Weak Measurement and (Bohmian) Conditional Wave Functions

TL;DR

This paper shows that weak measurement techniques can operationally define the wave function of a subsystem in quantum mechanics, aligning with Bohmian mechanics' concept of the conditional wave function, and discusses experimental implications.

Contribution

It establishes a connection between weak measurements and Bohmian conditional wave functions, providing an operational approach to defining subsystem wave functions in quantum mechanics.

Findings

Weak measurement can directly measure the conditional wave function.

The method can be extended to measure the conditional density matrix.

Experimental setups are proposed to demonstrate non-local dependencies.

Abstract

It was recently pointed out (and demonstrated experimentally) by Lundeen et al. that the wave function of a particle (more precisely, the wave function possessed by each member of an ensemble of identically-prepared particles) can be "directly measured" using weak measurement. Here it is shown that if this same technique is applied, with appropriate post-selection, to one particle from a (perhaps entangled) multi-particle system, the result is precisely the so-called "conditional wave function" of Bohmian mechanics. Thus, a plausibly operationalist method for defining the wave function of a quantum mechanical sub-system corresponds to the natural definition of a sub-system wave function which Bohmian mechanics (uniquely) makes possible. Similarly, a weak-measurement-based procedure for directly measuring a sub-system's density matrix should yield, under appropriate circumstances, the Bohmian "conditional density matrix" as opposed to the standard reduced density matrix. Experimental arrangements to demonstrate this behavior -- and also thereby reveal the non-local dependence of sub-system state functions on distant interventions -- are suggested and discussed.