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Laser Physics Letters

Jordan compatibility and Lie compatibility of triple positive operator-valued measures

Yuanyuan Duan, Liguang Wang

Published Aug 1, 2025• 0 citations
Publisher

Abstract

Abstract The Jordan product plays a pivotal role in quantum information theory, particularly within the framework of quantum compatibility. This research introduces the concept of the triple Jordan product and establishes its fundamental properties. Furthermore, for qubit systems, we characterize Jordan compatibility, Lie compatibility of three positive operator-valued measures and elucidate the interrelationships among Lie compatibility, joint compatibility, and Jordan compatibility.

DOI
10.1088/1612-202x/adf824

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Abstract The Jordan product plays a pivotal role in quantum information theory, particularly within the framework of quantum compatibility. This research introduces the concept of the triple Jordan product and establishes its fundamental properties. Furthermore, for qubit systems, we characterize Jordan compatibility, Lie compatibility of three positive operator-valued measures and elucidate the interrelationships among Lie compatibility, joint compatibility, and Jordan compatibility.
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.
In many a traditional physics textbook, a quantum measurement is defined as a projective measurement represented by a Hermitian operator. In quantum information theory, however, the concept of a measurement is dealt with in complete generality and we are therefore forced to confront the more general notion of positive-operator valued measures (POVMs), which suffice to describe all measurements that can be implemented in quantum experiments. We study the (in)compatibility of such POVMs and show that quantum theory realizes all possible (in)compatibility relations among sets of POVMs. This is in contrast to the restricted case of projective measurements for which commutativity is essentially equivalent to compatibility. Our result therefore points out a fundamental feature with respect to the (in)compatibility of quantum observables that has no analog in the case of projective measurements.
We start from the assumption that the real valued observables of a quantum system form a Jordan algebra which is equipped with a compatible Lie product characterizing infinitesimal symmetries, and ask whether two such systems can be considered as independent subsystems of a larger system. We show that this is possible if and only if the associator of the Jordan product is a fixed multiple of the associator of the Lie product. In this case it is known that the two products can be combined to an associative product in the Jordan algebra or its complexification, depending on the sign of the multiple.
Given two quantum channels, we examine the task of determining whether they are compatible-meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.
The POVM measurement assemblage consisting of a set of positive operator-value measurements (POVMs) is an important tool for describing and studying the non-locality, quantum steering and quantum correlation of composite quantum systems. In this paper, we first introduce the definition of joint measurability of the POVM measurement assemblage MS\mathcal{M}_SMS​ consisting of a finite number of POVMs and then prove that the joint measurability of MS\mathcal{M}_SMS​ is equivalent to its compatibility. We give a general form of a POVM measurement which is jointly measurable with the one-rank projection measurement. It is proved that the POVM measurement assemblage MS\mathcal{M}_SMS​ is jointly measurable if and only if the corresponding “row stochastic operator matrix" MS\textbf{M}_SMS​ composed of MS\mathcal{M}_SMS​ can be expressed as an “operator convex combination" of {0,IS}\{0,I_S\}{0,IS​}—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage M(n0 ,n1)\mathcal{M}(\textbf{n}_{0~},\textbf{n}_{1})M(n0 ​,n1​) induced by two three-dimensional real unit vectors ni(i=0,1)\textbf{n}_{i}(i=0,1)ni​(i=0,1) is jointly measurable if and only if n0=±n1\textbf{n}_{0}=\pm\textbf{n}_{1}n0​=±n1​. As an application, we obtain a necessary and sufficient condition under which a two-qubit quantum state is unsteerable with M(n0,n1)\mathcal{M}(\textbf{n}_{0},\textbf{n}_{1})M(n0​,n1​).