TY - JOUR

T1 - The Schmidt rank for the commuting operator framework

AU - van Luijk, Lauritz

AU - Schwonnek, René

AU - Stottmeister, Alexander

AU - Werner, Reinhard F.

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/6/18

Y1 - 2024/6/18

N2 - In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: the vacuum in quantum field theory, Araki–Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework.

AB - In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: the vacuum in quantum field theory, Araki–Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework.

KW - quant-ph

KW - math-ph

KW - math.MP

KW - math.OA

UR - http://www.scopus.com/inward/record.url?scp=85196296024&partnerID=8YFLogxK

U2 - 10.1007/s00220-024-05011-9

DO - 10.1007/s00220-024-05011-9

M3 - Article

VL - 405

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

M1 - 152

ER -