Quantum weak values are not unique; what do they actually measure?

Abstract

Precise definitions of "weak [quantum] measurements" and "weak value" [of a quantum observable] are offered, which seem to capture the meaning of the often vague ways that these terms are used in the literature. Simple finite dimensional examples are given showing that weak values of an observable are not unique, and in fact arbitrary weak values can be obtained by appropriate weak measurements. This implies that a "weak value" of an observable A, *by itself*, can furnish no unambiguous information about A; any information in a weak value is inextricably connected with the particular measurement procedure used to obtain that weak value. Moreover, arbitrary weak values can be obtained using a "meter space" of dimension as small as 2. A "Remarks" section questions the utility of "weak measurement".

Version 3 is mathematically identical to Version 2, but incorporates a referee's suggestions. A potentially ambiguous passage has been rewritten and an important overlapping reference added, which is discussed in a new Afterword section

References in corpus (2)

• Complex weak values in quantum measurement
• What do quantum "weak" measurements actually measure?

Cited by in corpus (8)

• Weak value controversy
• Sufficient conditions for uniqueness of the weak value
• A strange weak value in spontaneous pair productions via a supercritical step potential
• Counterexample to "Sufficient Conditions for uniqueness of the Weak Value" by J. Dressel and A. N. Jordan, arXiv:1106.1871v1
• "Contextual weak values" of quantum measurements with positive measurement operators are not limited to the traditional weak value
• Unusual Interaction of a Pre-and-Post-Selected Particle
• Proof gap in "Sufficient conditions for uniqueness of the Weak Value" by J. Dressel and A. N. Jordan, J. Phys. A 45 (2012) 015304
• Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling