A Werner state is a -dimensional bipartite quantum state density matrix that is invariant under all unitary operators. That is, it is a bipartite quantum state that satisfies

for all unitary operators U acting on d-dimensional Hilbert space. These states were first developed by Reinhard F. Werner in 1989.

General definition

is a mixture of projectors:

where

are the projectors and

is the permutation or flip operator that exchanges the two subsystems A and B.

Werner states are separable for p ≥ and entangled for p < . All entangled Werner states violate the PPT separability criterion, but for d ≥ 3 no Werner state violates the weaker reduction criterion. Werner states can be parametrized in different ways. One way of writing them is

where the new parameter α varies between −1 and 1 and relates to p as

Two-qubit example

Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state.

Werner-Holevo channels

A Werner-Holevo quantum channel is defined as

are defined as

denotes the partial transpose map on system A. Note that the Choi state is a Werner state:

.

Multipartite Werner states

Werner states can be generalized to the multipartite case. An N for any unitary U on a single subsystem. The Werner state is no longer described by a single parameter, but by N! − 1 parameters, and is a linear combination of the N! different permutations on N systems.

References