DOCSLIB.ORG

Psi $-Epistemic Interpretations of Quantum Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Psi $-Epistemic Interpretations of Quantum Theory ψ-epistemic interpretations of quantum theory have a mea- surement problem Joshua B. Ruebeck1, Piers Lillystone1, and Joseph Emerson2,3 1Institute for Quantum Computing and Department of Physics and Astronomy University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 2 Institute for Quantum Computing and Department of Applied Math University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 3 Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada February 11, 2020 ψ-epistemic interpretations of quantum theory tum state is merely a state of knowledge about the real maintain that quantum states only represent in- state of the system. A very thorough review of these complete information about the physical states two stances can be found in [24]. A third recently ar- of the world. A major motivation for this view ticulated category of ψ-doxastic interpretations [25{31] is the promise to provide a reasonable account argue that the quantum state is a state of belief, and of state update under measurement by assert- are distinguished from ψ-epistemic interpretations by ing that it is simply a natural feature of updat- the fact that they deny that a system has some `real ing incomplete statistical information. Here we state.' While not all interpretations conform to these demonstrate that all known ψ-epistemic onto- three descriptors, they are useful categories insofar as logical models of quantum theory in dimension they allow us to qualitatively discuss certain features d ≥ 3, including those designed to evade the con- separately from the particular interpretation in which clusion of the PBR theorem, cannot represent they are embedded. state update correctly. Conversely, interpreta- The ψ-epistemic class has garnered attention as a tions for which the wavefunction is real evade view which provides very appealing explanations of such restrictions despite remaining subject to otherwise paradoxical features of quantum theory like long-standing criticism regarding physical dis- the state update rule [17], the classical limit under continuity, indeterminism and the ambiguity of quantum chaos [20], no cloning [32], and entangle- the Heisenberg cut. This revives the possibil- ment [33]. For example, through this viewpoint state ity of a no-go theorem with no additional as- update is not a physical `collapse' process and therefore sumptions, and demonstrates that what is usu- not subject to paradoxes, indeterminism, and disconti- ally thought of as a strength of epistemic inter- nuity; rather it is understood as analogous to the non- pretations may in fact be a weakness. pardoxical `collapse' of a subjective probability distribu- tion via Bayes' rule upon consideration of new informa- tion. While several ψ-epistemic models have been pro- 1 Introduction posed [16, 22, 34{38], they generally have undesirable features or are restricted to a subtheory of full quan- There are many interpretations of quantum theory1. arXiv:1812.08218v3 [quant-ph] 9 Feb 2020 tum theory. None achieve all of the features that an Among the many differences between these interpreta- optimistic ψ-epistemicist would expect. tions, one that often takes center stage is the stance that This suggests the possibility that a fully satisfactory they take towards the wavefunction or quantum state. ψ-epistemic interpretation cannot actually explain all Three broad categories have been identified which cap- of quantum theory despite the qualitatively compelling ture a number of interpretations. Two of these cate- features of such a view2. This suspicion has led to a gories are more commonly juxtaposed: ψ-ontic inter- number of no-go theorems in recent years which estab- pretations [1{14] posit that the quantum state is a part of the real (physical) state of a system, whereas ψ- 2 epistemic interpretations [15{23] argue that the quan- As is well-known, the work of Bell [39] has shown that no ψ- epistemic interpretation can evade the non-locality that manifests Joshua B. Ruebeck: [email protected] trivially in ψ-ontic interpretations; this trivial manifestation of non-locality in ψ-ontic interpretations is an oft-forgotten insight 1This is an understatement. from Einstein [15, 18, 23]. Accepted in Quantum 2019-09-27, click title to verify 1 lish that, given at least one additional assumption, any ontic distinction4. Here we take some preliminary steps consistent interpretation of quantum theory cannot be in both of these directions, and argue that ψ-epistemic ψ-epistemic [36, 40{43]. These no-go theorems are gen- models are the natural arena in which to investigate in- erally proven within the ontological models formalism, teresting behavior of state update under measurement. which describes a large class of existing interpretations In Section2, we describe the ontological models for- of quantum theory [21, 44, 45]. malism, adding a description of state update under Within the ontological models formalism3, ψ- measurement and motivating its importance. Despite epistemic models can be given a precise mathematical this motivation, one might still argue that many dis- definition called the ψ-epistemic criterion [22]. This cri- tinctly quantum phenomena (e.g. Bell inequality vi- terion allows the possibility of conclusively ruling out olations) can be described without reference to state this type of model. Outside of this framework, it is un- update; thus, from an operationalist point of view, we likely that ψ-epistemic models can be precluded with shouldn't need to consider state update in order to in- any kind of certainty; ψ-doxastic interpretations, for vestigate these phenomena. However, we show in Sec- example, do not fit neatly into the ontological models tion3 that the consideration of state update actually framework and thus are not necessarily ruled out by places nontrivial restrictions on how one can represent these no-go theorems. This is despite the fact that they even a prepare-and-measure-once experiment. Thus our share many of the features which make ψ-epistemic in- results are directly applicable to models which have only terpretations appealing. In the present paper we restrict specified behavior for a single measurement. Section4 our attention to the ontological models framework. reviews a number of examples of ontological models The fact that an extra assumption is required to rule from the literature; in each case we either specify its out ψ-epistemic theories has purportedly been demon- state update rule (in dimension d = 2, for ψ-ontic mod- strated by the existence of ψ-epistemic models which, els, and for some models of subtheories) or prove its while being individually unsatisfactory for various rea- impossibility (for all known ψ-epistemic models in di- sons, do satisfy at least the bare minimum requirements mension d ≥ 3). Finally, we discuss the implications of a ψ-epistemic theory [35{37]. All of these models of our results and describe some open questions in Sec- were specified within a prepare-measure framework, so tion5. they have been proven to reproduce quantum statistics for all experiments that involve preparing a state and then measuring it once. In this paper we show that, 2 Defining measurement update in on- if we allow sequential measurements in the operational description, these models cannot reproduce operational tological models statistics. Our main contribution in this work is thus to demon- 2.1 The ontological models formalism strate that the state update rule imposes severe con- In the standard treatment, an operational theory [44] is straints on ψ-epistemic models. This is in contrast to described by a set of preparations P, a set of transfor- the prevailing view that, as articulated by Leifer, \a mations T , and a set of measurements M along with a straightforward resolution of the collapse of the wave- probability distribution function, the measurement problem, Schr¨odingerscat and friends is one of the main advantages of ψ-epistemic Pr(k|M, T, P ). (1) interpretations" [24]. As a consequence, we revive the possibility of a general no-go theorem for ψ-epistemic This quantity describes the probability of some mea- models that doesn't rely on an additional assumption surement outcome k ∈ given an experimenter's choice such as the locality assumption required in [40] which Z of P ∈ P, T ∈ T , and M ∈ M. When consider- conflicts with the non-locality that is implied by Bell's ing transformations this is called the prepare-transform- theorem [43,46]. See AppendixA for further discussion measure operational framework, and when we omit on this point. transformations it is the prepare-measure framework. Although state update under measurement has been Often we take P to be the set of pure quantum state described in a few specific models [22, 34, 47, 48] and preparations, T to be the full set unitary maps on a discussed with regards to contextuality [49], it has yet to Hilbert space, and M to be all projective measurements be treated generally or in relation to the ψ-epistemic/ψ- on this Hilbert space. In this case, we say we are de- 3A note on potentially confusing terminology—an ontological model assumes the reality of some kind of state (hence “onto- 4Although the Leggett-Garg inequalities [50, 51] might be con- logical”), but does not assume the reality of the quantum state strued as a general treatment of state update, it is more accurate specifically (and so is not necessarily ψ-ontic). to say that they are about the absence of state update. Accepted in Quantum 2019-09-27, click title to verify 2 scribing the full quantum theory5; in contrast, a sub- given the previous state λ and the choice T of trans- theory is described by taking subsets of P, T , M for the formation. Finally, measurements are represented by a full quantum theory. For example, in quantum infor- response function ξ(k|λ, M) which describes the prob- mation settings we often consider only measurements ability of an outcome k given the ontic state λ and the in the standard basis.
Load more

Recommended publications
  • Arxiv:2004.01992V2 [Quant-Ph] 31 Jan 2021 Ity [6–20]
    Hidden Variable Model for Universal Quantum Computation with Magic States on Qubits Michael Zurel,1, 2, ∗ Cihan Okay,1, 2, ∗ and Robert Raussendorf1, 2 1Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada 2Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC, Canada (Dated: February 2, 2021) We show that every quantum computation can be described by a probabilistic update of a proba- bility distribution on a finite phase space. Negativity in a quasiprobability function is not required in states or operations. Our result is consistent with Gleason's Theorem and the Pusey-Barrett- Rudolph theorem. It is often pointed out that the fundamental objects In Theorem 2, we apply this to quantum computation in quantum mechanics are amplitudes, not probabilities with magic states, showing that universal quantum com- [1, 2]. This fact notwithstanding, here we construct a de- putation can be classically simulated by the probabilistic scription of universal quantum computation|and hence update of a probability distribution. of all quantum mechanics in finite-dimensional Hilbert This looks all very classical, and therein lies a puzzle. spaces|in terms of a probabilistic update of a probabil- In fact, our Theorem 2 is running up against a number ity distribution. In this formulation, quantum algorithms of no-go theorems: Theorem 2 in [23] and the Pusey- look structurally akin to classical diffusion problems. Barrett-Rudolph (PBR) theorem [24] say that probabil- While this seems implausible, there exists a well-known ity representations for quantum mechanics do not exist, special instance of it: quantum computation with magic and [9{13] show that negativity in certain Wigner func- states (QCM) [3] on a single qubit.
    [Show full text]
  • Arxiv:1909.06771V2 [Quant-Ph] 19 Sep 2019
    Quantum PBR Theorem as a Monty Hall Game Del Rajan ID and Matt Visser ID School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand. (Dated: LATEX-ed September 20, 2019) The quantum Pusey{Barrett{Rudolph (PBR) theorem addresses the question of whether the quantum state corresponds to a -ontic model (system's physical state) or to a -epistemic model (observer's knowledge about the system). We reformulate the PBR theorem as a Monty Hall game, and show that winning probabilities, for switching doors in the game, depend whether it is a -ontic or -epistemic game. For certain cases of the latter, switching doors provides no advantage. We also apply the concepts involved to quantum teleportation, in particular for improving reliability. Introduction: No-go theorems in quantum foundations Furthermore, concepts involved in the PBR proof have are vitally important for our understanding of quantum been used for a particular guessing game [43]. physics. Bell's theorem [1] exemplifies this by showing In this Letter, we reformulate the PBR theorem into that locally realistic models must contradict the experi- a Monty Hall game. This particular gamification of the mental predictions of quantum theory. theorem highlights that winning probabilities, for switch- There are various ways of viewing Bell's theorem ing doors in the game, depend on whether it is a -ontic through the framework of game theory [2]. These are or -epistemic game; we also show that in certain - commonly referred to as nonlocal games, and the best epistemic games switching doors provides no advantage. known example is the CHSH game; in this scenario the This may have consequences for an alternative experi- participants can win the game at a higher probability mental test of the PBR theorem.
    [Show full text]
  • Ehrenfest Theorem
    Ehrenfest theorem Consider two cases, A=r and A=p: d 1 1 p2 1 < r > = < [r,H] > = < [r, + V(r)] > = < [r,p2 ] > dt ih ih 2m 2imh 2 [r,p ] = [r.p]p + p[r,p] = 2ihp d 1 d < p > ∴ < r > = < 2ihp > ⇒ < r > = dt 2imh dt m d 1 1 p2 1 < p > = < [p,H] > = < [p, + V(r)] > = < [p,V(r)] > dt ih ih 2m ih [p,V(r)] ψ = − ih∇V(r)ψ − ()− ihV(r)∇ ψ = −ihψ∇V(r) ⇒ [p,V(r)] = −ih∇V(r) d 1 d ∴ < p > = < −ih∇V(r) > ⇒ < p > = < −∇V(r) > dt ih dt d d Compare this with classical result: m vr = pv and pv = − ∇V dt dt We see the correspondence between expectation of an operator and classical variable. Poisson brackets and commutators Poisson brackets in classical mechanics: ⎛ ∂A ∂B ∂A ∂B ⎞ {A, B} = ⎜ − ⎟ ∑ ⎜ ⎟ j ⎝ ∂q j ∂p j ∂p j ∂q j ⎠ dA ⎛ ∂A ∂A ⎞ ∂A ⎛ ∂A ∂H ∂A ∂H ⎞ ∂A = ⎜ q + p ⎟ + = ⎜ − ⎟ + ∑ ⎜ & j & j ⎟ ∑ ⎜ ⎟ dt j ⎝ ∂q j ∂p j ⎠ ∂t j ⎝ ∂q j ∂q j ∂p j ∂p j ⎠ ∂t dA ∂A ∴ = {A,H}+ Classical equation of motion dt ∂ t Compare this with quantum mechanical result: d 1 ∂A < A > = < [A,H] > + dt ih ∂ t We see the correspondence between classical mechanics and quantum mechanics: 1 [A,H] → {A, H} classical ih Quantum mechanics and classical mechanics 1. Quantum mechanics is more general and cover classical mechanics as a limiting case. 2. Quantum mechanics can be approximated as classical mechanics when the action is much larger than h.
    [Show full text]
  • The Mathemathics of Secrets.Pdf
    THE MATHEMATICS OF SECRETS THE MATHEMATICS OF SECRETS CRYPTOGRAPHY FROM CAESAR CIPHERS TO DIGITAL ENCRYPTION JOSHUA HOLDEN PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright c 2017 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu Jacket image courtesy of Shutterstock; design by Lorraine Betz Doneker All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Holden, Joshua, 1970– author. Title: The mathematics of secrets : cryptography from Caesar ciphers to digital encryption / Joshua Holden. Description: Princeton : Princeton University Press, [2017] | Includes bibliographical references and index. Identifiers: LCCN 2016014840 | ISBN 9780691141756 (hardcover : alk. paper) Subjects: LCSH: Cryptography—Mathematics. | Ciphers. | Computer security. Classification: LCC Z103 .H664 2017 | DDC 005.8/2—dc23 LC record available at https://lccn.loc.gov/2016014840 British Library Cataloging-in-Publication Data is available This book has been composed in Linux Libertine Printed on acid-free paper. ∞ Printed in the United States of America 13579108642 To Lana and Richard for their love and support CONTENTS Preface xi Acknowledgments xiii Introduction to Ciphers and Substitution 1 1.1 Alice and Bob and Carl and Julius: Terminology and Caesar Cipher 1 1.2 The Key to the Matter: Generalizing the Caesar Cipher 4 1.3 Multiplicative Ciphers 6
    [Show full text]
  • The Case for Quantum State Realism
    Western University Scholarship@Western Electronic Thesis and Dissertation Repository 4-23-2012 12:00 AM The Case for Quantum State Realism Morgan C. Tait The University of Western Ontario Supervisor Wayne Myrvold The University of Western Ontario Graduate Program in Philosophy A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Morgan C. Tait 2012 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Philosophy Commons Recommended Citation Tait, Morgan C., "The Case for Quantum State Realism" (2012). Electronic Thesis and Dissertation Repository. 499. https://ir.lib.uwo.ca/etd/499 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. THE CASE FOR QUANTUM STATE REALISM Thesis format: Monograph by Morgan Tait Department of Philosophy A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada © Morgan Tait 2012 THE UNIVERSITY OF WESTERN ONTARIO School of Graduate and Postdoctoral Studies CERTIFICATE OF EXAMINATION Supervisor Examiners ______________________________ ______________________________ Dr. Wayne Myrvold Dr. Dan Christensen Supervisory Committee ______________________________ Dr. Robert Spekkens ______________________________ Dr. Robert DiSalle ______________________________ Dr. Chris Smeenk The thesis by Morgan Christopher Tait entitled: The Case for Quantum State Realism is accepted in partial fulfillment of the requirements for the degree of « Doctor of Philosophy» ______________________ _______________________________ Date Chair of the Thesis Examination Board ii Abstract I argue for a realist interpretation of the quantum state.
    [Show full text]
  • Reformulation of Quantum Mechanics and Strong Complementarity from Bayesian Inference Requirements
    Reformulation of quantum mechanics and strong complementarity from Bayesian inference requirements William Heartspring Abstract: This paper provides an epistemic reformulation of quantum mechanics (QM) in terms of inference consistency requirements of objective Bayesianism, which include the principle of maximum entropy under physical constraints. Physical constraints themselves are understood in terms of consistency requirements. The by-product of this approach is that QM must additionally be understood as providing the theory of theories. Strong complementarity - that dierent observers may live in separate Hilbert spaces - follows as a consequence, which resolves the rewall paradox. Other clues pointing to this reformu- lation are analyzed. The reformulation, with the addition of novel transition probability arithmetic, resolves the measurement problem completely, thereby eliminating subjectivity of measurements from quantum mechanics. An illusion of collapse comes from Bayesian updates by observer's continuous outcome data. Dark matter and dark energy pop up directly as entropic tug-of-war in the reformulation. Contents 1 Introduction1 2 Epistemic nature of quantum mechanics2 2.1 Area law, quantum information and spacetime2 2.2 State vector from objective Bayesianism5 3 Consequences of the QM reformulation8 3.1 Basis, decoherence and causal diamond complementarity 12 4 Spacetime from entanglement 14 4.1 Locality: area equals mutual information 14 4.2 Story of Big Bang cosmology 14 5 Evidences toward the reformulation 14 5.1 Quantum redundancy 15 6 Transition probability arithmetic: resolving the measurement problem completely 16 7 Conclusion 17 1 Introduction Hamiltonian formalism and Schrödinger picture of quantum mechanics are assumed through- out the writing, with time t 2 R. Whenever the word entropy is mentioned without addi- tional qualication, it refers to von Neumann entropy.
    [Show full text]
  • Antum Entanglement in Time
    antum Entanglement in Time by Del Rajan A thesis submied to the Victoria University of Wellington in fullment of the requirements for the degree of Doctor of Philosophy Victoria University of Wellington 2020 Abstract is thesis is in the eld of quantum information science, which is an area that reconceptualizes quantum physics in terms of information. Central to this area is the quantum eect of entanglement in space. It is an interdependence among two or more spatially separated quantum systems that would be impossible to replicate by classical systems. Alternatively, an entanglement in space can also be viewed as a resource in quantum information in that it allows the ability to perform information tasks that would be impossible or very dicult to do with only classical information. Two such astonishing applications are quantum com- munications which can be harnessed for teleportation, and quantum computers which can drastically outperform the best classical supercomputers. In this thesis our focus is on the theoretical aspect of the eld, and we provide one of the rst expositions on an analogous quantum eect known as entanglement in time. It can be viewed as an interdependence of quantum systems across time, which is stronger than could ever exist between classical systems. We explore this temporal eect within the study of quantum information and its foundations as well as through relativistic quantum information. An original contribution of this thesis is the design of one of the rst quantum information applications of entanglement in time, namely a quantum blockchain. We describe how the entanglement in time provides the quantum advantage over a classical blockchain.
    [Show full text]
  • Lecture 8 Symmetries, Conserved Quantities, and the Labeling of States Angular Momentum
    Lecture 8 Symmetries, conserved quantities, and the labeling of states Angular Momentum Today’s Program: 1. Symmetries and conserved quantities – labeling of states 2. Ehrenfest Theorem – the greatest theorem of all times (in Prof. Anikeeva’s opinion) 3. Angular momentum in QM ˆ2 ˆ 4. Finding the eigenfunctions of L and Lz - the spherical harmonics. Questions you will by able to answer by the end of today’s lecture 1. How are constants of motion used to label states? 2. How to use Ehrenfest theorem to determine whether the physical quantity is a constant of motion (i.e. does not change in time)? 3. What is the connection between symmetries and constant of motion? 4. What are the properties of “conservative systems”? 5. What is the dispersion relation for a free particle, why are E and k used as labels? 6. How to derive the orbital angular momentum observable? 7. How to check if a given vector observable is in fact an angular momentum? References: 1. Introductory Quantum and Statistical Mechanics, Hagelstein, Senturia, Orlando. 2. Principles of Quantum Mechanics, Shankar. 3. http://mathworld.wolfram.com/SphericalHarmonic.html 1 Symmetries, conserved quantities and constants of motion – how do we identify and label states (good quantum numbers) The connection between symmetries and conserved quantities: In the previous section we showed that the Hamiltonian function plays a major role in our understanding of quantum mechanics using it we could find both the eigenfunctions of the Hamiltonian and the time evolution of the system. What do we mean by when we say an object is symmetric?�What we mean is that if we take the object perform a particular operation on it and then compare the result to the initial situation they are indistinguishable.�When one speaks of a symmetry it is critical to state symmetric with respect to which operation.
    [Show full text]
  • Schrödinger and Heisenberg Representations
    p. 33 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mathematical formulation of the dynamics of a quantum system is not unique. So far we have described the dynamics by propagating the wavefunction, which encodes probability densities. This is known as the Schrödinger representation of quantum mechanics. Ultimately, since we can’t measure a wavefunction, we are interested in observables (probability amplitudes associated with Hermetian operators). Looking at a time-evolving expectation value suggests an alternate interpretation of the quantum observable: Atˆ () = ψ ()t Aˆ ψ ()t = ψ ()0 U † AˆUψ ()0 = ( ψ ()0 UA† ) (U ψ ()0 ) (4.1) = ψ ()0 (UA † U) ψ ()0 The last two expressions here suggest alternate transformation that can describe the dynamics. These have different physical interpretations: 1) Transform the eigenvectors: ψ (t) →U ψ . Leave operators unchanged. 2) Transform the operators: Atˆ ( ) →U† AˆU. Leave eigenvectors unchanged. (1) Schrödinger Picture: Everything we have done so far. Operators are stationary. Eigenvectors evolve under Ut( ,t0 ) . (2) Heisenberg Picture: Use unitary property of U to transform operators so they evolve in time. The wavefunction is stationary. This is a physically appealing picture, because particles move – there is a time-dependence to position and momentum. Let’s look at time-evolution in these two pictures: Schrödinger Picture We have talked about the time-development of ψ , which is governed by ∂ i ψ = H ψ (4.2) ∂t p. 34 in differential form, or alternatively ψ (t) = U( t, t0 ) ψ (t0 ) in an
    [Show full text]
  • Quantum PBR Theorem As a Monty Hall Game
    quantum reports Article Quantum PBR Theorem as a Monty Hall Game Del Rajan * and Matt Visser School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand; [email protected] * Correspondence: [email protected] Received: 21 November 2019; Accepted: 27 December 2019; Published: 31 December 2019 Abstract: The quantum Pusey–Barrett–Rudolph (PBR) theorem addresses the question of whether the quantum state corresponds to a y-ontic model (system’s physical state) or to a y-epistemic model (observer’s knowledge about the system). We reformulate the PBR theorem as a Monty Hall game and show that winning probabilities, for switching doors in the game, depend on whether it is a y-ontic or y-epistemic game. For certain cases of the latter, switching doors provides no advantage. We also apply the concepts involved in quantum teleportation, in particular for improving reliability. Keywords: Monty Hall game; PBR theorem; entanglement; teleportation; psi-ontic; psi-epistemic PACS: 03.67.Bg; 03.67.Dd; 03.67.Hk; 03.67.Mn 1. Introduction No-go theorems in quantum foundations are vitally important for our understanding of quantum physics. Bell’s theorem [1] exemplifies this by showing that locally realistic models must contradict the experimental predictions of quantum theory. There are various ways of viewing Bell’s theorem through the framework of game theory [2]. These are commonly referred to as nonlocal games, and the best known example is the CHSHgame; in this scenario, the participants can win the game at a higher probability with quantum resources, as opposed to having access to only classical resources.
    [Show full text]
  • Report of on the Reality of the Quantum State Article
    Report of On the reality of the quantum state article De´akNorbert December 6, 2016 1 Introduction Quantum states are the fundamental elements in quantum theory. How- ever, there are a lot of discussions and debates about what a quantum state actually represents. Is it corresponding directly to reality, or is it only a de- scription of what an experimenter knows of some aspect of reality, a physical system? There is a terminology to distinct the two theories: ontic state (state of reality) or epistemic state (state of knowledge). In [1], this is referred as the -ontic/epistemic distinction. The arguments started with the beginnings of quantum theory, with the debates between Bohr-Einstein, concerning the foundations of quantum me- chanics. In 1935, Einstein, Podolsky and Rosen published a paper stating that quantum mechanics is incomplete. As a repsonse, Bohr also published a paper, in which he defended the so called Copenhagen interpretation, a theory strongly supported by himself, Heisenberg and Pauli, that views the quantum state as epistemic [2]. One of the scientific works stating that the quantum state is less real is Spekken's toy theory [3]. It introduces a model, in which a toy bit as a system that can be in one of the four states: (-,-), (-,+), (+,-) and (+,+), represent- ing them on the xy plane, each of them being in a different quadrant. In this representation, the quantum states are epistemic, they are represented by probability distributions. Using this toy theory, Spekkens offers natural ex- planations for some features of quantum theory, like the non-distinguishibility of the non-orthogonal states and the no-cloning theorem.
    [Show full text]
  • Ph125: Quantum Mechanics
    Section 10 Classical Limit Page 511 Lecture 28: Classical Limit Date Given: 2008/12/05 Date Revised: 2008/12/05 Page 512 Ehrenfest’s Theorem Ehrenfest’s Theorem How do expectation values evolve in time? We expect that, as quantum effects becomes small, the fractional uncertainty in a physical observable Ω, given by q h(∆Ω)2i/hΩi, for a state |ψ i, becomes small and we need only consider the expectation value hΩi, not the full state |ψ i. So, the evolution of expectation values should approach classical equations of motion as ~ → 0. To check this, we must first calculate how expectation values time-evolve, which is easy to do: d „ d « „ d « „ dΩ « hΩi = hψ | Ω |ψ i + hψ |Ω |ψ i + hψ | |ψ i dt dt dt dt i „ dΩ « = − [− (hψ |H)Ω |ψ i + hψ |Ω(H|ψ i)] + hψ | |ψ i ~ dt i „ dΩ « = − hψ |[Ω, H]|ψ i + hψ | |ψ i ~ dt i fi dΩ fl = − h[Ω, H]i + (10.1) ~ dt Section 10.1 Classical Limit: Ehrenfest’s Theorem Page 513 Ehrenfest’s Theorem (cont.) If the observable Ω has no explicit time-dependence, this reduces to d i hΩi = − h[Ω, H]i (10.2) dt ~ Equations 10.1 and 10.2 are known as the Ehrenfest Theorem. Section 10.1 Classical Limit: Ehrenfest’s Theorem Page 514 Ehrenfest’s Theorem (cont.) Applications of the Ehrenfest Theorem To make use of it, let’s consider some examples. For Ω = P, we have d i hPi = − h[P, H]i dt ~ We know P commutes with P2/2 m, so the only interesting term will be [P, V (X )].
    [Show full text]