23 problems

An almost Hadamard matrix in the complex sense is the notion introduced in the cited work, while a complex Hadamard matrix is a unitary matrix whose entries have equal modulus afte…

Diagonalizing sequence conjecture. For every $U_0\in\operatorname{U}(2)$, the sequences satisfy

Let $U(d)$ denote a Haar-random $d\times d$ unitary matrix, and let $t$ be a positive integer. Hunter-Jones's conjecture. The permanent moment satisfies … The source attributes thi…

Egan–Ó Catháin conjecture. Is it true that

Let $U\in\sqrt{N}U_N$ be a rescaled unitary matrix. A strict almost Hadamard matrix, an isolated complex Hadamard matrix, and a complex Hadamard matrix with minimal AHM/CHM defect…

Let $\mathsf{U}(d)$ be the unitary group on a $d$-dimensional Hilbert space, and call a unitary matrix chaotic when it has the property studied in the paper, namely maximal PVM-dyn…

Diagonal-unitary scaling conjecture. There exist diagonal unitary matrices $D_1,D_2$ such that $D_1UD_2$ is doubly quasistochastic.

Let $\mathcal U_{\mathrm{unital}}$ denote the set of bipartite unitary operators whose associated quantum channels preserve tensor product algebras. The choice of the pattern matri…

Let $A$ be an $n\times n$ positive semidefinite matrix. Marcus–Minc max-per-unitary conjecture. The maximum of … over all $n\times n$ unitary matrices $U$ is attained when all diag…

Let $U$ be an arbitrary $n\times n$ unitary matrix such that each line sum equals $1$: … Let $P_m$ be the $n!$ permutation matrices of dimension $n$. De Vos–De Baerdemaecker conjec…

Let $U$ be a unitary matrix. A unitary diagonal matrix is a diagonal matrix whose diagonal entries have modulus $1$, and a unitary doubly stochastic matrix is a unitary matrix whos…

Unitary near-invariance conjecture. An analogous near-invariance result should hold for arbitrary unitary matrices under this relaxed condition, with the exceptional matrices broad…

Let $U\in U(n)$ be generic, and consider its normal forms $U=e^{i\varphi}D_1AD_2$, where $D_1$ and $D_2$ are unitary diagonal matrices with $(D_i)_{11}=1$ and $A$ has all row and c…

Let $U$ be an $n\times n$ unitary matrix. A matrix is doubly stochastic when all entries in each row and column sum to one, and a matrix is unitary diagonal when it is both diagona…

Convergence conjecture. The numerical algorithm converges to a unit line-sum matrix, that is, to a member of $\operatorname{XU}(n)$.

Numerical solvability conjecture. For every $n$ and every member of $U(n)$, the asymptotic scaling procedure provides a numerical solution.

Two-factor decomposition conjecture. Even the stronger bound $p\leq 2$ should hold.

Global-minimum conjecture. The global minimum of the $\Psi$ landscape is attained at a matrix $B$ with $\Psi=0$, and hence at a desired unit line-sum matrix.

Unitary matrix scaling conjecture. A similar procedure should allow, starting from any unitary matrix, finding a scaled matrix that is unitary and has all line sums equal to $1$.

Near-Haar spectral-distribution conjecture. The spectral distributions of $C_n$ should be close to the corresponding Haar-unitary spectral distributions when $U_n$ is distributed s…

Let $U$ be a unitary matrix with a fixed Kronecker product structure, meaning that the sequence of sizes of its Kronecker factors is fixed up to order. In either of the two cases c…

Edge convergence conjecture. The largest and smallest eigenvalues converge to the edge of the limiting support.

Let $d$ and $k$ be fixed, let ${\bf v}^{(1)},\ldots,{\bf v}^{(k)}$ be fixed vectors, and let $U$ vary over unitary matrices. Write ${\mathcal V}$ for the associated variety and let…