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Quantum Simulation Entanglement Theory Quantum Correlations

The Birkhoff theorem for unitary matrices of prime-power dimension

arXiv
Authors: Alexis De Vos, Stijn De Baerdemacker

Year

2018

Paper ID

39476

Status

Preprint

Abstract Read

~2 min

Abstract Words

122

Citations

N/A

Abstract

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 weights are equal to unity. If the dimension n of the unitary matrix equals a power of a prime p, i.e.\ if n=pw, then the Birkhoff decomposition does not need all n! possible permutation matrices, as the epicirculant permutation matrices suffice. This group of permutation matrices is isomorphic to the general affine group GA(w,p) of order only pw\(pw-1\)\(pw-p\)...\(pw-pw-1\) ll left\(pw right\)!.

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  • 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...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:1812.08833 [2] arXiv https://arxiv.org/abs/1812.08833 [3] Publisher https://arxiv.org/abs/1812.08833

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