Comment on some results of Erdahl and the convex structure of reduced density matrices

Abstract

In J. Math. Phys. 13, 1608-1621 (1972), Erdahl considered the convex structure of the set of $N$-representable 2-body reduced density matrices in the case of fermions. Some of these results have a straightforward extension to the $m$-body setting and to the more general quantum marginal problem. We describe these extensions, but can not resolve a problem in the proof of Erdahl's claim that every extreme point is exposed in finite dimensions. Nevertheless, we can show that when $2m\ge N$ every extreme point of the set of $N$-representable $m$-body reduced density matrices has a unique pre-image in both the symmetric and anti-symmetric setting. Moreover, this extends to the quantum marginal setting for a pair of complementary $m$-body and $(N-m)$-body reduced density matrices.

19 pages, 0 figures. Comments are welcome

References in corpus (6)

• N-representability is QMA-complete
• The Pauli principle revisited
• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Quantum marginal problem and representations of the symmetric group
• Three-party pure quantum states are determined by two two-party reduced states
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