Weak values in a classical theory with an epistemic restriction

Abstract

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 the classical theory. By using this analogous classical 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 minimised and the limits of weak measurement.

9 pages, 2 figures, comments welcome; v2 added some references; v3 published version

References in corpus (14)

• Complex weak values in quantum measurement
• Revisiting Hardy's Paradox: Counterfactual Statements, Real Measurements, Entanglement and Weak Values
• Anomalous Weak Values Are Proofs of Contextuality
• Weak value amplification is suboptimal for estimation and detection
• Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction
• How the result of a single coin toss can turn out to be 100 heads
• The significance of the imaginary part of the weak value
• Weak Values are Interference Phenomena
• Quantum Averages of Weak Values
• Strange Weak Values
• Optimal Probe Wavefunction of Weak-Value Amplification
• Quantum sensors based on weak-value amplification cannot overcome decoherence
• On the role of complex phases in the quantum statistics of weak measurements
• Comment on: How the result of a single coin toss can turn out to be 100 heads

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• Interpreting weak value amplification with a toy realist model
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