Work

Uniqueness of Purifications Is Equivalent to Haag Duality

Year: 2026

Type: article

Abstract: The uniqueness of purifications of quantum states on a system A up to local unitary transformations on a purifying system B is central to quantum information theory. We show that, if the two systems are modeled by commuting von Neumann algebras M_{A} and M_{B} on a Hilbert space H, then uniqueness of purifications is equivalent to Haag duality M_{A}=M_{B}^{'}. In particular, the uniqueness of purifications can fail in systems with infinitely many degrees of freedom-even when M_{A} and M_{B} are commuting factors that jointly generate B(H) and hence allow for local tomography of all density matrices on H. We present a simple argument showing that the uniqueness of purifications, and, hence, Haag duality, fail in ground state sectors of topologically ordered models at renormalization group fixed points on infinite two-dimensional lattices, partitioned into the union of two spatially separated cones and its complement.

Source: Physical Review Letters

DOI: 10.1103/d7nm-gx37

Authors: Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming

Institution: Leibniz University Hannover

Language: English

Topic: Advanced Operator Algebra Research

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,750

Funder: Niedersächsisches Ministerium für Wissenschaft und Kultur