From Birkhoff's Polytope to Petz Recovery: Unistochastic Matrices, Quantum Channels, and Approximate Markov Chains

A doubly stochastic matrix is unistochastic if its entries correspond to the squared moduli of a unitary matrix. Determining which n × n doubly stochastic matrices admit such a representation remains an open problem at the intersection of convex geometry, combinatorics, and quantum information. For 3 × 3 matrices, elegant triangle inequalities provide a complete characterization: the unistochastic set occupies approximately 75% of the Birkhoff polytope and exhibits deltoid cross-sections. For n ≥ 4, the characterization problem remains unresolved and is influenced in unexpected ways by the prime factorization of n via the defect of the Fourier matrix. This presentation surveys these results and then establishes a connection to a second, seemingly unrelated question: given a tripartite quantum state with small conditional mutual information, to what extent can one subsystem be recovered from the others? The Petz recovery map and its rotated variants offer a universal solution. These two topics are linked through coherification, which concerns when a classical stochastic process can be elevated to coherent quantum dynamics, and through the conditional mutual information as a continuous measure of non-unistochasticity. The talk concludes with open problems at this interface, including the star-shapedness conjecture for n = 4 and the pursuit of tighter recovery bounds.