On an extension problem for density matrices

Eric A. Carlen; Joel L. Lebowitz; Elliott H. Lieb

doi:10.1063/1.4808218

On an extension problem for density matrices

Eric A. Carlen, Joel L. Lebowitz, Elliott H. Lieb

Mathematics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

We investigate the problem of the existence of a density matrix ρ123 on a Hilbert space H₁⊗H₂⊗H₃ with given partial traces ρ12 = Tr3 ρ123 and ρ23 = Tr1 ρ123. While we do not solve this problem completely, we offer partial results in the form of some necessary and some sufficient conditions on ρ12 and ρ23. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, Tr1 ρ12 = Tr3ρ23.

Original language	English (US)
Article number	062103
Journal	Journal of Mathematical Physics
Volume	54
Issue number	6
DOIs	https://doi.org/10.1063/1.4808218
State	Published - Jun 3 2013

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1063/1.4808218

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title = "On an extension problem for density matrices",

abstract = "We investigate the problem of the existence of a density matrix ρ123 on a Hilbert space H1⊗H2⊗H3 with given partial traces ρ12 = Tr3 ρ123 and ρ23 = Tr1 ρ123. While we do not solve this problem completely, we offer partial results in the form of some necessary and some sufficient conditions on ρ12 and ρ23. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, Tr1 ρ12 = Tr3ρ23.",

author = "Carlen, {Eric A.} and Lebowitz, {Joel L.} and Lieb, {Elliott H.}",

note = "Funding Information: This work is partially supported by U.S. National Science Foundation Grant Nos. DMS 1201354 (E.A.C.), DMR 113501 and AFOSR FA9550-10-1-0131 (J.L.L.), and PHY 0965859 and a grant from the Simons Foundation (Grant No. 230207) (E.H.L.).",

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AU - Carlen, Eric A.

AU - Lebowitz, Joel L.

AU - Lieb, Elliott H.

N1 - Funding Information: This work is partially supported by U.S. National Science Foundation Grant Nos. DMS 1201354 (E.A.C.), DMR 113501 and AFOSR FA9550-10-1-0131 (J.L.L.), and PHY 0965859 and a grant from the Simons Foundation (Grant No. 230207) (E.H.L.).

PY - 2013/6/3

Y1 - 2013/6/3

N2 - We investigate the problem of the existence of a density matrix ρ123 on a Hilbert space H1⊗H2⊗H3 with given partial traces ρ12 = Tr3 ρ123 and ρ23 = Tr1 ρ123. While we do not solve this problem completely, we offer partial results in the form of some necessary and some sufficient conditions on ρ12 and ρ23. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, Tr1 ρ12 = Tr3ρ23.

AB - We investigate the problem of the existence of a density matrix ρ123 on a Hilbert space H1⊗H2⊗H3 with given partial traces ρ12 = Tr3 ρ123 and ρ23 = Tr1 ρ123. While we do not solve this problem completely, we offer partial results in the form of some necessary and some sufficient conditions on ρ12 and ρ23. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, Tr1 ρ12 = Tr3ρ23.

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ER -

On an extension problem for density matrices

Abstract

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