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arXiv:quant-ph/0207070 · International Studies in the Philosophy of Science

Shutters, Boxes, but No Paradoxes: Time Symmetry Puzzles in Quantum Theory

Ruth E. Kastner

Published Mar 1, 2004• 2 citations
arXivHTMLPDF

Abstract

The “N‐box experiment” is a much‐discussed thought experiment in quantum mechanics. It is claimed by some authors that a single particle prepared in a superposition of N+1 box locations and which is subject to a final “post‐selection” measurement corresponding to a different superposition can be said to have occupied “with certainty” N boxes during the intervening time. However, others have argued that under closer inspection, this surprising claim fails to hold. Aharonov and Vaidman have continued their advocacy of the claim in question by proposing a variation on the N‐box experiment, in which the boxes are replaced by shutters and the pre‐ and post‐selected particle is entangled with a photon. These authors argue that the resulting “N‐shutter experiment” strengthens their original claim regarding the N‐box experiment. It is argued in this article that the apparently surprising features of this variation are no more robust than those of the N‐box experiment and that it is not accurate to say that the particle is “with certainty” in all N shutters at any given time.Shutters, Boxes, but No Paradoxes: Time Symmetry Puzzles in Quantum TheoryAll authorsRuth E. Kastner https://doi.org/10.1080/02698590412331289279Published online:14 October 2010 Figure 1 Hilbert Space of the Shutter Particle. Display full size Figure 1 Hilbert Space of the Shutter Particle.

DOI
10.1080/02698590412331289279

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Abstract Since its publication, Aharonov and Vaidman’s three-box paradox has undergone three major advances: i). A non-counterfactual scheme by the same authors in 2003 with strong rather than weak measurements for verifying the particle’s subtle presence in two boxes. ii) A realization of the latter by Okamoto and Takeuchi in 2016. iii) A dynamic version by Aharonov et al. in 2017, with disappearance and reappearance of the particle. We now combine these advances together. Using photonic quantum routers the particle acts like a quantum “shutter.” It is initially split between Boxes A, B and C, the latter located far away from the former two. The shutter particle’s whereabouts can then be followed by a probe photon, split in both space and time and reflected by the shutter in its varying locations. Measuring the former is expected to reveal the following time-evolution: The shutter particle was, with certainty, in boxes A+C at t1, then only in C at t2, and finally in B+C at t3. Another branch of the split probe photon can show that boxes A+B were empty at t2. A Bell-like theorem applied to this experiment challenges any alternative interpretation that avoids disappearance-reappearance in favor of local hidden variables.

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An apparent paradox proposed by Aharonov and Vaidman in which a single particle can be found with certainty in two (or more) boxes is analyzed. It is found that the paradox arises from a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule. The paradox is resolved by observing that the counterfactual usage of the rule is, in general, not valid. 1. Background. The Aharonov-Bergmann-Lebowitz (ABL) rule is a well-known formula for calculating the probabilities of the possible outcomes of observables measured at an intermediate time t between pre- and post-selection measurements at times t1 and t2, respectively [Aharonov, Bergmann, and Lebowitz, 1964]. If an intermediate measurement of nondegenerate observable C with eigenvalues {ci} is performed at time t, the ABL rule states that the probability of outcome cj in the case of a preselection for the state |ψ1 〉 and a post-selection for the state |ψ2 〉 is given by: 1 PABL(cj|ψ1, ψ2) = |〈ψ2|cj〉 | 2 |〈cj||ψ1〉 | 2 i |〈ψ2|ci〉 | 2 |〈ci||ψ1〉 | 2 (1) It has recently been shown 2 that the ABL rule cannot be used in a counterfactual sense: i.e., in general, it cannot be used to calculate the probabilities of possible outcomes of observables that have not actually been measured at time t. 2. The three-box example.

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Time-symmetric quantum mechanics questioned and defended

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“A Time-Symmetric Formulation of Quantum Mechanics” by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is riddled with errors. That is not surprising, given that their starting point is an erroneous “postselection” process. Postselection ignores the measurement postulate of standard quantum physics, according to which a system’s quantum state abruptly switches, upon measurement, into an eigenstate of the measured observable.With that postulate abandoned, it’s to be expected that Aharonov and coauthors obtain results that persistently violate the standard quantum uncertainty principle. For example, in their discussion on page 28 of a spin-1⁄2 system, they incorrectly state that “the measurement of Sπ /4 [the spin component along the diagonal between the x-axis and z-axis] is effectively a simultaneous measurement of Sx and Sz .” If that were true, it would violate the uncertainty principle. But the statement is not true. A measurement of Sπ /4 could be made by positioning a Stern–Gerlach apparatus with its inhomogeneous magnetic field in the π/4 direction, without measuring either Sz or Sx . In fact, the measurement would put the system into an eigenstate of Sπ /4, leaving both Sz and Sx indeterminate, in agreement with the uncertainty principle.Aharonov and coauthors continue: “The idea that they [Sz and Sx ] are both well defined stems from the fact that measuring either one yields +1⁄2 with certainty.” But that supposed violation of the uncertainty principle is wrong. Using an obvious notation, in their example the system is claimed to be in the eigenstate |+Sz 〉 at time t. This eigenstate can also be written as (|+Sx 〉 + |−Sx 〉)/√2‾, showing that the spin component Sx is indeterminate whenever the system is in the state |+Sz 〉, in agreement with the uncertainty principle.One sentence later, the authors state, “If we first measure Sz and then Sx , . . . then, given the pre- and postselection, both measurements yield +1⁄2 with certainty.” That supposed violation of the uncertainty principle is wrong. If we first measure Sz , we’ll get +1⁄2 because the system was previously prepared in that state. But if we then measure Sx , we’ll find the result +1⁄2 with 0.5 probability and −1⁄2 with 0.5 probability, consistent with the uncertainty principle. To use the misleading “ensemble” language of Aharonov and coauthors, every member of the ensemble is in the state |+Sz 〉 = (|+Sx 〉 + |−Sx 〉)/√2‾. Contrary to the authors’ postselection process, it’s not true that 50% of the ensemble is in the state |+Sx 〉 (which would violate the uncertainty principle), and 50% is in |−Sx 〉. To misinterpret quantum superpositions such as (|+Sx 〉 + |−Sx 〉)/√2‾ in this manner is an elementary misconception. It also directly contradicts the experimental facts about measurements of the Stern–Gerlach type.Contrary to the assertion of Aharonov and coauthors on page 32 that they “have not modified quantum mechanics by one iota,” their postselection process would change the foundations of quantum mechanics. The fallacy in that process was pointed out by Asher Peres 16 years ago.11. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J , Google ScholarCrossref, ISI© 2011 American Institute of Physics.

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The interference between two paths of a single photon at a double slit is widely considered to be the most paradoxical result of quantum theory. Here is a new interesting question to the phenomenon: can a single shutter simultaneously close two slits by effectively being in a superposition of different locations? Aharonov and Vaidman have shown that it is indeed possible to construct a quantum shutter that can close two slits and reflect a probe photon perfectly when its initial and final states are appropriately selected. Here we report the experimental demonstration of their proposal overcoming the difficulty to realize a 'quantum shutter' by employing photonic quantum routers. The reflectance ratio of 0.61 ± 0.027 surpasses the classical limit with 4.1 standard deviation, shedding new light on the unusual physical properties of quantum operations. This experimental demonstration, where the strong measurement and non-local superposition seem co-existing, provides an alternative to weak measurements as a way to explore the nature of quantum physics.

Scientific Reports•Oct 14, 2016•20 citations
The “N‐box experiment” is a much‐discussed thought experiment in quantum mechanics. It is claimed by some authors that a single particle prepared in a superposition of N+1 box locations and which is subject to a final “post‐selection” measurement corresponding to a different superposition can be said to have occupied “with certainty” N boxes during the intervening time. However, others have argued that under closer inspection, this surprising claim fails to hold. Aharonov and Vaidman have continued their advocacy of the claim in question by proposing a variation on the N‐box experiment, in which the boxes are replaced by shutters and the pre‐ and post‐selected particle is entangled with a photon. These authors argue that the resulting “N‐shutter experiment” strengthens their original claim regarding the N‐box experiment. It is argued in this article that the apparently surprising features of this variation are no more robust than those of the N‐box experiment and that it is not accurate to say that the particle is “with certainty” in all N shutters at any given time.Shutters, Boxes, but No Paradoxes: Time Symmetry Puzzles in Quantum TheoryAll authorsRuth E. Kastner https://doi.org/10.1080/02698590412331289279Published online:14 October 2010 Figure 1 Hilbert Space of the Shutter Particle. Display full size Figure 1 Hilbert Space of the Shutter Particle.
Abstract Since its publication, Aharonov and Vaidman’s three-box paradox has undergone three major advances: i). A non-counterfactual scheme by the same authors in 2003 with strong rather than weak measurements for verifying the particle’s subtle presence in two boxes. ii) A realization of the latter by Okamoto and Takeuchi in 2016. iii) A dynamic version by Aharonov et al. in 2017, with disappearance and reappearance of the particle. We now combine these advances together. Using photonic quantum routers the particle acts like a quantum “shutter.” It is initially split between Boxes A, B and C, the latter located far away from the former two. The shutter particle’s whereabouts can then be followed by a probe photon, split in both space and time and reflected by the shutter in its varying locations. Measuring the former is expected to reveal the following time-evolution: The shutter particle was, with certainty, in boxes A+C at t1, then only in C at t2, and finally in B+C at t3. Another branch of the split probe photon can show that boxes A+B were empty at t2. A Bell-like theorem applied to this experiment challenges any alternative interpretation that avoids disappearance-reappearance in favor of local hidden variables.
An apparent paradox proposed by Aharonov and Vaidman in which a single particle can be found with certainty in two (or more) boxes is analyzed. It is found that the paradox arises from a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule. The paradox is resolved by observing that the counterfactual usage of the rule is, in general, not valid. 1. Background. The Aharonov-Bergmann-Lebowitz (ABL) rule is a well-known formula for calculating the probabilities of the possible outcomes of observables measured at an intermediate time t between pre- and post-selection measurements at times t1 and t2, respectively [Aharonov, Bergmann, and Lebowitz, 1964]. If an intermediate measurement of nondegenerate observable C with eigenvalues {ci} is performed at time t, the ABL rule states that the probability of outcome cj in the case of a preselection for the state |ψ1 〉 and a post-selection for the state |ψ2 〉 is given by: 1 PABL(cj|ψ1, ψ2) = |〈ψ2|cj〉 | 2 |〈cj||ψ1〉 | 2 i |〈ψ2|ci〉 | 2 |〈ci||ψ1〉 | 2 (1) It has recently been shown 2 that the ABL rule cannot be used in a counterfactual sense: i.e., in general, it cannot be used to calculate the probabilities of possible outcomes of observables that have not actually been measured at time t. 2. The three-box example.
What is the true meaning of quantum superposition? Can a particle genuinely reside in several places simultaneously? These questions lie at the heart of this paper which presents an updated survey of some important stages in the evolution of the three-boxes paradox, as well as novel conclusions drawn from it. We begin with the original thought experiment of Aharonov and Vaidman, and proceed to its non-counterfactual version. The latter was recently realized by Okamoto and Takeuchi using a quantum router. We then outline a dynamic version of this experiment, where a particle is shown to "disappear" and "re-appear" during the time evolution of the system. This surprising prediction based on self-cancellation of weak values is directly related to our notion of Quantum Oblivion. Finally, we present the non-counterfactual version of this disappearing-reappearing experiment. Within the near future, this last version of the experiment is likely to be realized in the lab, proving the existence of exotic hitherto unknown forms of superposition. With the aid of Bell's theorem, we prove the inherent nonlocality and nontemporality underlying such pre- and post-selected systems, rendering anomalous weak values ontologically real.
“A Time-Symmetric Formulation of Quantum Mechanics” by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is riddled with errors. That is not surprising, given that their starting point is an erroneous “postselection” process. Postselection ignores the measurement postulate of standard quantum physics, according to which a system’s quantum state abruptly switches, upon measurement, into an eigenstate of the measured observable.With that postulate abandoned, it’s to be expected that Aharonov and coauthors obtain results that persistently violate the standard quantum uncertainty principle. For example, in their discussion on page 28 of a spin-1⁄2 system, they incorrectly state that “the measurement of Sπ /4 [the spin component along the diagonal between the x-axis and z-axis] is effectively a simultaneous measurement of Sx and Sz .” If that were true, it would violate the uncertainty principle. But the statement is not true. A measurement of Sπ /4 could be made by positioning a Stern–Gerlach apparatus with its inhomogeneous magnetic field in the π/4 direction, without measuring either Sz or Sx . In fact, the measurement would put the system into an eigenstate of Sπ /4, leaving both Sz and Sx indeterminate, in agreement with the uncertainty principle.Aharonov and coauthors continue: “The idea that they [Sz and Sx ] are both well defined stems from the fact that measuring either one yields +1⁄2 with certainty.” But that supposed violation of the uncertainty principle is wrong. Using an obvious notation, in their example the system is claimed to be in the eigenstate |+Sz 〉 at time t. This eigenstate can also be written as (|+Sx 〉 + |−Sx 〉)/√2‾, showing that the spin component Sx is indeterminate whenever the system is in the state |+Sz 〉, in agreement with the uncertainty principle.One sentence later, the authors state, “If we first measure Sz and then Sx , . . . then, given the pre- and postselection, both measurements yield +1⁄2 with certainty.” That supposed violation of the uncertainty principle is wrong. If we first measure Sz , we’ll get +1⁄2 because the system was previously prepared in that state. But if we then measure Sx , we’ll find the result +1⁄2 with 0.5 probability and −1⁄2 with 0.5 probability, consistent with the uncertainty principle. To use the misleading “ensemble” language of Aharonov and coauthors, every member of the ensemble is in the state |+Sz 〉 = (|+Sx 〉 + |−Sx 〉)/√2‾. Contrary to the authors’ postselection process, it’s not true that 50% of the ensemble is in the state |+Sx 〉 (which would violate the uncertainty principle), and 50% is in |−Sx 〉. To misinterpret quantum superpositions such as (|+Sx 〉 + |−Sx 〉)/√2‾ in this manner is an elementary misconception. It also directly contradicts the experimental facts about measurements of the Stern–Gerlach type.Contrary to the assertion of Aharonov and coauthors on page 32 that they “have not modified quantum mechanics by one iota,” their postselection process would change the foundations of quantum mechanics. The fallacy in that process was pointed out by Asher Peres 16 years ago.11. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J , Google ScholarCrossref, ISI© 2011 American Institute of Physics.
Coherence as a resource for NNN-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 NNN-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.
The interference between two paths of a single photon at a double slit is widely considered to be the most paradoxical result of quantum theory. Here is a new interesting question to the phenomenon: can a single shutter simultaneously close two slits by effectively being in a superposition of different locations? Aharonov and Vaidman have shown that it is indeed possible to construct a quantum shutter that can close two slits and reflect a probe photon perfectly when its initial and final states are appropriately selected. Here we report the experimental demonstration of their proposal overcoming the difficulty to realize a 'quantum shutter' by employing photonic quantum routers. The reflectance ratio of 0.61 ± 0.027 surpasses the classical limit with 4.1 standard deviation, shedding new light on the unusual physical properties of quantum operations. This experimental demonstration, where the strong measurement and non-local superposition seem co-existing, provides an alternative to weak measurements as a way to explore the nature of quantum physics.