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arXiv:1310.5208 · Physical Review A

Improved error-tradeoff and error-disturbance relations in terms of measurement error components

Xiao-Ming Lu, Sixia Yu, Kazuo Fujikawa, C. H. Oh

Published Oct 20, 2014• 21 citations
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Abstract

Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report various error-disturbance tradeoff relations by decomposing the measurement errors and disturbance into two different components, namely, operator bias and fuzziness. Our uncertainty relations reveal the tradeoffs between these two components of errors, and imply various conditionally valid error-tradeoff relations for the unbiased and projective measurements. We also design a quantum circuit to measure the two components of the error and disturbance.

DOI
10.1103/physreva.90.042113

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Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report various error-disturbance tradeoff relations by decomposing the measurement errors and disturbance into two different components, namely, operator bias and fuzziness. Our uncertainty relations reveal the tradeoffs between these two components of errors, and imply various conditionally valid error-tradeoff relations for the unbiased and projective measurements. We also design a quantum circuit to measure the two components of the error and disturbance.
Heisenberg's uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: Although Heisenberg's first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg's intuition. Even if two incompatible observables cannot be measured together, one can still approximate their joint measurement, at the price of introducing some errors with respect to the ideal measurement of each of them. We present a tight relation characterizing the optimal tradeoff between the error on one observable vs. the error on the other. As a particular case, our approach allows us to characterize the disturbance of an observable induced by the approximate measurement of another one; we also derive a stronger error-disturbance relation for this scenario.
The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.
The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.
The indeterminacy of quantum mechanics was originally presented by Heisenberg through the tradeoff between the measuring error of the observable A and the consequential disturbance to the value of another observable B. This tradeoff now has become a popular interpretation of the uncertainty principle. However, the historic idea has never been exactly formulated previously and is recently called into question. A theory built upon operational and state-relevant definitions of error and disturbance is called for to rigorously reexamine the relationship. Here by putting forward such natural definitions, we demonstrate both theoretically and experimentally that there is no tradeoff if the outcome of measuring B is more uncertain than that of A. Otherwise, the tradeoff will be switched on and well characterized by the Jensen-Shannon divergence. Our results reveal the hidden effect of the uncertain nature possessed by the measured state, and conclude that the state-relevant relation between error and disturbance is not almosteverywhere a tradeoff as people usually believe.
It is shown that the well-defined unbiased measurement or disturbance of a dynamical variable is not maintained for the precise measurement of the conjugate variable, independently of uncertainty relations. The conditionally valid uncertainty relations on the basis of those additional assumptions, which include most of the familiar Heisenberg-type relations, thus become singular for the precise measurement. We clarify some contradicting conclusions in the literature concerning those conditionally valid uncertainty relations: The failure of a naive Heisenberg-type error-disturbance relation and the modified Arthurs-Kelly relation in the recent spin measurement is attributed to this singular behavior. The naive Heisenberg-type error-disturbance relation is formally preserved in quantum estimation theory, which is shown to be based on the strict unbiased measurement and disturbance, but it leads to unbounded disturbance for bounded operators such as spin variables. In contrast, the Heisenberg-type error-error uncertainty relation and the Arthurs-Kelly relation, as conditionally valid uncertainty relations, are consistently maintained.
It is shown that all the known uncertainty relations are the secondary consequences of Robertson's relation. The basic idea is to use the Heisenberg picture so that the time development of quantum mechanical operators incorporate the effects of the measurement interaction. A suitable use of triangle inequalities then gives rise to various forms of uncertainty relations. The assumptions of unbiased measurement and unbiased disturbance are important to simplify the resulting uncertainty relations and to give the familiar uncertainty relations, such as a naive Heisenberg error-disturbance relation. These simplified uncertainty relations are however, valid only conditionally. Quite independently of uncertainty relations, it is shown that the notion of precise measurement is incompatible with the assumptions of unbiased measurement and unbiased disturbance. We can thus naturally understand the failure of the naive Heisenberg's error-disturbance relation, as was demonstrated by the recent spin measurement by Erhart et al.