This work devoted to extension of majorization based on semi doubly stochastic operators(denoted by
SD(L1)) as a subclass of Markov operators but a larger class of doubly stochastic operator to
L1(X) when
X is a
σ-finite measure space. We fully describe why such as these operators are more suitable for generalizing majorization relation. By using semi doubly stochastic operators we refuse Hiai's conjecture\cite{Hiai} and answer affirmatively to Mirsky's question\cite{Mirsky} about extension of doubly stochastic matrix to infinite dimension as an infinite continuous counterpart of Hardy, Littlewood and P{\'o}lya's characterization of majorization. We collect a number of results of semi doubly stochastic operators such as equi-integrability of
{Sf: S∈SD(L1)} when
f is a fix element in
L1(X) and consequently relatively weakly compact, and also strong relation of semi doubly stochastic operators and integral stochastic operators. Motivation of our work is application in quantum information theory like Nielson's Theorem, Uhlmann's Theorem and convertibility via incoherent operation. We explain extension of Uhlmann's Theorem to infinite dimension by means of semi doubly stochastic operators which leads to a quantum interpretation of majorization preserver. We introduce three types of maps which preserve majorization.
The H-unistochastic matrices are a special class of symmetric bistochastic matrices obtained by taking the square of the absolute value of each entry of a Hermitian unitary matrix. We examine the geometric relationship of the convex hull of the n by n H-unistochastic matrices relative to the larger convex set of n by n symmetric bistochastic matrices. We show that any line segment in the convex set of the n by n symmetric bistochastic matrices which passes through the centroid of this convex set must spend at least two-thirds of its length in the convex hull of the n by n H-unistochastic matrices when n is either three or four and we prove a partial result for higher n. A class of completely positive linear maps called the self-dual doubly stochastic maps is useful for studying this problem. Some results on self-dual doubly stochastic maps are given including a self-dual version of the Laudau-Streater theorem.
While classical Sinkhorn scaling applies to nonnegative matrices, we show that any complex square matrix whose element-wise absolute value has total support can be mapped to a phased doubly stochastic matrix, or alternatively embedded into a larger doubly stochastic matrix via matrix completion.
Standard Birkhoff-von~Neumann and Pauli decompositions represent such matrices as linear combinations of
O(N2) permutation or Pauli terms, leading to a large ancilla overhead in a quantum Linear Combination of Unitaries (LCU) implementation. We prove that a bottleneck variant of Birkhoff's algorithm reduces the number of permutations to
O(Nlog(N/ε)), where
ε is the
ℓ1-norm approximation error of the reconstructed matrix, and demonstrate empirically that a largest-weight greedy variant requires only
≈2N terms for dense matrices (the exact average observed is
≈2.4N). The quadratic reduction in term count directly shrinks the ancilla register from
2log2N to
log2N qubits, shortens the SELECT circuit, and is especially valuable in fixed-Hadamard LCU architectures, where---under the standard uniform-coefficient assumption---the success probability scales as
1/K; our convex decomposition achieves
α=1 exactly and therefore avoids this penalty. The approach enables compact quantum implementations of dense operators appearing in optimal transport, non-Hermitian simulation, and other settings amenable to Sinkhorn preconditioning.
Furthermore, because the decomposition is a convex combination, the LCU normalization constant for the scaled matrix
S is exactly
αS=1, and the uniform superposition is an eigenvector of
S with eigenvalue~1. This structure can be exploited to achieve high success probability without amplitude amplification in many practical scenarios, including quantum walks and Markov chain simulations.