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arXiv:1202.5604

Proof gap in "Sufficient conditions for uniqueness of the Weak Value" by J. Dressel and A. N. Jordan, J. Phys. A 45 (2012) 015304

Stephen Parrott

Published Feb 25, 2012• 2 citations

Abstract

The article of the title attempts to prove a "General theorem" (GT) giving sufficient conditions under which a previously introduced "general conditioned average" "converges uniquely to the quantum weak value in the minimal disturbance limit." The "general conditioned average" is obtained from a positive operator valued measure (POVM) depending on a small "weakness" parameter g. We point out that unstated assumptions in the presentation of the "sufficient conditions" make them appear much more general than they actually are. Indeed, the stated "sufficient conditions" strengthened by these unstated assumptions seem very close to an assumption that the POVM operators be linear polynomials in g. Moreover, there appears to be a critical error or gap in the attempted proof, even assuming a linear POVM. A counterexample to the proof of the GT (though not to its conclusion) is given. Nevertheless, I conjecture that the conclusion is actually true for linear POVM's whose contextual values are chosen by the article's "pseudoinverse prescription".

DOI: 10.48550/arxiv.1202.5604

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The article of the title attempts to prove a "General theorem" (GT) giving sufficient conditions under which a previously introduced "general conditioned average" "converges uniquely to the quantum weak value in the minimal disturbance limit." The "general conditioned average" is obtained from a positive operator valued measure (POVM) depending on a small "weakness" parameter g. We point out that unstated assumptions in the presentation of the "sufficient conditions" make them appear much more general than they actually are. Indeed, the stated "sufficient conditions" strengthened by these unstated assumptions seem very close to an assumption that the POVM operators be linear polynomials in g. Moreover, there appears to be a critical error or gap in the attempted proof, even assuming a linear POVM. A counterexample to the proof of the GT (though not to its conclusion) is given. Nevertheless, I conjecture that the conclusion is actually true for linear POVM's whose contextual values are chosen by the article's "pseudoinverse prescription".

The abstract of "Contextual Values of Observables in Quantum Measurements" by J. Dressel, S. Agarwal, and A. N. Jordan [Phys. Rev. Lett. 104 240401 (2010)] (called DAJ below), states: "We introduce contextual values as a generalization of the eigenvalues of an observable that takes into account both the system observable and a general measurement procedure. This technique leads to a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit." A counterexample to the claim of the last sentence was presented in Version 1. Subsequently Dressel and Jordan placed in the arXiv the paper of the title (called DJ below) which attempts to prove the claim of DAJ quoted above under stronger hypotheses than given in DAJ, hypotheses which the counterexample does not satisfy. The present work (Version 6) presents a new counterexample to this revised claim of DJ. A brief introduction to "contextual values" is included. Also included is a critical analysis of DJ.

A recent Letter in Physical Review Letters, "Contextual Values of Observables in Quantum Measurements", by J. Dressel, S. Agarwal, and A. N. Jordan (abbreviated DAJ below), introduces the concept of "contextual values" and claims that they lead to "a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit". However, they do not define "minimal disturbance limit". The present paper is in part the saga of my search for a definition of "minimal disturbance limit" under which this claim could be proved. The search finally ended in what is probably a definitive counterexample to the claim.

We review and clarify the sufficient conditions for uniquely defining the generalized weak value as the weak limit of a conditioned average using the contextual values formalism introduced in Dressel, Agarwal and Jordan (2010 Phys. Rev. Lett. 104 240401 ). We also respond to criticism of our work by Parrott (arXiv:1105.4188v1) concerning a proposed counter-example to the uniqueness of the definition of the generalized weak value. The counter-example does not satisfy our prescription in the case of an underspecified measurement context. We show that when the contextual values formalism is properly applied to this example, a natural interpretation of the measurement emerges and the unique definition in the weak limit holds. We also prove a theorem regarding the uniqueness of the definition under our sufficient conditions for the general case. Finally, a second proposed counter-example by Parrott (arXiv:1105.4188v6) is shown not to satisfy the sufficiency conditions for the provided theorem.

We introduce contextual values as a generalization of the eigenvalues of an observable that takes into account both the system observable and a general measurement procedure. This technique leads to a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit. As such, we address the controversy in the literature regarding the theoretical consistency of the quantum weak value by providing a more general theoretical framework and giving several examples of how that framework relates to existing experimental and theoretical results.

We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM, a sufficient statistic called a Lehmann-Scheffé-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.

We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM, a sufficient statistic called a Lehmann-Scheffé-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.