Invariance of reduced density matrices under Local Unitary operations

2013/07/03 by Ana María Guerra Martins, Martins, A. M.
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Physical sciences #Matrix Theory and Algorithms #Quantum Information and Cryptography #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.1307.0948

openalex publication_date 2013/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of \it cyclic local operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group SU(2)^{⊗ n}. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.

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