© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2206.02673 · Physical Review A

No contextual advantage in nonparadoxical scenarios of the two-state vector formalism

Jaskaran Singh, Rajendra Singh Bhati, Arvind

Published Jan 5, 2023• 2 citations

Abstract

The two-state vector formalism (TSVF) proposed by Aharonov, Bergmann, and Lebowitz [A. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964)] allows a counterfactual assignment of probabilities of outcomes of contemplated (but unperformed) measurements on quantum systems. The probabilities assigned by the Aharonov-Bergmann-Lebowitz (ABL) rule and the associated weak values have been used to provide insights into quantum situations and to unearth underlying quantum contextuality. We apply the principle of exclusivity on ABL probabilities which are assigned to mutually orthogonal projectors to define paradoxical and nonparadoxical scenarios. Given a pre- and a postselected pair of states, we consider the nonparadoxical sector with a view to explore the demonstration of quantum contextuality. For the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenario, we numerically show that the ABL probabilities of the TSVF in the nonparadoxical sector do not offer any contextual advantage. Our approach can be easily generalized to other contextual scenarios as well. We thus argue that several previous proofs of the emergence of contextuality in pre- and postselected scenarios are only possible if the principle of exclusivity is violated and are therefore classified as paradoxical.

DOI: 10.1103/physreva.107.012206

Related papers

Pre- and postselected quantum systems, counterfactual measurements, and consistent histories

Oliver Cohen

We examine some surprising results that have been obtained for pre- and postselected quantum systems. We show that these results depend on a counterfactual interpretation of the rule of Aharonov, Bergmann, and Lebowitz [Phys. Rev. 134, B1410 (1964)] (the ABL rule) and that this interpretation is not valid in general. We then argue, with the help of the consistent histories interpretation of quantum mechanics, that there is a special class of situations where application of the counterfactual interpretation of the ABL rule can be justified. We then reexamine the aforementioned surprising results.

Physical Review A•Jun 1, 1995•45 citations

Resolving the Three-Box Paradox

R. E. Kastner

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.

•Jul 14, 1998•0 citations

Closing a Loophole in the Case Against the Counterfactual Usage of the ABL Rule

Ruth E. Kastner

A currently discussed interpretation of quantum theory, Time-Symmetrized Quantum Theory, makes certain claims about the properties of systems between pre- and post- selection measurements. These claims are based on a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule for calculating the probabilities of measurement outcomes between such measurements. It has been argued by several authors that the counterfactual usage of the ABL rule is, in general, incorrect. This paper examines what might appear to be a loophole in those arguments and shows that this apparent loophole cannot be used to support a counterfactual interpretation of the ABL rule. It is noted that the invalidity of the counterfactual usage of the ABL rule implies that the characterization of those outcomes receiving probability 1 in a counterfactual application of the rule as `elements of reality' is, in general, unfounded.

arXiv:quant-ph/9807015•Jul 5, 1998•1 citation

The Two-State Vector Formalism

Lev Vaidman A

The two-state vector formalism (TSVF), the time-symmetric description of the standard quantum mechanics originated by Aharonov, Bergmann and Lebowitz is reviewed. The TSVF describes a quantum system at a particular time by two quantum states: the usual one, evolving forward in time, defined by the results of a complete measurement at the earlier time, and by the quantum state evolving backward in time, defined by the results of a complete measurement at a later time.

arXiv:0706.1347•Jun 10, 2007•3 citations

Coherence as a resource for $N$-box and quantum pigeonhole paradoxes

Som Kanjilal, Rafael Wagner, Ernesto F. Galvão

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.

quant-pharXiv:2607.24282•Jul 28, 2026•0 citations

The Counterfactual Meaning of the ABL Rule

‎Louis Marchildon

The Aharonov-Bergmann-Lebowitz rule assigns probabilities to quantum measurement results at time t on the condition that the system is prepared in a given way at t_1 < t and found in a given state at t_2 > t. The question whether the rule can also be applied counterfactually to the case where no measurement is performed at the intermediate time t has recently been the subject of controversy. I argue that the counterfactual meaning may be understood in terms of the true value of an observable at t. Such a value cannot be empirically determined for, by stipulation, the measurement that would yield it is not performed. Nevertheless, it may or may not be well-defined depending on one's proposed interpretation of quantum mechanics. Various examples are discussed illustrating what can be asserted at the intermediate time without running into contradictions.

arXiv:quant-ph/0307082•Jul 10, 2003•2 citations

The two-state vector formalism (TSVF) proposed by Aharonov, Bergmann, and Lebowitz [A. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964)] allows a counterfactual assignment of probabilities of outcomes of contemplated (but unperformed) measurements on quantum systems. The probabilities assigned by the Aharonov-Bergmann-Lebowitz (ABL) rule and the associated weak values have been used to provide insights into quantum situations and to unearth underlying quantum contextuality. We apply the principle of exclusivity on ABL probabilities which are assigned to mutually orthogonal projectors to define paradoxical and nonparadoxical scenarios. Given a pre- and a postselected pair of states, we consider the nonparadoxical sector with a view to explore the demonstration of quantum contextuality. For the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenario, we numerically show that the ABL probabilities of the TSVF in the nonparadoxical sector do not offer any contextual advantage. Our approach can be easily generalized to other contextual scenarios as well. We thus argue that several previous proofs of the emergence of contextuality in pre- and postselected scenarios are only possible if the principle of exclusivity is violated and are therefore classified as paradoxical.

We examine some surprising results that have been obtained for pre- and postselected quantum systems. We show that these results depend on a counterfactual interpretation of the rule of Aharonov, Bergmann, and Lebowitz [Phys. Rev. 134, B1410 (1964)] (the ABL rule) and that this interpretation is not valid in general. We then argue, with the help of the consistent histories interpretation of quantum mechanics, that there is a special class of situations where application of the counterfactual interpretation of the ABL rule can be justified. We then reexamine the aforementioned surprising results.

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.

A currently discussed interpretation of quantum theory, Time-Symmetrized Quantum Theory, makes certain claims about the properties of systems between pre- and post- selection measurements. These claims are based on a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule for calculating the probabilities of measurement outcomes between such measurements. It has been argued by several authors that the counterfactual usage of the ABL rule is, in general, incorrect. This paper examines what might appear to be a loophole in those arguments and shows that this apparent loophole cannot be used to support a counterfactual interpretation of the ABL rule. It is noted that the invalidity of the counterfactual usage of the ABL rule implies that the characterization of those outcomes receiving probability 1 in a counterfactual application of the rule as `elements of reality' is, in general, unfounded.

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.

The Aharonov-Bergmann-Lebowitz rule assigns probabilities to quantum measurement results at time t on the condition that the system is prepared in a given way at t_1 < t and found in a given state at t_2 > t. The question whether the rule can also be applied counterfactually to the case where no measurement is performed at the intermediate time t has recently been the subject of controversy. I argue that the counterfactual meaning may be understood in terms of the true value of an observable at t. Such a value cannot be empirically determined for, by stipulation, the measurement that would yield it is not performed. Nevertheless, it may or may not be well-defined depending on one's proposed interpretation of quantum mechanics. Various examples are discussed illustrating what can be asserted at the intermediate time without running into contradictions.