s + x c c 2 2 s Φ = i=1,2 α =a,b,c x α α i i s
Classical vs Quantum Mechanical 28 In the classical world once the initial conditions (precisely measured values of the positions and momenta, q i and p i, of the system) are given, local differential equations through Newton s equations, derivable from a more general Hamiltonian function, determine the states of the system at all future times, barring numerical inaccuracies in calculations and computations. In the quantum world, simultaneously precise measurements of the positions and momenta are limited by nature due to Heisenberg s Uncertainty Principle (not because of imprecision of the experimental apparatus or human inaccuracies). However, once one of the states of the system is determined through the measurement process, it evolves deterministically into the future from then on via a time evolution operator, so long as other environmental perturbations do not interfere.
Classical vs Quantum Mechanical 29 In the words of Julian Schwinger, Quantum Mechanics is a causal, statistically deterministic theory. Heisenberg s Uncertainty Principle: Δ q i Δ p i h 4π with h Planck s constant given by h = 6.626068 10-34 m 2 kg /s (or J - s)
Two Relevant Asides 30 Feynman recasts the statement of the uncertainty principle in terms of alternatives mentioned earlier: Any determination of the alternative taken by a process capable of following more than one alternative destroys the interference between alternatives. The EPR work reduces to the demonstration that if certain rational arguments are to be true then the uncertainty principle is violated (i.e., simultaneous knowledge of the measured values of the conjugate variables, x and p becomes possible) in a carefully designed quantum mechanical example. Therefore, leading to the conclusion that quantum mechanics is incomplete.
More on Quantum Mechanics Unitary operators play an important role in quantum mechanics because they transform the state of a system in such a way that the probability amplitude is not altered (they preserve inner products). A Hamiltonian (or the Lagrangian) in classical physics embodies the dynamics of the system and allows derivation of Newtonian equations of motion that yield the positions and momenta of the system: dq i dt dp i dt 31 = H p i = H q i
More on Quantum Mechanics 32 In quantum mechanics Schrodinger s equation for non-relativistic cases and Dirac s equation or the Klein-Gordon equation for relativistic cases, yield the probability amplitudes through a Hamiltonian: # ih & % ( $ 2π ' t ψ = H ψ [The equations of motion in both classical and quantum physics are derivable from an action principle.]
Differences in the Two Worlds 33 In the classical theories of physics, direct statements are made, or relationships established, among objects and events, localized in space and time. The mathematical symbols of the theory have direct bearing on the reality of physical objects and relationships. For example, time and space are integral parts of the narrative of the theory. Symbols representing the dynamics of the systems either in mechanics, such as F for forces; or in electrodynamics, such as E and B for electric and magnetic fields are functions of space and time variables, and are considered directly measurable quantities.
Differences in the Two Worlds 34 In quantum theory, space and time are secondary concepts. The observables are designated by abstract operators. The statements of the theory are more like meta-statements on the direct measurements of the classical world. The parameters x and t as they appear in the QM expressions do not refer to particles or events, rather to the probability amplitude for the events or objects (which has meaning over repeated experiments, unless it happens to be for events of probability 1 ). This immediately deprives the theory of adherence to local characteristics, to some degree, unless interest exists only in a very limited aspect of nature.
Sharp Differences Between the Classical and Quantum Mechanical Worlds 35 To depict in as simple and as stark a manner as possible the differences in the logical underpinnings of the two formalisms, we rely on a gedanken experiment, developed by Mermin based on Lucien Hardy s twoparticle quantum states. Mermin s Gedanken demonstration is based on the analysis of the findings of black boxes that act as detectors, but in an arrangement that excludes any connection between the boxes or between the boxes and the source. Mermin suggests two boxes each having two switches that may be set randomly (inside of the box). Each switch can flash a light upon detection that is either Green (G) or Red (R), indicating the outcome of the detection.
A Gedanken Experiment 36 S1 S1 S2 Left Black Box Source S2 Right Black Box The arrangement for Mermin s Gedanken experiment for two-particle Hardy states
Particulars of the gedanken experiment 37 The source is arranged to be in line with the boxes, with each box on the opposite side of the source, and far enough from it. At regular predetermined time intervals, the source emits two particles each directed toward one of the detectors (black boxes). While the particles are in flight, a random choice is made (say, flip of a coin) to set the switches in each box to one of the two possible positions (S1 or S2). For each run a definite detection occurs, and the color of the flashed light in each box is recorded, G (Green) or R (Red). With this arrangement any observed correlation between the results may be attributed to the correlation between the particles that originated from the same source.
Particulars of the gedanken experiment 38 For each run of the experiment the settings and the outcome of each switch are designated as follows: For example, 21GR indicates the switch on the left box was set to 2, the switch on the right box was set to 1, the outcome for the left box was Green, and the outcome for the right box was Red, etc. Many runs of the experiment are made and the results recorded according to the above nomenclature. The experimental results are then analyzed to look for any correlations/conclusions. The following three main observations are made.
Observations from the gedanken experiments 39 (a) In runs in which the detectors end up with different settings, they never both flash green: 21GG and 12GG never occur. (b) In runs in which both detectors end up set to 2, one occasionally finds both flashing green: 22GG sometimes occurs. (c) In runs in which both detectors end up set to 1, they never both flash red: 11RR never occurs.
Intuitively Logical Interpretation of the Data Given the stated features, it is hard not to interpret the data as follows: 40 From (a) one could reach the conclusion that whenever one of the particles is of a variety that allows type 2 detector (S2 in the diagram) to flash G, the other particle must be of a variety that requires a type 1 detector to flash R, since (21GG and 12GG never occur). From (b) it follows that on those occasional 22 runs when both detectors flash G, both particles must be of a variety that requires a type 1 detector to flash R. Based on the previous conclusion, if the random setting of the detectors, instead of 22, had resulted in a 11 run, then both detectors would have flashed R. But this is impossible since it contradicts (c). [Note the statistical nature of this experiment and the argument due to the finite, albeit large, number of runs, as opposed to the GHZ type experiments]*
Mysterious Results!!! 41 The regularities observed in (a)-(c) analysis of the data are inconsistent with a very common sense (intuitive classical logic), and apparently unavoidable, explanation of the results of the experiment. The observed correlations, therefore, seem deeply mysterious, since (b) appears to be incompatible with (a) and (c)! Utilizing Hardy s two-particle states, Mermin demonstrates that in a quantum mechanical framework it is possible to have a consistent gedanken experiment, incorporating features (a), (b), and (c) of the data. In fact, under suitable conditions a maximum probability of approximately 9% for the 22GG state can be obtained.
Not So Mysterious in QM 42 Let s construct a quantum mechanical state corresponding to the experimental set up described by Mermin, representing S1 and S2 as the position of the switches (detector types) in each box, G (Green) and R (Red) as the values of the measurements for each switch. Note that, in fact, S1 and S2 correspond to two different quantum mechanical operators (for example, in the GHZ type experiments they represented different components of the Pauli operators). With the above in mind, there is, indeed, quite a bit of freedom in building quantum mechanical states representing the two particles in the experiment. One such approach is presented below. In the context of quantum mechanics there is NO mystery to the observed correlations in the gedanken experiment.
QM of the experiment 43 If we label the particle going to the left black box with 1 and the particle to the right black box with 2 then expressing the quantum state of the system in terms of the superposition of three two-particle states, which are eigenstates of operator S1, we d have the following: Eq. 1
QM of the experiment 44 Note that feature (c) of the observed data is taken into account by excluding the S1,R; S1,R> state from the superposition explicitly. Feature (a) of the observations leads to the following orthogonality relations (Eq. 2): Note that the above are obtained by considering the following orthonormality conditions that hold for each particle, 1 0r 2.
QM of the experiment 45 Feature (b) of the observations, apparently incompatible with features (a) and (c), requires only that our two-particle state not to be orthogonal to the S2,G; S2,G> state, i.e., Eq. 3:
QM of the experiment 46 Utilizing Eq. 2 in Eq. 3, we obtain the following expression for p: This tells us that the amplitude g must be non-zero, which in turn implies that a and b from Eq. 2 must be non-zero, too. Therefore, any such set of non-zero parameters with a Y given by Eq. 1 would demonstrate/resolve the quantum mystery!
QM of the experiment 47 With a little algebra, using the normalization of the wave function, an expression for g is found that would enable us to evaluate how large p can be: This gives p a maximum value of 0.09017 which may be obtained from the following equations where i= 1, 2 corresponding to each particle:
The Case for a New Logic 48 Here we have a stark contradiction between our intuitive logical inferences and what can happen within a system that follows a consistent mathematical formalism (in this case of state vectors of a Hilbert space). Where did we go wrong? What logical missteps were made in the analysis of the classical gedanken experiment? [I don t know, I am really asking!] In the context of physicists analyzing experiments, we can state that our intuitive concepts based on the requirements of locality and separability may be at fault! What are the precise translations of these intuitive notions in terms of where the difficulties are in logical reasoning (if there are any)?
The Case for a New Logic 49 We seem to get in trouble with quantum mechanical results whenever we extrapolate from the results of our local experiments to other situations, such as what would have happened if switches were set to 11 in case b of the analysis; or on the case of the double-slit experiment imagining/extrapolating that each electron has to go through one of the holes labeled as 1 or 2! Whether or not logic enjoys an empirical root is an interesting question (philosophical), but it is worth asking: if nature s observed and welldocumented laws seem to violate rationality, shouldn t we reconsider the roots of our logical reasoning?
Classical Computing and Logic 50 All the logical operations in a classical computer are carried out through a few binary operational relationships (NOT, OR, AND, XOR) carried among classical bits, 1s and 0s (Boolean Logic). I contend that these operations are of a sequential nature that inherently carry the concept of sequential time or exclusive alternatives (as soon as a decision point is arrived) as discussed earlier. In the classical world, advances in logic allowed us to think through various finite-state machines, or their probabilistic counterparts to arrive at the concept of a universal Turing machine, useful both to theory and practice. It doesn t matter what hardware, we would achieve the same result! In software, similarly, logic in the form of models and languages has granted us the luxury of encoding our algorithms with universality.
Quantum Computing and Logic 51 Although a finite set of quantum gates allows us to construct quantum algorithms, their universal character is not obvious, neither is there a sense that a universal quantum machine, similar to the universal Turing machine exists! What is the Boolean-like logic for quantum computers? I contend that advances in logic would be helpful, even crucial, in advancing our understanding of the physical theories at the deepest level. Can one develop logic that leads to operations representing the parallel nature of quantum theory based on interfering alternatives, as discussed earlier, and leading to the resolution of seemingly logical paradoxes that are at the heart of non-local features of quantum theory.
Concluding Remarks Epilogue 52 I hope I have been successful in demonstrating an inevitable sense of mystery and puzzlement that appears to haunt our intuition when learning the most successful scientific theory of nature! Note the very important role that probabilities play in the development and understanding of the theory. Comment: QM ends with probabilities, but it is much more than that! However, probability is the starting point for the experts in probability/statistics in their work! I also hope I have had some success in convincing that there is a case to be made for new developments in logic that would bring all the consequences of quantum theory into the fold of rationality. For example, a consistent new logic would have its own definitions and terms that would warn us about logical problems we would get into if we argued classically. It would have terms similar to, say, non-sequitur, etc. Questions???
A modified version of this presentation was given at the 60 th World Statistics Congress in Rio, Brazil, July 28, 2015. R. P. Feynman, R. B. Leighton, and M. Sands, Feynman Lectures in Physics, Volume III (Addison-Wesley, 1964). J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements (Springer, 2001). A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777-780 (1935). D. Bohm and Y. Aharanov, Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky, Phys. Rev. 108, 1070 (1952). J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics I, 195-200 (1964). J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge U.P., Cambridge, 1987). P. A. M. Dirac, The Principles of Quantum Mechanics (4 th edition, Oxford University Press, 1958). N. D. Mermin, Quantum mysteries refined, Am. J. Phys., 62 (10), 880-887 (1994). Lucien Hardy, Nonlocality for Two Particles Without Inequalities for Almost All Entangled states, Phys. Rev. Lett., Volume 71, 11, 1665-1668 (1993). N. D. Mermin, Quantum mysteries revisited, Am. J. Phys., 58 (8), 731-734 (1990). Lecture Notes in Logic: Logic and Algebraic Structures in Quantum Computing, edited by J. Chubb, A. Eskandarian, and V. Harizanov, Cambridge University Press, February 2016. References 53
Appendix: Details of the double-slit 54
Appendix: Details of the double-slit Φ 1 = x 1 1 s Φ 2 = x 2 2 s 55 Electron hole1_ Photon D1= x 1 a 1 s = aφ 1 Electron hole2 _ Photon D1= x 2 b 2 s = bφ 2 Electron at x any hole_ Photon D1= aφ 1 + bφ 2 Assuming Symmetry in devices and geometry Electron at x any hole_ Photon D2 = aφ 2 + bφ 1
Appendix: Details of the double-slit 56 Pr(Electron at x _ Photon D1) = aφ 1 + bφ 2 2 Pr(Electron at x _ Photon D2) = aφ 2 + bφ 1 2 What is the probability of finding an electron at x, irrespective of whether a photon was detected at D1 or D2? Pr(Electron at x ) = aφ 1 + bφ 2 2 + aφ 2 + bφ 1 2 Note: Here the addition rule of the classical probability theory holds, because in principle the two alternatives are distinguishable, even if we choose not to care which detector the photon was detected in! Nature doesn t care about our actions!