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A Class of Distance-Based Incompatibility Measures for Quantum Measurements

Prabha Mandayam, M. D. Srinivas

Published Dec 1, 2015• 0 citations
Publisher

Abstract

We discuss a recently proposed class of incompatibility measures for quantum measurements, which is based on quantifying the effect of measurements of one observable on the statistics of the outcome of another. We summarize the properties of this class of measures, and present a tight upper bound for the incompatibility of any set of projective measurements in finite dimensions. We also discuss non-projective measurements, and give a non-trivial upper bound on the mutual incompatibility of a pair of Luders instruments. Using the example of incompatible observables that commute on a subspace, we elucidate how this class of measures goes beyond uncertainty relations in quantifying the mutual incompatibility of quantum measurements.

DOI
10.18520/cs/v109/i11/1997-2001

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We discuss a recently proposed class of incompatibility measures for quantum measurements, which is based on quantifying the effect of measurements of one observable on the statistics of the outcome of another. We summarize the properties of this class of measures, and present a tight upper bound for the incompatibility of any set of projective measurements in finite dimensions. We also discuss non-projective measurements, and give a non-trivial upper bound on the mutual incompatibility of a pair of Luders instruments. Using the example of incompatible observables that commute on a subspace, we elucidate how this class of measures goes beyond uncertainty relations in quantifying the mutual incompatibility of quantum measurements.
We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. Specifically, for a pair of observables AAA and BBB with purely discrete spectra, we compare the following two probability distributions: one resulting from a measurement of AAA followed by a measurement of BBB on a given state and the other obtained from a measurement of BBB alone on the same state. We show that maximizing the distance between these two distributions over all states yields a valid measure of the incompatibility of observables AAA and BBB, which is zero if and only if they commute and is strictly greater than zero (and less than or equal to one) otherwise. For finite-dimensional systems, we obtain a tight upper bound on the incompatibility of any pair of observables and show that the bound is attained when the observables are totally nondegenerate and associated with mutually unbiased bases. In the process, we also establish an important relation between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of nonprojective measurements. In particular, we obtain a nontrivial upper bound on the incompatibility of one L\"uders instrument with another.
Uncertainty relations are often considered to be a measure of incompatibility of noncommuting observables. However, such a consideration is not valid in general, motivating the need for an alternate measure that applies to any set of noncommuting observables. We present an operational approach to quantifying incompatibility without invoking uncertainty relations. Our measure aims to capture the incompatibility of noncommuting observables as manifest in the nonorthogonality of their eigenstates. We prove that this measure has all the desired properties: It is zero when the observables commute, strictly greater than zero when they do not, and is maximum when they are mutually unbiased. We also obtain tight upper bounds on this measure for any NNN noncommuting observables and compute it exactly when the observables are mutually unbiased.
We consider the question of characterizing the incompatibility of sets of high-dimensional quantum measurements. We introduce the concept of measurement incompatibility in subspaces. That is, starting from a set of measurements that is incompatible, one considers the set of measurements obtained by projection onto any strict subspace of fixed dimension. We identify three possible forms of incompatibility in subspaces: (i) incompressible incompatibility---measurements that become compatible in every subspace, (ii) fully compressible incompatibility---measurements that remain incompatible in every subspace, and (iii) partly compressible incompatibility---measurements that are compatible in some subspace and incompatible in another. For each class, we discuss explicit examples. Finally, we present some applications of these ideas. First, we show that joint measurability and coexistence are two inequivalent notions of incompatibility in the simplest case of qubit systems. Second, we highlight the implications of our results for tests of quantum steering.
We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when the pair of incompatible observables commute over a subspace, unlike EURs which give a zero lower bound in such cases. Here, we explicitly show how these measures go beyond EURs in quantifying incompatibility: For any set of quantum observables, we show that both incompatibility measures are bounded from below by the corresponding EURs for the Tsallis (T2T_{2}T2​) entropy. We explicitly evaluate the incompatibility of a pair of qubit observables in both operational scenarios. We also obtain an efficiently computable lower bound for the mutually incompatibility of a general set of 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.
For a pair of observables, they are called ``incompatible'' if and only if their commutator does not vanish, which represents one of the key features in quantum mechanics. The question is, how can we characterize the incompatibility among three or more observables? Here, we explore one possible route towards this goal through uncertainty relations, which impose fundamental constraints on the measurement precisions for incompatible observables. Specifically, we propose to measure the incompatibility by the optimal state-independent bounds of additive variance-based uncertainty relations. In this way, the degree of incompatibility becomes an intrinsic property among the operators, i.e., state independent. In particular, we focus on the incompatibility of spin-1/21/21/2 systems as an illustration. For an arbitrary, including nonorthogonal, setting of a finite number Pauli-spin operators, the incompatibility is analytically solved; the spins are maximally incompatible if and only if they are orthogonal to each other. On the other hand, our measure of incompatibility represents a versatile tool for applications such as testing the entanglement of bipartite states, and EPR-steering criteria.