Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.
Consider a statistical model with an epistemic restriction such that, unlike in classical mechanics, the allowed distribution of positions is fundamentally restricted by the form of an underlying momentum field. Assume an agent (observer) who wishes to estimate the momentum field given information on the conjugate positions. We discuss a classically consistent, weakly unbiased, best estimation scheme for the momentum field (by minimizing the mean-squared error) based on which the abstract mathematical rules of quantum mechanics can be derived. The results suggest that a quantum wave function is not an objective agent-independent attribute of reality but represents the agent's best estimations of the momentum, given the positions, under epistemic restriction. Quantum uncertainty and complementarity between momentum and position find their epistemic origin from the trade-off between the mean-squared errors of simultaneous estimations of momentum field and mean position, with the Gaussian wave function representing the simultaneous efficient estimations, achieving the Cram\'er-Rao bounds of the associated mean-squared errors. We then argue that unitary time evolution and wave-function collapse in measurement are normative rules for an agent to update her estimations given information on the experimental settings.
What does it take for real-deterministic
c-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic
c-valued variables out of the weak values obtained via a nonperturbing weak measurement of quantum operators with a postselection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to convert the incompatibility between the quantum operator and the basis into the fluctuation of an ``error term,'' and then superimpose it onto the real part of the weak value. We show that this class of ``
c-valued physical quantities'' provides a real-deterministic contextual hidden-variable model for the quantum expectation value of a certain class of operators. We then show that the Schr\"odinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the
c-valued physical quantities associated with a pair of Hermitian operators. Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two
c-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty principle is captured by a specific irreducible epistemic restriction between the two
c-valued physical quantities, foreign in classical mechanics, constraining the allowed form of their joint distribution. We then suggest an epistemic interpretation of the two terms decomposing the
c-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.
The wave function is the cornerstone of quantum mechanics, although it is still open to different interpretations. It has been shown that the amplitude and the phase of a wave function are manifested in the imaginary and real parts of the momentum weak value, respectively. Based on this, we develop a weak measurement scheme to reconstruct the wave function through measuring the momentum weak value. A proof-of-principle experiment is made to confirm the feasibility of our measurement scheme, in which the polarization states and the momentum of light are coupled. We believe the experimental protocol has the potential to be applied in diverse fields such as wave-front sensing, edge detection, and optical communication, and measurement of the momentum weak value may offer valuable insight to better understand quantum mechanics.
How would the world appear to us if its ontology was that of classical mechanics but every agent faced a restriction on how much they could come to know about the classical state? We show that in most respects it would appear to us as quantum. The statistical theory of classical mechanics, which specifies how probability distributions over phase space evolve under Hamiltonian evolution and under measurements, is typically called Liouville mechanics, so the theory we explore here is Liouville mechanics with an epistemic restriction. The particular epistemic restriction we posit as our foundational postulate specifies two constraints. The first constraint is a classical analog of Heisenberg's uncertainty principle; the second-order moments of position and momentum defined by the phase-space distribution that characterizes an agent's knowledge are required to satisfy the same constraints as are satisfied by the moments of position and momentum observables for a quantum state. The second constraint is that the distribution should have maximal entropy for the given moments. Starting from this postulate, we derive the allowed preparations, measurements, and transformations and demonstrate that they are isomorphic to those allowed in Gaussian quantum mechanics and generate the same experimental statistics. We argue that this reconstruction of Gaussian quantum mechanics constitutes additional evidence in favor of a research program wherein quantum states are interpreted as states of incomplete knowledge and that the phenomena that do not arise in Gaussian quantum mechanics provide the best clues for how one might reconstruct the full quantum theory.
Abstract It is demonstrated with a particle in the one-dimensional configuration space (OCS) that knowledge of the wave function supposes more than statistical restrictions on the results of measurements. In particular, apart from the probability (density) field in the OCS, the wave function also suggests the existence of two fields that predict two (equiprobable) particle momentum values for each OCS point, and the average of these two momenta at each point is related only to the wave function phase, while their difference (coinciding with the Bohm quantum-mechanical potential) is related only to the wave function amplitude. An analogue of the Heisenberg uncertainty relation is obtained for both fields.