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arXiv:2608.04354

The Born Representation Theorem and the Unistochastic Theorem

Jacob A. Barandes

quant-phmath-phmath.PRmath.RASubmitted Aug 5, 2026• Announced Aug 6, 2026• 0 citations
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Abstract

This paper presents self-contained, constructive proofs of two new theorems about stochastic matrices, with direct relevance to quantum theory. The first theorem, herein called the Born Representation Theorem, shows that each entry of any stochastic matrix can be expressed as the trace of a pairwise product of matrices, where the first factor in the pairwise product belongs to a positive-operator-valued measure (POVM) and the second factor belongs to a projection-valued measure (PVM). As its name suggests, this theorem entails that the entries of any stochastic matrix can be expressed in terms of a generalized version of the quantum-theoretic Born rule. It follows as a corollary that if the POVM in this first theorem is a PVM, then the stochastic matrix is unistochastic, meaning that its entries are each the modulus square of the corresponding entry of a unitary matrix of the same size. The second theorem proved in this paper, called the Unistochastic Theorem, then shows that by dilating the underlying vector space by a bounded number of additional dimensions if necessary, each entry of any stochastic matrix can be expressed in terms of the trace of a pairwise product for which both factors belong to PVMs, and can thus be derived via marginalization from a larger unistochastic matrix. This second theorem therefore establishes a kind of primacy of unistochastic matrices over stochastic matrices, and hints at a close connection with unitary time evolution in quantum theory. The paper concludes with a brief discussion of potential applications to discrete-time deterministic processes and Markov chains.

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This paper presents self-contained, constructive proofs of two new theorems about stochastic matrices, with direct relevance to quantum theory. The first theorem, herein called the Born Representation Theorem, shows that each entry of any stochastic matrix can be expressed as the trace of a pairwise product of matrices, where the first factor in the pairwise product belongs to a positive-operator-valued measure (POVM) and the second factor belongs to a projection-valued measure (PVM). As its name suggests, this theorem entails that the entries of any stochastic matrix can be expressed in terms of a generalized version of the quantum-theoretic Born rule. It follows as a corollary that if the POVM in this first theorem is a PVM, then the stochastic matrix is unistochastic, meaning that its entries are each the modulus square of the corresponding entry of a unitary matrix of the same size. The second theorem proved in this paper, called the Unistochastic Theorem, then shows that by dilating the underlying vector space by a bounded number of additional dimensions if necessary, each entry of any stochastic matrix can be expressed in terms of the trace of a pairwise product for which both factors belong to PVMs, and can thus be derived via marginalization from a larger unistochastic matrix. This second theorem therefore establishes a kind of primacy of unistochastic matrices over stochastic matrices, and hints at a close connection with unitary time evolution in quantum theory. The paper concludes with a brief discussion of potential applications to discrete-time deterministic processes and Markov chains.
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes, collectively encompass many important kinds of stochastic processes, including Markov chains and random dynamical systems. This paper then states and proves a new theorem that establishes a precise correspondence between any indivisible stochastic process and a unitarily evolving quantum system. This theorem therefore leads to a new formulation of quantum theory, alongside the Hilbert-space, path-integral, and quasi-probability formulations. The theorem also provides a first-principles explanation for why quantum systems are based on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the Born rule. In addition, the theorem suggests that by selecting a suitable Hilbert space, together with an appropriate choice of unitary evolution, one can simulate any indivisible stochastic process on a quantum computer, thereby potentially opening up an extensive set of novel applications for quantum computing.
Gauge Freedom and the Born Rule: Extending the Unitary Refinement Program to Generalized Quantum Measurements This preprint develops a generalized refinement framework for mixed quantum states and positive operator-valued measures (POVMs), extending an earlier representation theorem derived for pure states and projection-valued measurements (PVMs). The central result is a Representation Invariance Theorem, showing that physically equivalent purification and Naimark-dilation realizations induce identical abundance assignments, allowing the measure constructed in an enlarged projective space to descend consistently to the operational pair. The paper argues that the generalized Born rule can be understood as the compressed representation of a previously constructed abundance measure rather than an assumed probabilistic postulate. The framework preserves the distinction between primitive branch abundance, induced event measure, and rational credence established in the earlier work. In addition to extending the theory to mixed states and POVMs, the paper proves a finite-dimensional robustness theorem showing that approximately orthogonal record structures induce only controlled deviations from the ideal projective measure. The resulting framework links quantum foundations, generalized measurement theory, decoherence, and modern quantum information science. This manuscript is the second stage of a broader research program investigating: finite projective refinements and branch abundance; representation invariance for generalized measurements (this work); robustness, continuous realization spaces, and gauge-like structures of approximate branches; and operator-algebraic extensions involving Type III von Neumann algebras.
Finite probability theory is enriched by introducing the mathematical notion (no physics involved) of a superposition event ΣS\Sigma SΣS--in addition to the usual discrete event SSS (subset of the outcome space U=(u1,...,un)U=\left( u_{1},...,u_{n}\right)U=(u1​,...,un​)). Mathematically, the two types of events are distinguished using n×nn\times nn×n density matrices. The density matrix ρ(S)\rho\left( S\right)ρ(S) for a discrete event is diagonal and the density matrix ρ(ΣS)\rho\left( \Sigma S\right)ρ(ΣS) is obtained as an outer product ∣s⟩⟨s∣\left\vert s\right\rangle \left\langle s\right\vert∣s⟩⟨s∣ of a normalized vector ∣s⟩∈Rn\left\vert s\right\rangle \in\mathbb{R}^{n}∣s⟩∈Rn. Probabilities are defined using density matrices as Pr⁡(T∣ρ)=tr⁡[PTρ]\Pr\left( T|\rho\right) =\operatorname*{tr}\left[ P_{T}\rho\right]Pr(T∣ρ)=tr[PT​ρ] where T⊆UT\subseteq UT⊆U and PTP_{T}PT​ is the diagonal projection matrix with diagonal entries χT(ui)\chi_{T}\left( u_{i}\right)χT​(ui​). Then for the singleton {ui}⊆U\left\{ u_{i}\right\} \subseteq U{ui​}⊆U, the probability of the outcome uiu_{i}ui​ conditioned by the _superposition_ event ΣS\Sigma SΣS is Pr⁡({ui}∣ΣS)=⟨ui∣s⟩2\Pr\left( \left\{ u_{i}\right\} |\Sigma S\right) =\left\langle u_{i}|s\right\rangle ^{2}Pr({ui​}∣ΣS)=⟨ui​∣s⟩2, the Born Rule. Thus the Born Rule arises naturally from the mathematics of superposition when superposition events are added to ordinary finite probability theory. No further explanation is required when the mathematics uses Cn\mathbb{C}^{n}Cn instead of Rn\mathbb{R}^{n}Rn except that the square ⟨ui∣s⟩2\left\langle u_{i}|s\right\rangle ^{2}⟨ui​∣s⟩2 is the absolute square ∣⟨ui∣s⟩∣2\left\vert \left\langle u_{i}|s\right\rangle \right\vert ^{2}∣⟨ui​∣s⟩∣2.
The classical embeddability problem asks whether a given stochastic matrix T , describing transition probabilities of a d -level system, can arise from the underlying homogeneous continuous-time Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given T . More precisely, we aim at characterising the set of quantum-embeddable stochastic matrices that arise from memoryless continuous-time quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantum-embeddable but not classically-embeddable, as well as families of stochastic matrices that are not quantum-embeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a non-zero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantum-embeddable.
We study two relevant characterizations of a commutative positive operator valued measure (POVM) F . The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ . We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ . Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.
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.