Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physics Today

A time-symmetric formulation of quantum mechanics

Yakir Aharonov, Sandu Popescu, Jeff Tollaksen

Published Nov 1, 2010• 222 citations
Publisher

Abstract

Quantum mechanics allows one to independently select both the initial and final states of a single system. Such pre- and postselection reveals novel effects that challenge our ideas about what time is and how it flows.
DOI
10.1063/1.3518209

Related papers

Pre and post-selection, weak measurements and the flow of time in quantum mechanics

Jeff Tollaksen

AIP conference proceedings•Jan 1, 2011•1 citation

Pre and post-selection, weak measurements and the flow of time in quantum mechanics

Jeff Tollaksen, Andrei Yu. Khrennikov, Gregg S. Jaeger +2 more

AIP conference proceedings•Jan 1, 2011•0 citations

Pre- and Post-Selected Ensembles and Time-Symmetry in Quantum Mechanics

D. J. Miller

arXiv:quant-ph/0702231•International Journal of Theoretical Physics•Oct 22, 2008•0 citations

Weak Values and Quantum Nonlocality

Yakir Aharonov, Eliahu Cohen

Entanglement and nonlocality are studied in the framework of pre-/postselected ensembles with the aid of weak measurements and the two-state-vector formalism. In addition to the EPR–Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose entangled pre- or postselected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting “emerging correlations” between independent preand postselected ensembles of particles. This can be viewed as a quantum violation of the classical “pigeonhole principle.” Introduction It is seldom acknowledged that seven years before the celebrated Bell paper [1], Bohm and Aharonov [2] published an analysis of the EPR paradox [3]. They suggested an experimental setup, based on Compton scattering, for testing nonlocal correlations between the polarizations of two annihilation photons. In 1964, Bell proposed his general inequality, thereby excluding local realism. During the same time, Aharonov et al. constructed the foundations of a time-symmetric formalism of quantum mechanics [4]. While Bell's proof utilizes entanglement to demonstrate nonlocal correlations,wewill describe inwhat follows the emergence of nonlocal correlations between product states. For this purpose, however, we shall invoke weak measurements of pre- and postselected ensembles. In classical mechanics, initial conditions of position and velocity for every particle fully determine the time evolution of the system. Therefore, trying to impose an additional final condition would lead either to redundancy or to inconsistency with the initial conditions. This is radically different in quantum mechanics. Because of the uncertainty principle, an initial state vector does not fully determine, in general, the outcome of a future measurement. However, adding a final (backward-evolving) state vector results in a more complete description of the quantum system in between these two boundary conditions, which has a bearing on the determination of measurement outcomes. The basis for this time-symmetric formulation of quantum mechanics was laid down by Aharonov, Bergman, and Lebowitz (ABL), who derived a symmetric probability rule concerning measurements performed on systems, while taking into account the final state of the system, in addition to the usual initial state [4].

arXiv:1504.03797•Cambridge University Press eBooks•Aug 31, 2016•11 citations

A Time-Symmetric Variational Reformulation of Nonrelativistic Quantum Mechanics

Lance H. Carter

Standard quantum mechanics relies on two distinct dynamical principles: unitary evolution and collapse. A mathematically self-contained variational framework is presented that replaces this dualism with a single principle, in which nonrelativistic Schrödinger dynamics are not postulated but emerge as an admissible optimality condition of a primal-dual boundary-value problem. By expressing the state in terms of hydrodynamic variables $(ρ,\mathbf{j})$ subject to a continuity constraint, it is shown that Fisher-information regularization yields the linear Schrödinger equation within the admissible single-valued variational class. Rather than evolving an initial state forward in time, the dynamics arise from minimizing a global action that connects the initial and final boundary constraints, with the selected solution corresponding to a specific hydrodynamic flow within an ensemble of admissible histories. A von Neumann pointer model illustrates how Born-rule statistics for recorded outcomes arise without introducing a separate collapse law. Within this formulation, quantum uncertainty is interpreted as effective randomness over boundary-compatible histories rather than as a fundamental stochastic postulate. The resulting framework provides a nonrelativistic proof of concept for how a single time-symmetric variational reformulation can recover key features of quantum theory.

quant-pharXiv:2512.22320•Dec 29, 2025•0 citations

Time-symmetric quantum mechanics questioned and defended

Michael Nauenberg

In their feature article "A Time-Symmetric Formulation of Quantum Mechanics" (PHYSICS TODAY, November 2010, page 27), Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen state that there exists a "freedom to impose independent initial and final conditions on the evolution of a quantum system" without having to modify quantum mechanics "by an iota." The supporting illustrations they give, however, are based on an inadequate analysis of the measurement process in quantum mechanics.Consider their gedanken experiment in which measurements are made at two successive times, t and t 1, after the system has been prepared in a state Ψ at t 0 < t < t 1. Now suppose that the experiment is repeated, but without any measurements made at t 1. Then the standard statistical prediction of quantum mechanics for the outcome at the intermediate time t is identical in both experiments, contradicting the authors' claim that "the results at [the intermediary time] t depend not only on what happened earlier at t 0, but also on what happens later at t 1."To illustrate their arguments, the authors describe some measurements of polarization with spin-1⁄2 particles as follows: "We could, for example, start at t 0 with an ensemble of spin-1⁄2 particles, each one polarized 'up' in the z-direction. Then at t 1 we measure each spin in the x-direction and select only the particles for which the spin turned out to be up again, but in the new direction. Thus, at any intermediate time t, the spin components in both the z and the x directions—two noncommuting observables—would seem to be completely determined." The rationale given for that strange conclusion is that "if at t we measure the spin along x, we must also find it up [along x], because otherwise the measurement at t 1 wouldn't find it up [along x]." But that claim is incorrect. Selecting, after a measurement, a subset of particles with spin up along a given axis does not imply that before the measurement such particles had spin up along that axis. On the contrary, if some particles at t 1 also emerge with spin down along x, then, according to quantum mechanics, the state Ψ at t < t 1 does not represent particles polarized along the x-direction. To find that state, the axis of the measurement device—for example, a Stern–Gerlach magnet—must be rotated until all particles emerge with the same direction of polarization. For the example under consideration, that would be the direction of the z-axis, which corresponds to the polarization at t 0 and t. Hence, a measurement at any later time t 1 is not lost information; instead, it is redundant information about the outcome of a measurement at an intermediate time t when at the initial time t 0 the particles are in a state Ψ.The claim by Aharonov and coauthors that at various stages of the measurement process ensembles can be separated into subensembles that can be associated with quantum states leads to contradictions with the principles of quantum mechanics, and gives rise to the paradoxes of "impossible ensembles" discussed in the article. Their unphysical description of the measurement process leads them to the false conclusion that "quantum mechanics offers a place to specify both an initial and an independent final state," and to such outlandish statements as the idea "that quantum mechanics lets one impose . . . a putative final state of the universe."© 2011 American Institute of Physics.

Physics Today•May 1, 2011•10 citations
Entanglement and nonlocality are studied in the framework of pre-/postselected ensembles with the aid of weak measurements and the two-state-vector formalism. In addition to the EPR–Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose entangled pre- or postselected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting “emerging correlations” between independent preand postselected ensembles of particles. This can be viewed as a quantum violation of the classical “pigeonhole principle.” Introduction It is seldom acknowledged that seven years before the celebrated Bell paper [1], Bohm and Aharonov [2] published an analysis of the EPR paradox [3]. They suggested an experimental setup, based on Compton scattering, for testing nonlocal correlations between the polarizations of two annihilation photons. In 1964, Bell proposed his general inequality, thereby excluding local realism. During the same time, Aharonov et al. constructed the foundations of a time-symmetric formalism of quantum mechanics [4]. While Bell's proof utilizes entanglement to demonstrate nonlocal correlations,wewill describe inwhat follows the emergence of nonlocal correlations between product states. For this purpose, however, we shall invoke weak measurements of pre- and postselected ensembles. In classical mechanics, initial conditions of position and velocity for every particle fully determine the time evolution of the system. Therefore, trying to impose an additional final condition would lead either to redundancy or to inconsistency with the initial conditions. This is radically different in quantum mechanics. Because of the uncertainty principle, an initial state vector does not fully determine, in general, the outcome of a future measurement. However, adding a final (backward-evolving) state vector results in a more complete description of the quantum system in between these two boundary conditions, which has a bearing on the determination of measurement outcomes. The basis for this time-symmetric formulation of quantum mechanics was laid down by Aharonov, Bergman, and Lebowitz (ABL), who derived a symmetric probability rule concerning measurements performed on systems, while taking into account the final state of the system, in addition to the usual initial state [4].
Standard quantum mechanics relies on two distinct dynamical principles: unitary evolution and collapse. A mathematically self-contained variational framework is presented that replaces this dualism with a single principle, in which nonrelativistic Schrödinger dynamics are not postulated but emerge as an admissible optimality condition of a primal-dual boundary-value problem. By expressing the state in terms of hydrodynamic variables (ρ,j)(ρ,\mathbf{j})(ρ,j) subject to a continuity constraint, it is shown that Fisher-information regularization yields the linear Schrödinger equation within the admissible single-valued variational class. Rather than evolving an initial state forward in time, the dynamics arise from minimizing a global action that connects the initial and final boundary constraints, with the selected solution corresponding to a specific hydrodynamic flow within an ensemble of admissible histories. A von Neumann pointer model illustrates how Born-rule statistics for recorded outcomes arise without introducing a separate collapse law. Within this formulation, quantum uncertainty is interpreted as effective randomness over boundary-compatible histories rather than as a fundamental stochastic postulate. The resulting framework provides a nonrelativistic proof of concept for how a single time-symmetric variational reformulation can recover key features of quantum theory.
In their feature article "A Time-Symmetric Formulation of Quantum Mechanics" (PHYSICS TODAY, November 2010, page 27), Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen state that there exists a "freedom to impose independent initial and final conditions on the evolution of a quantum system" without having to modify quantum mechanics "by an iota." The supporting illustrations they give, however, are based on an inadequate analysis of the measurement process in quantum mechanics.Consider their gedanken experiment in which measurements are made at two successive times, t and t 1, after the system has been prepared in a state Ψ at t 0 < t < t 1. Now suppose that the experiment is repeated, but without any measurements made at t 1. Then the standard statistical prediction of quantum mechanics for the outcome at the intermediate time t is identical in both experiments, contradicting the authors' claim that "the results at [the intermediary time] t depend not only on what happened earlier at t 0, but also on what happens later at t 1."To illustrate their arguments, the authors describe some measurements of polarization with spin-1⁄2 particles as follows: "We could, for example, start at t 0 with an ensemble of spin-1⁄2 particles, each one polarized 'up' in the z-direction. Then at t 1 we measure each spin in the x-direction and select only the particles for which the spin turned out to be up again, but in the new direction. Thus, at any intermediate time t, the spin components in both the z and the x directions—two noncommuting observables—would seem to be completely determined." The rationale given for that strange conclusion is that "if at t we measure the spin along x, we must also find it up [along x], because otherwise the measurement at t 1 wouldn't find it up [along x]." But that claim is incorrect. Selecting, after a measurement, a subset of particles with spin up along a given axis does not imply that before the measurement such particles had spin up along that axis. On the contrary, if some particles at t 1 also emerge with spin down along x, then, according to quantum mechanics, the state Ψ at t < t 1 does not represent particles polarized along the x-direction. To find that state, the axis of the measurement device—for example, a Stern–Gerlach magnet—must be rotated until all particles emerge with the same direction of polarization. For the example under consideration, that would be the direction of the z-axis, which corresponds to the polarization at t 0 and t. Hence, a measurement at any later time t 1 is not lost information; instead, it is redundant information about the outcome of a measurement at an intermediate time t when at the initial time t 0 the particles are in a state Ψ.The claim by Aharonov and coauthors that at various stages of the measurement process ensembles can be separated into subensembles that can be associated with quantum states leads to contradictions with the principles of quantum mechanics, and gives rise to the paradoxes of "impossible ensembles" discussed in the article. Their unphysical description of the measurement process leads them to the false conclusion that "quantum mechanics offers a place to specify both an initial and an independent final state," and to such outlandish statements as the idea "that quantum mechanics lets one impose . . . a putative final state of the universe."© 2011 American Institute of Physics.