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arXiv:1312.1857 · Physical Review A

Deriving tight error-trade-off relations for approximate joint measurements of incompatible quantum observables

Cyril Branciard

Published Feb 24, 2014• 36 citations
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Abstract

The quantification of the ``measurement uncertainty''aspect of Heisenberg's uncertainty principle---that is, the study of trade-offs between accuracy and disturbance, or between accuracies in an approximate joint measurement on two incompatible observables---has regained a lot of interest recently. Several approaches have been proposed and debated. In this paper we consider Ozawa's definitions for inaccuracies (as root-mean-square errors) in approximate joint measurements, and study how these are constrained in different cases, whether one specifies certain properties of the approximations---namely their standard deviations and/or their bias---or not. Extending our previous work [C. Branciard, Proc. Natl. Acad. Sci. USA 110, 6742 (2013)], we derive error-trade-off relations, which we prove to be tight for pure states. We show explicitly how all previously known relations for Ozawa's inaccuracies follow from ours. While our relations are in general not tight for mixed states, we show how these can be strengthened and how tight relations can still be obtained in that case.

DOI
10.1103/physreva.89.022124

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The quantification of the ``measurement uncertainty''aspect of Heisenberg's uncertainty principle---that is, the study of trade-offs between accuracy and disturbance, or between accuracies in an approximate joint measurement on two incompatible observables---has regained a lot of interest recently. Several approaches have been proposed and debated. In this paper we consider Ozawa's definitions for inaccuracies (as root-mean-square errors) in approximate joint measurements, and study how these are constrained in different cases, whether one specifies certain properties of the approximations---namely their standard deviations and/or their bias---or not. Extending our previous work [C. Branciard, Proc. Natl. Acad. Sci. USA 110, 6742 (2013)], we derive error-trade-off relations, which we prove to be tight for pure states. We show explicitly how all previously known relations for Ozawa's inaccuracies follow from ours. While our relations are in general not tight for mixed states, we show how these can be strengthened and how tight relations can still be obtained in that case.
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.
Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [P. Busch et al., Phys. Rev. Lett. 111, 160405 (2013); P. Busch et al., Phys. Rev. A 89, 012129 (2014)]. Here we reformulate their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin system. The relation reveals that the worst-case inaccuracy is tightly bounded from below by the incompatibility of target observables, and is verified by the experiment employing joint measurement in which two compatible observables designed to approximate two incompatible observables on one qubit are measured simultaneously.
Recent years have witnessed a controversy over Heisenberg's famous error-disturbance relation. Here the conflict is resolved by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. Two approaches to adapting the classic notion of root-mean-square error to quantum measurements are discussed. One is based on the concept of a noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.
Heisenberg's uncertainty principle was originally formulated in 1927 as a quantitative relation between the "mean error" of a measurement of one observable and the disturbance thereby caused on another observable. Heisenberg derived this famous relation under an additional assumption on quantum measurements that has been abandoned in the modern theory, and its universal validity was questioned in a debate on the sensitivity limit to gravitational-wave detectors in 1980s. A universally valid form of the error-disturbance relation was shown to be derived in the modern framework of general quantum measurements in 2003. We have experienced a considerable progress in theoretical and experimental study of error-disturbance relations in the last decade. In 2013 Branciard showed a new stronger form of universally valid error-disturbance relations, one of which is proved tight for spin measurements carried out in "pure" states. Nevertheless, a recent information-theoretical study of error-disturbance relations has suggested that Branciard relations can be considerably strengthened for measurements in mixed states. Here, we show a method for strengthening Branciard relations in mixed states and derive several new universally valid and stronger error-disturbance relations in mixed states. In particular, it is proved that one of them gives an ultimate error-disturbance relation for spin measurements, which is tight in any state. The new relations will play an important role in applications to state estimation problems including quantum cryptographic scenarios.
Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenberg’s error-disturbance relation. In contrast, a Heisenberg-type tradeoff relation for joint measurements of position and momentum has been formulated and proven in [Phys. Rev. Lett. 111, 160405 (2013)]. Here I show how the apparent conflict is resolved by a careful consideration of the quantum generalization of the notion of root-mean-square error. The claim of a violation of Heisenberg’s principle is untenable as it is based on a historically wrong attribution of an incorrect relation to Heisenberg, which is in fact trivially violated. We review a new general trade-off relation for the necessary errors in approximate joint measurements of incompatible qubit observables that is in the spirit of Heisenberg’s intuitions. The experiments mentioned may directly be used to test this new error inequality.
The quantum multiparameter estimation is very different from the classical multiparameter estimation due to Heisenberg's uncertainty principle in quantum mechanics. When the optimal measurements for different parameters are incompatible, they cannot be jointly performed. We find a correspondence relationship between the inaccuracy of a measurement for estimating the unknown parameter with the measurement error in the context of measurement uncertainty relations. Taking this correspondence relationship as a bridge, we incorporate Heisenberg's uncertainty principle into quantum multiparameter estimation by giving a trade-off relation between the measurement inaccuracies for estimating different parameters. For pure quantum states, this trade-off relation is tight, so it can reveal the true quantum limits on individual estimation errors in such cases. We apply our approach to derive the trade-off between attainable errors of estimating the real and imaginary parts of a complex signal encoded in coherent states and obtain the joint measurements attaining the trade-off relation. We also show that our approach can be readily used to derive the trade-off between the errors of jointly estimating the phase shift and phase diffusion without explicitly parametrizing quantum measurements.