Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate a corresponding version of Birkhoff–von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory.
Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate a corresponding version of Birkhoff–von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory.
Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
Quantum magic squares are generalizations of doubly stochastic matrices to arbitrary, possibly non-commutative algebras, more specifically C∗-algebras. Birkhoff-von Neumann Theorem states that every doubly stochastic matrix is a convex combination of permutation matrices. One inevitable question is whether a quantum magic square is a convex combinations of quantum permutation matrices. It has been proven by Cuevas, Drescher, and Netzer that the generalization of the Birkhoff-von Neumann Theorem in the quantum setting is not true in general. A quantum magic square is a convex combination of Arveson extreme points but not necessarily a convex combination of quantum permutation matrices. Additionally, the relationships between different subsets of quantum magic squares, including their inclusion properties, have been studied in this context.
Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff–von Neumann theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is as follows: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here, we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.
Abstract The entropy principle asserts that spontaneous processes drive systems toward states of higher disorder. Operational origin of such randomness is a question of utmost importance. The seminal Birkhoff theorem (also known as Birkhoff–von Neumann theorem) identifies doubly stochastic transformations as convex mixtures of reversible transformations, thereby providing an epistemic account of disorder. Quantum theory, however, does not satisfy this equivalence in general. In this work, we investigate this question within the framework of Generalized Probabilistic Theories (GPTs). We prove a violation of the Birkhoff theorem in the square-bit model, a canonical non-quantum GPT, and further show that this violation persists asymptotically under tensor products of any finite, yet arbitrarily large, number of copies of even-polygon models. This provides evidence that the failure of the Birkhoff theorem is an intrinsic operational feature of a wide class of GPTs, rather than a peculiarity of quantum theory. Furthermore, we show that, beyond quantum theory, the preorder over states induced by mixtures of reversible transformations can differ from that induced by doubly stochastic transformations, leading to inequivalent notions of state convertibility and distinct resource theories of purity. Beyond its foundational significance, we show that the violation of the Birkhoff theorem admits a concrete operational interpretation: in the square-bit theory, it serves as a genuine resource for enhancing the performance of Random Access Codes. Finally, we discuss the construction of entropy functions in GPTs that remain consistent with thermodynamic principles, in the spirit of von Neumann’s formulation.
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 consisting of a finite number of POVMs and then prove that the joint measurability of MS 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 is jointly measurable if and only if the corresponding “row stochastic operator matrix" MS composed of MS can be expressed as an “operator convex combination" of {0,IS}—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage M(n0,n1) induced by two three-dimensional real unit vectors ni(i=0,1) is jointly measurable if and only if 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).
Abstract A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators summing to identity. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. Discrete transformations in the set of quantum measurements can be described by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares) and study their properties. Analyzing the dynamics they induce we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set.