Unistochastic Matrices

• Unistochastic matrices are doubly stochastic matrices formed by squaring the moduli of a unitary matrix, linking linear algebra with quantum theory.
• They exhibit non-convexity and intricate structure in dimensions three and higher, characterized by chain-link (bracelet) inequalities within the Birkhoff polytope.
• Generalized block-unistochastic matrices extend the concept via block unitary matrices, offering new insights for quantum channels, frame theory, and statistical physics.

A unistochastic matrix is a bistochastic (or doubly stochastic) matrix that arises as the entrywise squared moduli of a unitary matrix. The study of unistochastic matrices provides a natural bridge between linear algebra, convex geometry, quantum information theory, and statistical physics, with deep connections to combinatorics and the geometry of the Birkhoff polytope. Despite their seemingly simple definition, the structure and characterization of unistochastic matrices exhibit striking complexity, especially in dimensions three and higher, where the set becomes non-convex and admits intricate internal geometry and inclusions.

1. Definitions, Basic Structure, and Hierarchy

Let $B = (B_{ij})$ be a real $n \times n$ matrix. $B$ is called bistochastic if $B_{ij} \geq 0$ for all $i,j$, $\sum_j B_{ij} = 1$ for each row $i$, and $\sum_i B_{ij} = 1$ for each column $j$. The set of all $n \times n$ bistochastic matrices forms the Birkhoff polytope $\mathsf{B}_n$, whose vertices are the $n!$ permutation matrices.

A bistochastic matrix $B$ is unistochastic if there exists a unitary $U \in U(n)$ such that $B_{ij} = |U_{ij}|^2$ for all $i,j$: $$\mathsf{U}_n = \{ U \circ \overline{U} : U \in U(n) \} \subseteq \mathsf{B}_n.$$ Similarly, $B$ is orthostochastic if it arises from a real orthogonal $O \in O(n)$, $B_{ij} = (O_{ij})^2$. The inclusions are strict for $n\ge3$: $$\{\text{permutation matrices}\} \subsetneq \{\text{orthostochastic}\} \subsetneq \{\text{unistochastic}\} \subsetneq \{\text{bistochastic}\}.$$ For $n=2$, the sets of orthostochastic, unistochastic, and bistochastic matrices coincide: $\mathsf{U}_2 = \mathsf{B}_2$. For $n \geq 3$, the inclusions are proper, and the set $\mathsf{U}_n$ is generally non-convex and of full dimension but with a rich semialgebraic boundary (Nechita et al., 2023, Goyeneche et al., 2016, Rajchel-Mieldzioć et al., 2021, Gutkin, 2013).

2. Geometric and Algebraic Properties inside the Birkhoff Polytope

The Birkhoff–von Neumann theorem characterizes $\mathsf{B}_n$ as the convex hull of permutation matrices. However, $\mathsf{U}_n \subsetneq \mathsf{B}_n$ is non-convex for $n \geq 3$ (Nechita et al., 2023, Rajchel-Mieldzioć et al., 2021).

To address membership in $\mathsf{U}_n$, strong necessary conditions exist. The bracelet condition (also called chain-link inequalities) applies to any pair of probability vectors $\alpha, \beta \in \Delta_n$: $$2\max_j \sqrt{\alpha_j \beta_j} \leq \sum_{j=1}^n \sqrt{\alpha_j \beta_j}.$$ A matrix is called a bracelet matrix if every pair of rows/columns satisfies these inequalities. The set of bracelet matrices $\mathsf{L}_n$ obeys $$\mathsf{U}_n \subseteq \mathsf{L}_n \subseteq \mathsf{B}_n$$, with equality $\mathsf{U}_3 = \mathsf{L}_3$, but $\mathsf{U}_n \subsetneq \mathsf{L}_n$ for $n \geq 4$ (Nechita et al., 2023, Rajchel-Mieldzioć et al., 2021, Gutkin, 2013). In three dimensions, the chain-link inequalities are both necessary and sufficient: a $3 \times 3$ bistochastic matrix is unistochastic if and only if the associated triangle inequalities hold for the constructed side lengths $\ell_i = \sqrt{B_{ij} B_{kl}}$ (Gutkin, 2013).

The set of factorisable matrices $\mathsf{F}_n$ (products of $2\times2$ T-transforms) always lies inside $\mathsf{L}_n$ and is closed under left and right multiplication within $\mathsf{L}_n$ (Rajchel-Mieldzioć et al., 2021).

For circulant matrices (invariant under cycles), detailed characterizations exist for small dimensions: for $n=3$, the circulant unistochastic matrices form a monoid under multiplication; for $n=4$, $$\mathsf{U}_4 \cap \mathsf{C}_4 = \mathsf{L}_4 \cap \mathsf{C}_4$$ (Rajchel-Mieldzioć et al., 2021).

3. Generalized and Block-Unistochastic Matrices

A significant generalization is obtained by considering block-matrices. For integers $n, s$, given a unitary $U \in U(ns)$, write $U$ as an $n \times n$ block matrix with $s \times s$ blocks $U_{ij}$. Define the map

$$\Phi_{n,s}(U) = \left( \frac{1}{s} \Vert U_{ij} \Vert_F^2 \right)_{i,j=1}^n,$$

where $\Vert \cdot \Vert_F$ denotes the Frobenius norm. This defines the set of $s$-unistochastic (generalized unistochastic) matrices $$\mathsf{U}_{n,s} = \Phi_{n,s}(U(ns)) \subseteq \mathsf{B}_n$$. For $s=1$ this recovers ordinary unistochastic matrices, $\mathsf{U}_{n,1} = \mathsf{U}_n$.

These sets satisfy: $$\mathsf{U}_n = \mathsf{U}_{n,1} \subseteq \mathsf{U}_{n,s} \subseteq \mathsf{B}_n\quad \forall s\geq1.$$ A key structural property is block-convexity: if $B \in \mathsf{U}_{n,s}$ and $C \in \mathsf{U}_{n,t}$, then the convex combination $\frac{s}{s+t}B + \frac{t}{s+t}C$ lies in $\mathsf{U}_{n,s+t}$ via the direct sum of block unitaries.

The closure property holds: $$\overline{ \bigcup_{s\geq1} \mathsf{U}_{n,s} } = \mathsf{B}_n,$$ i.e. any bistochastic matrix can be approximated arbitrarily well by sequences of generalized unistochastic matrices of higher block size (Nechita et al., 2023).

Special structural results along edges and faces of the Birkhoff polytope describe the precise location of $s$-unistochastic points as combinations (with rational denominators) of permutation matrices.

There is an analogous construction for generalized orthostochastic matrices via real orthogonal block matrices and an embedding $\mathsf{U}_{n,s} \subseteq \mathsf{O}_{n,2s}$ (Nechita et al., 2023).

4. Unistochastic Matrices and Quantum Information: Frame and Entanglement Connections

Unistochastic matrices play a central role in frame theory and quantum measurement. The existence of a complex equiangular tight frame (ETF) of $N$ unit vectors in $\mathbb{C}^d$ is equivalent to the existence of a Hermitian unitary $U_N(\theta)$ such that the matrix $\mathcal{B}_N(\theta)$

$\mathcal{B}_N(\theta)_{ij} = |U_N(\theta)_{ij}|^2$

is unistochastic, with the parameter $\theta$ related to $d$ via $d = N \sin^2(\theta/2)$ (Goyeneche et al., 2016). The Gram matrix structure and resulting symmetric POVMs are determined by the properties of the underlying unistochastic matrix. These connections generalize upon taking Kronecker products of so-called "complex conference" matrices.

Explicit algorithms exist for reconstructing the generating unitary matrix from a candidate unistochastic matrix, e.g., fixed-point alternations between modulus-imposition and column orthogonalization, often with symmetrization if ETF constraints are imposed (Goyeneche et al., 2016).

In quantum information and mathematical physics, unistochastic matrices are the "classical shadows" of quantum channels represented by unitary evolution. In particular, the study of such matrices is central to the theory of quantum Markov chains, quantum-classical correspondence, eigenvalue majorization, and connections to physical invariants such as the Jarlskog parameter in CP violation (Gutkin, 2013, Rajchel-Mieldzioć et al., 2021).

5. Probabilistic and Measure-Theoretic Aspects

The Haar measure on $U(ns)$ pushes forward via the block map $\Phi_{n,s}$ to probability measures $\mu_{n,s}$ supported on $\mathsf{U}_{n,s}$. For random $B \sim \mu_{n,s}$, entrywise moments satisfy $\mathbb{E}[B_{ij}] = 1/n$. The variance

$$\operatorname{Var}(B_{ij}) = \frac{(n-1)^2}{n^2(N^2-1)} \to 0 \quad \text{as } s \to \infty, \ N = ns,$$

so as block size increases, $\mu_{n,s}$ concentrates on the flat matrix $J/n$. Thus, $\{\mu_{n,s}\}$ interpolates between the "fully quantum" measure $\mu_{n,1}$ (non-convex support) and the Dirac measure at $J/n$ as $s \to \infty$ (Nechita et al., 2023).

Random unistochastic matrices via Householder constructions (using random vectors or random orthogonal/unitary matrices) have applications in statistical hypothesis testing for mixing in dynamical systems, such as estimating critical values for ergodicity and weak-mixing properties based on the spectral gap (second largest eigenvalue) (Smith, 2012).

6. Examples, Explicit Families, and Geometry

Certain rays and faces within $\mathsf{B}_n$ are populated by explicit unistochastic matrices constructed from robust Hadamard matrices. For even $n \leq 20$, rays joining the flat matrix $W_n$ to a permutation $P$,

$B(t) = (1-t)W_n + t P,$

are unistochastic for all $t \in [0,1]$, realized by unitary matrices constructed from robust (or, for certain $n$, real or conference) Hadamards. If the Hadamard is real, these rays are orthostochastic (Rajchel-Mieldzioć et al., 2018). The approach yields explicit parameterizations of equi-entangled bases in composite Hilbert spaces.

For small $n$, geometry and inclusion are sharply characterized. In $n=3$, the entire unistochastic region admits a triangle inequality criterion; for $n=4$, $\mathsf{U}_4$ is star-shaped about the flat matrix, occupying roughly $61\%$ of $\mathsf{B}_4$ by volume, whereas the polygonal (chain-link) inequalities cover about $71\%$. For large $n$, a plausible implication is that while most bistochastic matrices satisfy the necessary bracelet inequalities, the actual unistochastic region becomes a vanishingly small subset (Rajchel-Mieldzioć et al., 2018).

7. Open Problems and Extensions

The full characterization of unistochastic matrices for $n \geq 5$ remains unresolved. Known sufficient conditions cover rays and certain triangles inside $\mathsf{B}_n$ but general criteria in higher dimensions are lacking. The study of generalized unistochastic (block) matrices achieves a form of density in the Birkhoff polytope but does not yield a complete explicit description of $\mathsf{U}_n$.

These problems are central to quantum information theory, matrix analysis, combinatorics, and the study of convex polytopes, with broad applications ranging from quantum channels, frame theory, and mixing in dynamical systems to the geometry of majorization and spectral theory (Nechita et al., 2023, Rajchel-Mieldzioć et al., 2021, Rajchel-Mieldzioć et al., 2018, Gutkin, 2013, Smith, 2012).