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Birkhoffs theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group {bf E}$_{2^{2w+1}}^+$ of order $2^{2w+1}$. An associated projective group of order $2^{2w}$ equally suffices.
It was shown recently that Birkhoffs theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.
The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called scaled matrix which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equa
The Birkhoffs theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. It can be proven that a similar theorem holds for unitary matrices with equal line sums for prime dimensions.
Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into two diagon
The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the