What quantum measurements measure

A solution to the second measurement problem, determining what prior microscopic properties can be inferred from measurement outcomes (“pointer positions”), is worked out for projective and generalized (POVM) measurements, using consistent histories. The result supports the idea that equipment properly designed and calibrated reveals the properties it was designed to measure. Applications include Einstein's hemisphere and Wheeler's delayed choice paradoxes, and a method for analyzing weak measurements without recourse to weak values. Quantum measurements are noncontextual in the original sense employed by Bell and Mermin: if $[A,B]=[A,C]=0,[B,C]\ne 0$, the outcome of an $A$ measurement does not depend on whether it is measured with $B$ or with $C$. An application to Bohm's model of the Einstein-Podolsky-Rosen situation suggests that a faulty understanding of quantum measurements is at the root of this paradox.

Figure 1

Einstein paradox. (a) Spherical wave. (b) Particle moving on straight line through collimator. (c) Quantum wave packet passing through collimator.

Figure 2

(a) Collimator with two holes. (b) Fluorescent screen a large distance to the right of the collimator. Due to constructive interference of waves coming from the two holes a particle can sometimes be observed in a region which is classically forbidden.

Figure 3

Mach-Zehnder interferometer (a) with a source $S_1$, two beam splitters $B{S}_1$ and $B{S}_2$, and detectors $D^{+}$ and $D^{-}$; (b) with the second beam splitter removed.

Figure 4

Mach-Zehnder interferometer with two inputs (a) arranged to determine relative phase between the two arms; (b) arranged to measure which path (which arm).

Figure 5

Apparatus to measure $A$ along with $B$ ($U_{\beta }$), or with $C$ ($U_{\gamma }$).