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Chinese Physics B

Quantification of Jordan incompatibility for triple positive operator-valued measures

Yuan-Yuan Duan, Li-Guang Wang, Zheng-Li Chen

Published Apr 8, 2026• 0 citations
Publisher

Abstract

Abstract We investigate the compatibility of three positive operator-valued measures via the triple Jordan product. First, we establish the necessary and sufficient conditions for Jordan compatibility of three unbiased qubit measurements. Then, based on the negativity of the triple Jordan product, we propose a measure to quantify the incompatibility of three POVMs and elaborate on several key properties. Subsequently, we study the scenario of maximal incompatibility for three unbiased qubit measurements and prove that this maximum is attained when three POVMs are constructed from three mutually unbiased bases. Finally, we illustrate our quantification with two concrete examples.

DOI
10.1088/1674-1056/ae5c77

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Abstract We investigate the compatibility of three positive operator-valued measures via the triple Jordan product. First, we establish the necessary and sufficient conditions for Jordan compatibility of three unbiased qubit measurements. Then, based on the negativity of the triple Jordan product, we propose a measure to quantify the incompatibility of three POVMs and elaborate on several key properties. Subsequently, we study the scenario of maximal incompatibility for three unbiased qubit measurements and prove that this maximum is attained when three POVMs are constructed from three mutually unbiased bases. Finally, we illustrate our quantification with two concrete examples.
A fundamental feature of quantum mechanics is the incompatibility between quantum measurements, which leads to the Heisenberg uncertainty relations and is intrinsically related to detection of many other quantum features such as nonlocality and steering. Given the significance and ubiquity of incompatibility, it is desirable to quantify the degree of incompatibility between two quantum measurements which are mathematically described by positive operator valued measures. The commutator (Lie product) between operators has widely been used in quantifying incompatibility (noncommutativity) and plays a crucial role in quantum mechanics. In contrast, the anticommutator (Jordan product) between operators has been relatively less employed in the study of quantum mechanics. In this work, we explore the role of the Jordan product in characterizing measurement incompatibility. The key observation lies in that while the product of two non-negative classical observables is automatically non-negative, this is not the case for quantum observables. Indeed the Jordan product of two non-negative observables is not necessarily non-negative. This negativity of the Jordan product is a purely quantum phenomenon and is here exploited to quantify a kind of measurement incompatibility, which interpolates between noncommutativity and joint measurability. We reveal basic features of this approach, make some comparisons with existing quantifiers of incompatibility and suggest the issue of investigating its applications and implications in quantum information tasks.
Mutually unbiased bases capture perfect complementarity between two quantum measurements. Extensions beyond pairwise unbiasedness have been proposed, but essentially no non-trivial higher-order constructions are known. We introduce kkk-fold unbiased measurements (kkk-UMs), extending the kkk-fold unbiased bases notion of [arXiv:1706.04446] from rank-one basis measurements to arbitrary-rank projective measurements, and show that this higher-rank setting supports a much richer theory. We develop the notions of algebraic and spectral kkk-UMs and prove that they coincide for rank-one measurements and triples of measurements (3-UMs). We establish strong no-go results for higher-order rank-one constructions and three-outcome 3-UMs, but obtain infinitely many higher-rank triples using Hadamard matrices and Clifford algebras. We then give these structures an exact operational interpretation in terms of measurement incompatibility. For 3-UMs with any number of outcomes, we determine their generalised incompatibility robustness exactly and construct an explicit joint measurement for their noisy versions at the compatibility threshold. Finally, we implement the symmetry reduction of the sum-of-squares hierarchy recently introduced in [New J. Phys. 28, 064509 (2026)] and give numerical evidence that the noise thresholds arising from the kkk-UM analysis may characterise the asymptotic behaviour of this hierarchy. In particular, with very high precision, we numerically show that our constructed four-outcome 3-UMs are among the most incompatible triples of four-outcome measurements.
Quantum measurements based on mutually unbiased bases are commonly used in quantum information processing, as they are generally viewed as being maximally incompatible and complementary. Here we quantify precisely the degree of incompatibility of mutually unbiased bases (MUB) using the notion of noise robustness. Specifically, for sets of k MUB in dimension d, we provide upper and lower bounds on this quantity. Notably, we get a tight bound in several cases, in particular for complete sets of k=d+1 MUB (using the standard construction for d being a prime power). On the way, we also derive a general upper bound on the noise robustness for an arbitrary set of quantum measurements. Moreover, we prove the existence of sets of k MUB that are operationally inequivalent, as they feature different noise robustness, and we provide a lower bound on the number of such inequivalent sets up to dimension 32. Finally, we discuss applications of our results for Einstein-Podolsky-Rosen steering.
The incompatibility between quantum measurements (as mathematically represented by positive operator-valued measures, i.e., POVMs) is a key feature of quantum mechanics and is intrinsically related to the noncommutativity of operators. For both theoretical and practical considerations, it is desirable to quantify the degree of incompatibility between quantum measurements, and considerable effort has been devoted to this issue. In this paper, we provide a novel approach to measurement incompatibility by exploiting the Lüders channels derived from POVMs and employing the measurement disturbance to quantify incompatibility. This is achieved by constructing an approximately joint measurement for a pair of POVMs, which is an enlarged POVM with the correct marginal property for one of the two POVMs but not necessarily for the other. The degree of failure of the marginal property for the other POVM is a kind of measurement disturbance and can be naturally interpreted as a quantifier of the incompatibility between the two POVMs. We reveal basic properties of this quantifier of measurement incompatibility, identify its maximal value in some cases, compare it with several popular measures in the literature, and illustrate it with some typical examples. Some related open issues are also discussed.
We study when a noisy qubit channel renders all positive-operator-valued measures (POVMs) jointly measurable. For every unital qubit channel, we derive the exact incompatibility-breaking criterion and prove that projective measurements already determine the boundary for arbitrary POVMs. Through the steering--joint-measurability correspondence, this result gives the exact POVM-steering boundary for all two-qubit states with maximally mixed marginals, with the Werner states recovered as a special case. For nonunital qubit channels, we construct a generally asymmetric parent POVM and obtain an explicit sufficient incompatibility-breaking condition, which in turn yields a sufficient unsteerability criterion for arbitrary two-qubit states.