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arXiv:1005.0472 · Physical Review A

Approximating incompatible von Neumann measurements simultaneously

Teiko Heinosaari, Maria Anastasia Jivulescu, Daniel Reitzner, Mário Ziman

Published Sep 28, 2010• 9 citations
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Abstract

We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the L\"uders instrument of the optimal joint observable.

DOI
10.1103/physreva.82.032328

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We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the L"uders instrument of the optimal joint observable.
Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measurements. The marginal observables found to be optimal approximators are generally not among the coarse-grainings of the observables to be approximated. This yields scope for the improvement of existing joint measurement schemes. Both the quality of the approximations and the intrinsic unsharpness of the approximators are shown to be subject to Heisenberg-type uncertainty relations.
It is well known that jointly measurable observables cannot lead to a violation of any Bell inequality--independent of the state and the measurements chosen at the other site. In this Letter we prove the converse: every pair of incompatible quantum observables enables the violation of a Bell inequality and therefore must remain incompatible within any other no-signaling theory. While in the case of von Neumann measurements it is sufficient to use the same pair of observables at both sites, general measurements can require different choices. The main result is obtained by showing that for arbitrary dimension the Clauser-Horne-Shimony-Holt inequality provides the Lagrangian dual of the characterization of joint measurability. This leads to a simple criterion for joint measurability beyond the known qubit case.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
Joint or simultaneous measurements of non-commuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for defining what an "optimal" joint measurement is, with corresponding different tradeoff relations for the measurements. Understanding the limitations of such measurements is of fundamental interest and relevant for quantum technology. Here, we experimentally test a tradeoff relation for the sharpness of qubit measurements, a relation which refers directly to the form of the measurement operators, rather than to errors in estimates. We perform the first optical implementation of the simplest possible optimal joint measurement, requiring less quantum resources than have previously often been employed. Using a heralded single-photon source, we demonstrate quantum-limited performance of the scheme on single quanta.
Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.