Eigenvalue Distributions of Reduced Density Matrices

Matthias Christandl; Brent Doran; Stavros  Kousidis; Michael Walter

doi:10.1007/s00220-014-2144-4

Eigenvalue Distributions of Reduced Density Matrices

Matthias Christandl, Brent Doran, Stavros Kousidis, Michael Walter

Department of Mathematical Sciences

24 Citations (Scopus)

Abstract

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution’s support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

Original language	English
Journal	Communications in Mathematical Physics
Volume	332
Pages (from-to)	1-52
ISSN	0010-3616
DOIs	https://doi.org/10.1007/s00220-014-2144-4
Publication status	Published - Nov 2014

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title = "Eigenvalue Distributions of Reduced Density Matrices",

abstract = "Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution{\textquoteright}s support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.",

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T1 - Eigenvalue Distributions of Reduced Density Matrices

AU - Christandl, Matthias

AU - Doran, Brent

AU - Kousidis, Stavros

AU - Walter, Michael

PY - 2014/11

Y1 - 2014/11

N2 - Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution’s support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

AB - Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution’s support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

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DO - 10.1007/s00220-014-2144-4

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SN - 0010-3616

VL - 332

SP - 1

EP - 52

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

ER -

Eigenvalue Distributions of Reduced Density Matrices

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