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arXiv:1512.07050 · Physical Review A

Heisenberg's error-disturbance relations: A joint measurement-based experimental test

Yuanyuan Zhao, Paweł Kurzyński, Guo‐Yong Xiang, Chuan‐Feng Li, Guang‐Can Guo

Published Apr 7, 2017• 8 citations
arXivHTMLPDF

Abstract

The original Heisenberg error-disturbance relation was recently shown to be not universally valid and two different approaches to reformulate it were proposed. The first one focuses on how the error and disturbance of two observables $A$ and $B$ depend on a particular quantum state. The second one asks how a joint measurement of $A$ and $B$ affects their eigenstates. Previous experiments focused on the first approach. Here we focus on the second one. First, we propose and implement an extendible method of quantum-walk-based joint measurements of noisy Pauli operators to test the error-disturbance relation for qubits introduced in the work of Busch et al. [Phys. Rev. A 89, 012129 (2014)], where the polarization of the single photon, corresponding to a walker's auxiliary degree of freedom that is commonly known as a coin, undergoes a position- and time-dependent evolution. Then we formulate and experimentally test a universally valid state-dependent relation for three mutually unbiased observables. We therefore establish a method of testing error-disturbance relations.

DOI
10.1103/physreva.95.040101

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The original Heisenberg error-disturbance relation was recently shown to be not universally valid and two different approaches to reformulate it were proposed. The first one focuses on how the error and disturbance of two observables AAA and BBB depend on a particular quantum state. The second one asks how a joint measurement of AAA and BBB affects their eigenstates. Previous experiments focused on the first approach. Here we focus on the second one. First, we propose and implement an extendible method of quantum-walk-based joint measurements of noisy Pauli operators to test the error-disturbance relation for qubits introduced in the work of Busch et al. [Phys. Rev. A 89, 012129 (2014)], where the polarization of the single photon, corresponding to a walker's auxiliary degree of freedom that is commonly known as a coin, undergoes a position- and time-dependent evolution. Then we formulate and experimentally test a universally valid state-dependent relation for three mutually unbiased observables. We therefore establish a method of testing error-disturbance relations.
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, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of $\ifmmode\pm\else\textpm\fi{}1$-valued observables A,BA,BA,B are tightly bounded from below by a quantity that measures the degree of incompatibility of AAA and BBB. The claim of a violation of Heisenberg is shown to fail because it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.
Recently there have been claims of experimental violations of Heisenberg's error-disturbance relation [Rozema et al, PRL 109, 200404 (2012), Erhard et al, Nature Phys. 8, 185 (2012)]. These claims were contradicted by a proof of a form of such relations [Busch et al, PRL 111, 160405 (2013)], showing that the validity of any formalization of Heisenberg's principle depends on the choice of appropriate measures of error and disturbance. While the error quantity underlying the experiments was shown to be conceptually problematic in general, it turns out to coincide, in the qubit experiments in question, with the alternative error measure used by Busch et al. Here we analyze this coincidence and discover a technique, hitherto deemed unavailable, for determining the disturbance measure. We show that this procedure reflects directly the operational meaning of the disturbance measure, in contrast to the weak-value method used in the experiments of Rozema et al and several other groups. The latter is thus not fundamental for an understanding of the quantum mechanical error-disturbance trade-off. The strong scheme is also shown to be applicable to a wider class of joint measurement 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 uncertainty principle formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable such that their product should be no less than the limit set by Planck's constant. However, Ozawa in 1988 showed a model of position measurement that breaks Heisenberg's relation and in 2003 revealed an alternative relation for error and disturbance to be proven universally valid. Here, we report an experimental test of Ozawa's relation for a single-photon polarization qubit, exploiting a more general class of quantum measurements than the class of projective measurements. The test is carried out by linear optical devices and realizes an indirect measurement model that breaks Heisenberg's relation throughout the range of our experimental parameter and yet validates Ozawa's relation.
The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that, in nearly 90 years, there have been no direct tests of measurement uncertainty relations. This lacuna was due to the absence of two essential ingredients: appropriate measures of measurement error (and disturbance) and precise formulations of such relations that are universally valid and directly testable. We formulate two distinct forms of direct tests, based on different measures of error. We present a prototype protocol for a direct test of measurement uncertainty relations in terms of value deviation errors (hitherto considered nonfeasible), highlighting the lack of universality of these relations. This shows that the formulation of universal, directly testable measurement uncertainty relations for state-dependent error measures remains an important open problem. Recent experiments that were claimed to constitute invalidations of Heisenberg's error-disturbance relation, are shown to conform with the spirit of Heisenberg's principle if interpreted as direct tests of measurement uncertainty relations for error measures that quantify distances between observables.
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.