“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.1