The Born Representation Theorem and the Unistochastic Theorem

• URL: http://arxiv.org/abs/2608.04354v1
• Date: Wed, 05 Aug 2026 01:55:06 GMT
• Title: The Born Representation Theorem and the Unistochastic Theorem
• Abstract summary: The paper presents self-contained, constructive proofs of two new theorems about matrices, with direct relevance to quantum theory.<n>The Born Representation Theorem shows that each entry of any matrix can be expressed as the trace of a pairwise product.<n>The Unistochastic Theorem shows that by dilating the underlying vector space by a bounded number of additional dimensions, each entry of any matrix can be expressed in terms of the trace of a pairwise product.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Structure of matrix product locally purifiable density operators [21.292344323864516]
  We study a fundamental theorem for matrix product locally purifiable density operators (LPDOs)<n>We prove that, under suitable invertibility or cyclic conditions, two sLPDO representations generate the same density matrix for arbitrary system sizes.<n>We provide a counterexample that suggests an obstruction to a general fundamental theorem for LPDOs with periodic boundary conditions.
  arXiv  Detail & Related papers  (2026-08-03T18:00:02Z)
• Spectral Estimation with Free Decompression [47.81955761814048]
  We introduce a novel method of "free decompression" to estimate the spectrum of very large (impalpable) matrices.<n>Our method can be used to extrapolate from the empirical spectral densities of small submatrices to infer the eigenspectrum of extremely large (impalpable) matrices.
  arXiv  Detail & Related papers  (2025-06-13T17:49:25Z)
• Quantitative Tsirelson's Theorems via Approximate Schur's Lemma and Probabilistic Stampfli's Theorems [1.1470070927586018]
  We show that each operator in $mathcalB$ is $O(d2epsilon)$-close in operator norm to an operator in the commutant $mathcalA'$.<n>As an application of our results to quantum information theory, we obtain a quantitative Tsirelson's theorem.
  arXiv  Detail & Related papers  (2025-05-28T12:51:27Z)
• Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry [63.694184882697435]
  Global Covariance Pooling (GCP) has been demonstrated to improve the performance of Deep Neural Networks (DNNs) by exploiting second-order statistics of high-level representations.<n>This paper provides a comprehensive and unified understanding of the matrix logarithm and power from a Riemannian geometry perspective.
  arXiv  Detail & Related papers  (2024-07-15T07:11:44Z)
• Discrete dynamics in the set of quantum measurements [0.0]
  A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators $P_j=P_jdaggeq 0$ summing to identity.<n>We describe discrete transformations in the set of quantum measurements by em blockwise matrices.<n>We formulate our main result: a quantum analog of the Ostrowski description of the classical Birkhoff polytope.
  arXiv  Detail & Related papers  (2023-08-10T19:34:04Z)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• Connecting classical finite exchangeability to quantum theory [45.76759085727843]
  Exchangeability is a fundamental concept in probability theory and statistics.<n>It allows to model situations where the order of observations does not matter.<n>It is well known that both theorems do not hold for finitely exchangeable sequences.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality [69.62715388742298]
  We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs) We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT.
  arXiv  Detail & Related papers  (2022-03-08T14:47:39Z)
• Riemannian statistics meets random matrix theory: towards learning from high-dimensional covariance matrices [2.352645870795664]
  This paper shows that there seems to exist no practical method of computing the normalising factors associated with Riemannian Gaussian distributions on spaces of high-dimensional covariance matrices. It is shown that this missing method comes from an unexpected new connection with random matrix theory. Numerical experiments are conducted which demonstrate how this new approximation can unlock the difficulties which have impeded applications to real-world datasets.
  arXiv  Detail & Related papers  (2022-03-01T03:16:50Z)
• Statistics of the Spectral Form Factor in the Self-Dual Kicked Ising Model [0.0]
  We show that at large enough times the probability distribution agrees exactly with the prediction of Random Matrix Theory. This behaviour is due to a recently identified additional anti-unitary symmetry of the self-dual kicked Ising model.
  arXiv  Detail & Related papers  (2020-09-07T16:02:29Z)
• J-matrix method of scattering in one dimension: The relativistic theory [0.0]
  We make a relativistic extension of the one-dimensional J-matrix method of scattering. The relativistic potential matrix is a combination of vector, scalar, and pseudo-scalar components.
  arXiv  Detail & Related papers  (2020-01-14T19:02:15Z)
• Joint measurability meets Birkhoff-von Neumann's theorem [77.34726150561087]
  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. We also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
  arXiv  Detail & Related papers  (2018-09-19T18:57:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.