Epistemically restricted phase-space representation, weak momentum value, and reconstruction of the quantum wave function

Abstract

A phase space distribution associated with a quantum state was previously proposed, which incorporates a specific epistemic restriction parameterized by a global random variable on the order of Planck constant, transparently manifesting quantum uncertainty in phase space. Here we show that the epistemically restricted phase space (ERPS) distribution can be determined via weak measurement of momentum followed by post-selection on position. In the ERPS representation, the phase and amplitude of the wave function are neatly captured respectively by the positiondependent (conditional) average and variance of the epistemically restricted momentum fluctuation.They are in turn respectively determined by the real and imaginary parts of the weak momentum value, permitting a reconstruction of wave function using weak momentum value measurement, and an interpretation of momentum weak value in term of epistemically restricted momentum fluctuations. The ERPS representation thus provides a transparent and rich framework to study the deep conceptual links between quantum uncertainty embodied in epistemic restriction, quantum wave function, and weak momentum measurement with position post-selection, which may offer useful insight to better understand their meaning.

Efficient classical computation of expectation values in a class of quantum circuits with an epistemically restricted phase space representation

“…We summarise the phase space representation of quantum mechanics based on the ER ensemble of trajectories proposed in Refs. 40 , 41 . Consider a system of N spatial degrees of freedom , arranged as N -elements vector in where the superscript T denotes transposition, with the corresponding canonical conjugate momentum .…”

“…We have argued in Refs. 40 , 41 that the abstract formulas of quantum mechanics can be expressed as a specific modification of the above classical statistical mechanics of ensemble of trajectories in phase space. First, we introduce an “ontic extension” in the form of a global-nonseparable variable : it is real-valued with the dimension of action and depends only on time (i.e., spatially uniform).…”

“…We can however read Theorem 1 the other way around. Namely, the ER classical model can be seen as a ‘representation’ of quantum statistics in the ER classical phase space 41 . In this reading, first we are given a quantum wave function describing the preparation.…”

“…First, we emphasize that the epistemic restriction in classical phase space directly reflects the quantum uncertainty relation 40 , 41 , and the ER phase space distribution defined in Eq. ( 7 ) is always non-negative.…”

“…To this end, it has been shown that a significant large set of phenomena traditionally seen as specifically quantum could in fact be explained within classical statistical models with some kinds of “epistemic (statistical) restrictions” 36 – 39 . Partly inspired by these remarkable results, recently, we have developed a novel phase space representation of quantum mechanics, showing that its opaque formalism in the complex Hilbert space can be phrased more intuitively as a classical statistical mechanics of ensemble of trajectories subjected to a specific epistemic restriction parameterized by a global-nonseparable variable fluctuating randomly on the order of Planck constant 40 , 41 . The epistemic restriction manifests directly the quantum uncertainty relation between noncommuting (quantum) positon and momentum operators acting on the abstract Hilbert space into a more intuitive statistical constraint on the allowed distributions over the (classical) phase space.…”