The article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is stimulating and raises some interesting and profound issues regarding the foundations of quantum mechanics and quantum measurements. One point the authors make is that a measurement of spin component √2‾/2 of a spin-1⁄2 particle is unphysical and can be attributed only to errors in weak measurements performed on a collection of N spins. I offer a more mundane interpretation that does not require the introduction of any error or postselection concepts. The physical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 with some distribution of orientations (that is, polar coordinates θ and φ). A conventional single measurement of Sz can yield only one of the eigenvalues ±1⁄2. This may be interpreted in the classical vector model by envisioning that the spin lies on a cone with θ defined by cosθ = √3‾/3. The expectation value of Sx then vanishes due to the uniform distribution of φ. Having some control over φ can yield a finite value of Sx . Thus I would argue that only an expectation value greater than √3‾/2 is unphysical; a value of √2‾/2 is quite physical and can even be interpreted classically in terms of some distribution of φ and θ.
I would phrase the important observation of Aharonov and coauthors in a different way. While spin-1⁄2 has a magnitude of √3‾/2, a fundamental limitation of conventional single-point quantum measurements is that they can yield only the values +1⁄2 and −1⁄2 and any expectation value must therefore lie between those two extremes. Classically, therefore, the spin may not be fully aligned along the z-axis with θ = 0. Such alignment is impossible since it would yield well-defined values of the three noncommuting variables Sz = √3‾/2 and Sx = Sy = 0. However, once multiple time measurements are performed, one can define a reduced distribution of some of the measurements that is conditional on the outcome in the other measurements; such a distribution can be interpreted in terms of Sz greater than 1⁄2 but less than or equal to √3‾/2. That interpretation does not violate any of quantum mechanics’ fundamental rules, which do not usually consider such quantities.
Rather than a new, time-symmetric formulation of quantum mechanics that involves pre- and postselection, one can simply treat the scheme of the authors’ figure 1b as a three-point correlation function, whereas figures 2a and 2b show a four-point correlation function, each panel having its own set of conditional probabilities. Error, time reversal, and postselection need not be invoked. Instead, one may think in multiple dimensions and develop the right language for the interpretation of the observables.
The combination of pre- and postselection is an attempt to create an artificial ensemble that reproduces the results of multiple-point measurements in a single one-point measurement. As shown by Aharonov and coauthors, that is possible by introducing errors. An alternative physical picture is obtained by retaining the multipoint analysis. Multidimensional thinking is well developed in coherent nonlinear spectroscopy, where a system of spins or optical chromophores are subjected to sequences of short impulsive pulses.1 , 2 Similar ideas may be applied for the interpretation of multiple measurements. One can think of various types of n-point observables obtained by combining n − m perturbations and m measurements. The nonlinear n-point response functions in spectroscopy represent n − 1 impulsive perturbations followed by a single measurement. The objects Aharonov and colleagues considered correspond to n = m. Proper multiple-distribution functions could then be naturally used for the interpretation of such generalized measurements.