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.”
