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arXiv:1505.05014

Comment on "Proof of Heisenberg's error-disturbance principle"

Masanao Ozawa

Published May 19, 2015• 0 citations
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Abstract

Recently, Kosugi [arXiv:1504.03779v2 [quant-ph]] argued that Heisenberg's error-disturbance relation (EDR) must be interpreted as being between the resolution, the preparational error for the post-measurement observable, and the disturbance. He further claimed that Heisenberg's EDR can be proven to hold true in general, when the meter observable is modified as one of its functions. Here, some comments are given to suggest that the above claims are not supported.

DOI
10.48550/arxiv.1505.05014

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Recently, Kosugi [arXiv:1504.03779v2 [quant-ph]] argued that Heisenberg's error-disturbance relation (EDR) must be interpreted as being between the resolution, the preparational error for the post-measurement observable, and the disturbance. He further claimed that Heisenberg's EDR can be proven to hold true in general, when the meter observable is modified as one of its functions. Here, some comments are given to suggest that the above claims are not supported.
In Heisenberg's error-disturbance relation for electron position measurement, the measurement error must be the one that determines the uncertainty in the electron position just after the measurement. It is the resolution ε(xt)ε(x_t)ε(xt​) i.e., the measurement error of the post-measurement observable x^t\hat{x}_tx^t​, not the precision ε(x0)ε(x_0)ε(x0​) i.e., that of the pre-measurement observable x^0\hat{x}_0x^0​ that determines the uncertainty of the observable x^t\hat{x}_tx^t​. Therefore, Heisenberg's relation must be interpreted as one between the resolution ε(xt)ε(x_t)ε(xt​) and the disturbance η(y0)η(y_0)η(y0​). The magnitude of the disturbance η(y0)η(y_0)η(y0​) is independent of the definition of the measurement value of the observable x^t\hat{x}_tx^t​. Heisenberg's error-disturbance relation is proven to hold true in general, when the measurement value of x^t\hat{x}_tx^t​ is defined in order to minimize the resolution ε(xt)ε(x_t)ε(xt​).
Heisenberg originally envisioned the error-disturbance relation under the premise that the state after measurement must remain consistent with the Kennard-Robertson uncertainty relation. This implies that the measurement error must be formulated via a posterior observable x^t\hat{x}_tx^t​, rather than a prior observable x^0\hat{x}_0x^0​, because the posterior measurement error determines the post-measurement uncertainty of the electron. Building upon the preliminary conceptual foundation reported in arXiv:1504.03779, this paper presents a rigorous derivation of the error-disturbance principle formulated from the unitary transformation equations of observables. Each readout XXX of a posterior probe observable X^t\hat{X}_tX^t​ completes a single measurement event, producing a specific conditional object state. The Kennard-Robertson uncertainty relation must hold for these states. Specifically, we show that the uncertainty relation between the error εX(x^t)ε_{X}(\hat{x}_t)εX​(x^t​) and the disturbance ηX(p^0)η_{X}(\hat{p}_0)ηX​(p^​0​) holds strictly at the level of individual measurements, characterized by the specific readout XXX. We also verify that conventional error-disturbance relations, where the errors are evaluated by averaging over the unconditioned state, hold true. Our results provide a refined theoretical basis for understanding the fundamental trade-off in individual measurement outcomes.
Abstract Although Heisenberg's uncertainty principle is represented by a rigorously proven relation about intrinsic indeterminacy in quantum states, Heisenberg's error-disturbance relation (EDR) has been commonly believed as another aspect of the principle. However, recent developments of quantum measurement theory made Heisenberg's EDR testable to observe its violations. Here, we study the EDR for Stern-Gerlach measurements. In a previous report, it has been pointed out that their EDR is close to the theoretical optimal. The present letter reports that even the original Stern-Gerlach experiment in 1922, as the available experimental data show, violates Heisenberg's EDR. The results suggest that Heisenberg's EDR is more ubiquitously violated than it has long been supposed.
In a recent publication [PRL 111, 160405 (2013)] we proved a version of Heisenberg's error-disturbance tradeoff. This result was in apparent contradiction to claims by Ozawa of having refuted these ideas of Heisenberg. In a direct reaction [arXiv:1308.3540] Ozawa has called our work groundless, and has claimed to have found both a counterexample and an error in our proof. Here we answer to these allegations. We also comment on the submission [arXiv:1307.3604] by Rozema et al, in which our approach is unfavourably compared to that of Ozawa.
Recently, Busch, Lahti, and Werner (arXiv:1306.1565v1 [quant-ph]) claimed that Heisenberg's error-disturbance relation can be proved in its original form with new formulations of error and disturbance, in contrast to the theory proposed by the present author and confirmed by recent experiments. Despite their claim, it is shown here that a class of solvable models of position measurement with explicit interaction Hamiltonians escape the Busch-Lahti-Werner relation. It is also made clear where their proof fails. Those models have unambiguously defined zero root-mean-square error and finite root-mean-square disturbance in every input state and are naturally considered to violate Heisenberg's error-disturbance relation in any conceivable formulation.
In this comment on the paper by F. Kaneda, S.-Y. Baek, M. Ozawa and K. Edamatsu [Phys. Rev. Lett. 112, 020402, 2014, arXiv:1308.5868], we point out that the claim of having refuted Heisenberg's error-disturbance relation is unfounded since it is based on the choice of unsuitable and operationally problematical quantifications of measurement error and disturbance. As we have shown elsewhere [PRL 111, 160405, 2013], for appropriate choices of operational measures of error and disturbance, Heisenberg's heuristic relation can be turned into a precise inequality which is a rigorous consequence of quantum mechanics.