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arXiv:1506.07850 · Electronic Proceedings in Theoretical Computer Science

Logical pre- and post-selection paradoxes are proofs of contextuality

Matthew F. Pusey, Matthew Leifer

Published Nov 4, 2015• 25 citations

Abstract

If a quantum system is prepared and later post-selected in certain states, "paradoxical" predictions for intermediate measurements can be obtained. This is the case both when the intermediate measurement is strong, i.e. a projective measurement with Luders-von Neumann update rule, or with weak measurements where they show up in anomalous weak values. Leifer and Spekkens [quant-ph/0412178] identified a striking class of such paradoxes, known as logical pre- and post-selection paradoxes, and showed that they are indirectly connected with contextuality. By analysing the measurement-disturbance required in models of these phenomena, we find that the strong measurement version of logical pre- and post-selection paradoxes actually constitute a direct manifestation of quantum contextuality. The proof hinges on under-appreciated features of the paradoxes. In particular, we show by example that it is not possible to prove contextuality without Luders-von Neumann updates for the intermediate measurements, nonorthogonal pre- and post-selection, and 0/1 probabilities for the intermediate measurements. Since one of us has recently shown that anomalous weak values are also a direct manifestation of contextuality [arXiv:1409.1535], we now know that this is true for both realizations of logical pre- and post-selection paradoxes.

DOI: 10.4204/eptcs.195.22

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If a quantum system is prepared and later post-selected in certain states, "paradoxical" predictions for intermediate measurements can be obtained. This is the case both when the intermediate measurement is strong, i.e. a projective measurement with Luders-von Neumann update rule, or with weak measurements where they show up in anomalous weak values. Leifer and Spekkens [quant-ph/0412178] identified a striking class of such paradoxes, known as logical pre- and post-selection paradoxes, and showed that they are indirectly connected with contextuality. By analysing the measurement-disturbance required in models of these phenomena, we find that the strong measurement version of logical pre- and post-selection paradoxes actually constitute a direct manifestation of quantum contextuality. The proof hinges on under-appreciated features of the paradoxes. In particular, we show by example that it is not possible to prove contextuality without Luders-von Neumann updates for the intermediate measurements, nonorthogonal pre- and post-selection, and 0/1 probabilities for the intermediate measurements. Since one of us has recently shown that anomalous weak values are also a direct manifestation of contextuality [arXiv:1409.1535], we now know that this is true for both realizations of logical pre- and post-selection paradoxes.

Many seemingly paradoxical effects are known in the predictions for outcomes of intermediate measurements made on pre- and post-selected quantum systems. Despite appearances, these effects do not demonstrate the impossibility of a noncontextual hidden variable theory, since an explanation in terms of measurement disturbance is possible. Nonetheless, we show that for every paradoxical effect wherein all the pre- and post-selected probabilities are 0 or 1 and the pre- and post-selected states are nonorthogonal, there is an associated proof of the impossibility of a noncontextual hidden variable theory. This proof is obtained by considering all the measurements involved in the paradoxical effect--the preselection, the post-selection, and the alternative possible intermediate measurements--as alternative possible measurements at a single time.

Coherence as a resource for

N

-box and quantum pigeonhole paradoxes

Pre- and post-selection (PPS) paradoxes are striking demonstrations of quantum nonclassicality. Logical PPS paradoxes, where inferences made with the Aharonov-Bergmann-Lebowitz (ABL) rule are exactly 0 or 1, are linked to contextuality. Non-logical paradoxes lack this strong signature. In this work, we analyse more general, non-logical PPS scenarios involving mixed pre- and post-selected states. We show that two such scenarios, the

N

-box and quantum pigeonhole paradoxes, require coherence of both pre- and post-selected states in the basis of the intermediate measurement. This is done by showing each paradox holds if and only if there is weak-value anomaly for a single, paradox-specific operator, together with the fact that weak-value anomaly requires coherence. This clarifies the role of different notions of nonclassicality in these scenarios, highlighting the required quantitative departures from (strictly) classical explanations provided by incoherent sub-theories of quantum theory.

By analysing the concept of contextuality (Bell–Kochen–Specker) in terms of pre- and post-selection, it is possible to assign definite values to observables in a new and surprising way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated. We also prove that every PPS-paradox with definite predictions directly implies 'quantum contextuality' which is introduced as the analogue of contextuality at the level of quantum mechanics rather than at the level of hidden variable theories. Finally, we argue that certain results of these measurements (e.g. eccentric weak values outside the eigenvalue spectrum) cannot be explained by a 'classical-like' hidden variable theory.

By analyzing the concept of contextuality (Bell-Kochen-Specker) in terms of pre-and-post-selection (PPS), it is possible to assign definite values to observables in a new way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated including a proof that every PPS-paradox with definite predictions implies contextuality. Certain results of these measurements (eccentric weak values with e.g. negative values outside the spectrum), however, cannot be explained by a classical-like hidden variable theory. Surprising theoretical implications are discussed.

By analyzing the concept of contextuality (Bell-Kochen-Specker) in terms of pre-and-post-selection (PPS), it is possible to assign definite values to observables in a new way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated including a proof that every PPS-paradox with definite predictions implies contextuality. Certain results of these measurements (eccentric weak values with e.g. negative values outside the spectrum), however, cannot be explained by a "classical-like" hidden variable theory. Surprising theoretical implications are discussed.

Abstract Many features of single-partite quantum walks can be simulated by classical waves. However, it was recently experimentally shown that some temporal sequences of measurements on a quantum walker do not admit a classical description in terms of macro-realistic theories. Here, we extend this observation and present pre- and post-selection scenarios in quantum walks leading to logical paradoxes. Such paradoxes were recently shown to be equivalent to proofs of contextuality, therefore we provide an additional argument for non-classicality of a quantum walk model. The assumptions behind the claims of non-classicality (including contextuality and violations of macro-realism) are also discussed.