Invariance of reduced density matrices under Local Unitary operations

TL;DR

This paper characterizes local unitary operators that leave the set of one-qubit reduced density matrices invariant in multi-qubit states, identifying a subgroup structure and providing methods to determine state equivalence classes.

Contribution

It derives necessary and sufficient conditions for LU invariance of reduced states and characterizes the subgroup of operators with this property, including their dependence on reduced state types.

Findings

LU operators with invariance form a subgroup of the local unitary group.

The subgroup's dimension varies with the purity of reduced states.

Method to compute LU operators that fix reduced density matrices and classify states.

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 $SU(2)^{\otimes n}$. 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.