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arXiv:quant-ph/9803046

The Concept of Experimental Accuracy and Simultaneous Measurements of Position and Momentum

D. M. Appleby

Published Mar 17, 1998• 9 citations
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Abstract

The concept of experimental accuracy is investigated in the context of the unbiased joint measurement processes defined by Arthurs and Kelly. A distinction is made between the errors of retrodiction and prediction. Four error-disturbance relationships are derived, analogous to the single error-disturbance relationship derived by Braginsky and Khalili in the context of single measurements of position only. A retrodictive and a predictive error-error relationship are also derived. The connection between these relationships and the extended Uncertainty Principle of Arthurs and Kelly is discussed. The similarities and differences between the quantum mechanical and classical concepts of experimental accuracy are explored. It is argued that these relationships provide grounds for questioning Uffink's conclusion, that the concept of a simultaneous measurement of non-commuting observables is not fruitful.

DOI
10.48550/arxiv.quant-ph/9803046

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The concept of experimental accuracy is investigated in the context of the unbiased joint measurement processes defined by Arthurs and Kelly. A distinction is made between the errors of retrodiction and prediction. Four error-disturbance relationships are derived, analogous to the single error-disturbance relationship derived by Braginsky and Khalili in the context of single measurements of position only. A retrodictive and a predictive error-error relationship are also derived. The connection between these relationships and the extended Uncertainty Principle of Arthurs and Kelly is discussed. The similarities and differences between the quantum mechanical and classical concepts of experimental accuracy are explored. It is argued that these relationships provide grounds for questioning Uffink's conclusion, that the concept of a simultaneous measurement of non-commuting observables is not fruitful.
The problem of characterising the accuracy of, and disturbance caused by a joint measurement of position and momentum is investigated. In a previous paper the problem was discussed in the context of the unbiased measurements considered by Arthurs and Kelly. It is now shown, that suitably modified versions of these results hold for a much larger class of simultaneous measurements. The approach is a development of that adopted by Braginsky and Khalili in the case of a single measurement of position only. A distinction is made between the errors of retrodiction and the errors of prediction. Two error-error relationships and four error-disturbance relationships are derived, supplementing the Uncertainty Principle usually so-called. In the general case it is necessary to take into account the range of the measuring apparatus. Both the ideal case, of an instrument having infinite range, and the case of a real instrument, for which the range is finite, are discussed.
The accuracy of the Arthurs-Kelly model of a simultaneous measurement of position and momentum is analysed using concepts developed by Braginsky and Khalili in the context of measurements of a single quantum observable. A distinction is made between the errors of retrodiction and prediction. It is shown that the distribution of measured values coincides with the initial-state Husimi function when the retrodictive accuracy is maximized (this being the result first obtained by Busch, using somewhat different methods). It is further shown that the distribution of measured values is related to the final-state anti-Husimi function (the P representation of quantum optics) when the predictive accuracy is maximized. The disturbance of the system by the measurement is also discussed. A class of minimally disturbing measurements is characterized. It is shown that the distribution of measured values then coincides with one of the smoothed Wigner functions described by Cartwright.
The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty \nrelations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. \nSpecifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances \nin terms of the commutators. \nFurthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality \nis found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive \nvarious types of inequalities. \nFinally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of \nthree canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.
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.
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.