Werner state

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
 * $$\rho_{AB} = (U \otimes U) \rho_{AB} (U^\dagger \otimes U^\dagger)$$

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

where
 * $$P^\text{sym}_{AB} = \frac{1}{2}(I_{AB}+F_{AB}),$$
 * $$P^\text{as}_{AB} = \frac{1}{2}(I_{AB}-F_{AB}),$$

are the projectors and
 * $$F_{AB} = \sum_{ij} |i\rangle \langle j|_A \otimes |j\rangle \langle i|_B$$

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$$W_{AB}^{(p,2)} = \frac{p}{6} \begin{pmatrix}2 & 0 & 0 & 0 \\ 0&1 & 1 &0 \\0&1&1&0\\0&0&0&2\end{pmatrix} + \frac{(1-p)}{2} \begin{pmatrix}0 & 0 & 0 & 0 \\ 0&1 & -1 &0 \\0&-1&1&0\\0&0&0&0\end{pmatrix} = \begin{pmatrix} \frac{p}{3} & 0 & 0 & 0 \\ 0 & \frac{3-2p}{6} & \frac{-3+4p}{6} & 0 \\ 0 & \frac{-3+4p}{6} & \frac{3-2p}{6} & 0\\ 0 & 0 & 0 & \frac{p}{3} \end{pmatrix}. $$Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state: $$W_{AB}^{(\lambda,2)} = \lambda |\Psi^-\rangle\!\langle\Psi^-| + \frac{1-\lambda}{4}I_{AB}, \qquad |\Psi^-\rangle\equiv \frac{1}{\sqrt2}(|01\rangle-|10\rangle),$$ 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 $$\lambda \leq 1/3$$ and entangled for $$\lambda > 1/3$$.

Werner-Holevo channels
A Werner-Holevo quantum channel $$\mathcal{W}_{A\rightarrow B}^{\left( p,d\right) }$$ with parameters $$p\in\left[  0,1\right]  $$ and integer $$d\geq2$$ is defined as

\mathcal{W}_{A\rightarrow B}^{\left( p,d\right)  } = p \mathcal{W}_{A\rightarrow B}^{\text{sym}  }+\left(  1-p\right)\mathcal{W}_{A\rightarrow B}^{\text{as} }, $$ where the quantum channels $$\mathcal{W}_{A\rightarrow B}^{\text{sym} }$$ and $$\mathcal{W}_{A\rightarrow B}^{\text{as} }$$ are defined as

\mathcal{W}_{A\rightarrow B}^{\text{sym} }(X_{A})   = \frac{1}{d+1}\left[\operatorname{Tr}[X_{A}]I_{B}+\operatorname{id}_{A\rightarrow B} (T_{A}(X_{A}))\right],$$
 * $$\mathcal{W}_{A\rightarrow B}^{\text{as} }(X_{A})   =

\frac{1}{d-1}\left[\operatorname{Tr}[X_{A}]I_{B}-\operatorname{id}_{A\rightarrow B} (T_{A}(X_{A}))\right], $$ and $$T_{A}$$ denotes the partial transpose map on system A. Note that the Choi state of the Werner-Holevo channel $$\mathcal{W}_{A\rightarrow B}^{p,d}$$ is a Werner state:

\mathcal{W}_{A\rightarrow B}^{\left( p,d\right)  }(\Phi_{RA})=p  \frac{2}{d\left(  d+1\right)  }P_{RB}^{\text{sym}}+ \left( 1-p\right)\frac{2}{d\left( d-1\right)  }P_{RB}^{\text{as}}, $$ where $$\Phi_{RA} = \frac{1}{d} \sum_{i,j} |i\rangle \langle j|_R \otimes |i\rangle \langle 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.