Werner state

Template:Short description A Werner state is a $d^2$ × $d^2$-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form $U\otimes U$. That is, it is a bipartite quantum state ${\rho }_{AB}$ that satisfies

${\rho }_{AB}=(U\otimes U){\rho }_{AB}(U^{†}\otimes U^{†})$

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

General definition

Every Werner state $W_{AB}^{(p,d)}$ is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight $p\in [0,1]$ being the main parameter that defines the state, in addition to the dimension $d\ge 2$:

$W_{AB}^{(p,d)}=p\frac{2}{d(d+1)}{P}_{AB}^{\text{sym}}+(1-p)\frac{2}{d(d-1)}{P}_{AB}^{\text{as}},$

where

$P_{AB}^{\text{sym}}=\frac{1}{2}(I_{AB}+F_{AB}),$

$P_{AB}^{\text{as}}=\frac{1}{2}(I_{AB}-F_{AB}),$

are the projectors and

$F_{AB}=\sum _{ij}|i⟩⟨j{|}_A\otimes |j⟩⟨i{|}_B$

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

Werner states are separable for p ≥ 1⁄2 and entangled for p < 1⁄2. 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

${\rho }_{AB}=\frac{1}{d^2-d\alpha }(I_{AB}-\alpha F_{AB}),$

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

$\alpha =((1-2p)d+1)/(1-2p+d).$

Two-qubit example

Two-qubit Werner states, corresponding to $d=2$ above, can be written explicitly in matrix form as$$W_{AB}^{(p,2)}=\frac{p}{6}\left(\begin{array}{cccc}2& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 2\end{array}\right)+\frac{(1-p)}{2}\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& -1& 0\\ 0& -1& 1& 0\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}\frac{p}{3}& 0& 0& 0\\ 0& \frac{3-2p}{6}& \frac{-3+4p}{6}& 0\\ 0& \frac{-3+4p}{6}& \frac{3-2p}{6}& 0\\ 0& 0& 0& \frac{p}{3}\end{array}\right).$$Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state: $$W_{AB}^{(\lambda ,2)}=\lambda |{\Psi }^{-}⟩⟨{\Psi }^{-}|+\frac{1-\lambda }{4}{I}_{AB},|{\Psi }^{-}⟩\equiv \frac{1}{\sqrt{2}}(|01⟩-|10⟩),$$ where $\lambda \in [-1/3,1]$ (or, confining oneself to positive values, $\lambda \in [0,1]$) is related to $p$ by $\lambda =(3-4p)/3$. Then, two-qubit Werner states are separable for $\lambda \le 1/3$ and entangled for $\lambda >1/3$.

Werner-Holevo channels

A Werner-Holevo quantum channel ${𝒲}_{A\to B}^{(p,d)}$ with parameters $p\in [0,1]$ and integer $d\ge 2$ is defined as

${𝒲}_{A\to B}^{(p,d)}=p{𝒲}_{A\to B}^{\text{sym}}+(1-p){𝒲}_{A\to B}^{\text{as}},$

where the quantum channels ${𝒲}_{A\to B}^{\text{sym}}$ and ${𝒲}_{A\to B}^{\text{as}}$ are defined as

${𝒲}_{A\to B}^{\text{sym}}(X_A)=\frac{1}{d+1}[\mathrm{Tr}[X_A]I_B+{\mathrm{id}}_{A\to B}(T_A(X_A))],$

${𝒲}_{A\to B}^{\text{as}}(X_A)=\frac{1}{d-1}[\mathrm{Tr}[X_A]I_B-{\mathrm{id}}_{A\to B}(T_A(X_A))],$

and $T_A$ denotes the partial transpose map on system A. Note that the Choi state of the Werner-Holevo channel ${𝒲}_{A\to B}^{p,d}$ is a Werner state:

${𝒲}_{A\to B}^{(p,d)}({\Phi }_{RA})=p\frac{2}{d(d+1)}{P}_{RB}^{\text{sym}}+(1-p)\frac{2}{d(d-1)}{P}_{RB}^{\text{as}},$

where ${\Phi }_{RA}=\frac{1}{d}\sum _{i,j}|i⟩⟨j{|}_R\otimes |i⟩⟨j{|}_A$.

Multipartite Werner states

Werner states can be generalized to the multipartite case. An N-party Werner state is a state that is invariant under $U\otimes U\otimes \cdots \otimes U$ 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.