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arXiv:0805.1538 · Physical Review A

Joint measurement of two unsharp observables of a qubit

Sixia Yu, Nai-Le Liu, Li Li, C. H. Oh

Published Jun 14, 2010• 90 citations
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Abstract

We derive a single inequality as the sufficient and necessary condition for two unsharp observables of a two-level system to be jointly measurable in a single apparatus and construct explicitly the joint observables of two jointly measurable observables. By introducing a generalized distinguishability as a measure of the unsharpness of an unsharp measurement, we derive a complementarity inequality, which generalizes Englert's duality inequality, from the condition of joint measurement of two orthogonal unsharp observables.

DOI
10.1103/physreva.81.062116

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We derive a single inequality as the sufficient and necessary condition for two unsharp observables of a two-level system to be jointly measurable in a single apparatus and construct explicitly the joint observables of two jointly measurable observables. By introducing a generalized distinguishability as a measure of the unsharpness of an unsharp measurement, we derive a complementarity inequality, which generalizes Englert's duality inequality, from the condition of joint measurement of two orthogonal unsharp observables.
Whereas complementarity manifests itself via two incompatible observables, quantum contextuality can only be revealed via the joint measurements among at least three observables. By incorporating unsharp measurements and joint measurements into a realistic model, we reestablish an inequality due to Liang, Spekkens, and Wiseman rigorously based on the assumption of noncontextuality alone. Its violation therefore unambiguously pinpoints the quantum contextuality of a two-level system. The maximal violation is attained by three triplewise jointly measurable observables that are pairwise jointly measured in an incompatible way. We also present the necessary and sufficient condition of triplewise joint measurability of three unbiased observables of a qubit.
Bohr's complementarity principle is one of the central concepts in quantum mechanics which restricts joint measurement for certain observables. Of course, later development shows that joint measurement could be possible for such observables with the introduction of a certain degree of unsharpness or fuzziness in the measurement. In this paper, we show that the optimal degree of unsharpness, which guarantees the joint measurement of all possible pairs of dichotomic observables, determines the degree of nonlocality in quantum mechanics as well as in more general no-signaling theories.
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.
We consider a joint measurement of two and three unsharp qubit observables through an Arthur–Kelly-type joint measurement model for qubits. We investigate the effect of the initial state of the detectors on the unsharpness of the measurement as well as the post-measurement state of the system. Particular emphasis is given on a physical understanding of the POVM to PVM transition in the model and entanglement between the system and the detectors. Two approaches for characterizing the unsharpness of the measurement and the resulting measurement uncertainty relations are considered. The corresponding measures of unsharpness are connected for the case where both the measurements are equally unsharp. The connection between the POVM elements and symmetries of the underlying Hamiltonian of the measurement interaction is made explicit and used to perform the joint measurement in arbitrary directions. Finally, in the case of three observables, we derive a necessary condition for the approximate joint measurement and use it to show the relative freedom available when the observables are non-orthogonal.
We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence, predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to {\em validate} a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix -- which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables -- could be realized if and only if compatible POVMs are employed.