Werner State

A Werner state is a -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 $\rho_$ that satisfies
:$\rho_ = (U \otimes U) \rho_ (U^\dagger \otimes U^\dagger)$
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 $W_^$ is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight p \in ,1/math> being the main parameter that defines the state, in addition to the dimension $d \geq 2$:
:$W_^ = p \frac P^\text_ + (1-p) \frac P^\text_,$
where
:$P^\text_ = \frac(I_+F_),$
:$P^\text_ = \frac(I_-F_),$
are the projectors and
:$F_ = \sum_ , i\rangle \langle j, _A \otimes , j\rangle \langle i, _B$
is the permutation or flip operator that exchanges the two subsystems ''A'' and ''B''.

Werner states are separable for ''p'' ≥ 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
:$\rho_ = \frac(I_ - \alpha F_),$
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$W_^ = \frac \begin2 & 0 & 0 & 0 \\ 0&1 & 1 &0 \\0&1&1&0\\0&0&0&2\end + \frac \begin0 & 0 & 0 & 0 \\ 0&1 & -1 &0 \\0&-1&1&0\\0&0&0&0\end = \begin \frac & 0 & 0 & 0 \\ 0 & \frac & \frac & 0 \\ 0 & \frac & \frac & 0\\ 0 & 0 & 0 & \frac \end.$Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state: $W_^ = \lambda , \Psi^-\rangle\!\langle\Psi^-, + \fracI_, \qquad , \Psi^-\rangle\equiv \frac(, 01\rangle-, 10\rangle),$ where \lambda\in 1/3,1/math> (or, confining oneself to positive values, \lambda\in ,1/math>) 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 $\mathcal_^$ with parameters p\in\left 0,1\right and integer $d\geq2$
is defined as

:$\mathcal_^ = p \mathcal_^+\left( 1-p\right)\mathcal_^,$
where the quantum channels $\mathcal_^$ and
$\mathcal_^$ are defined as
:
\mathcal_^(X_) =
\frac\left operatorname[X__+\operatorname_
(T_(X_))\right">_.html" ;"title="operatorname[X_">operatorname[X__+\operatorname_
(T_(X_))\right
:\mathcal_^(X_) =
\frac\left operatorname[X__-\operatorname_
(T_(X_))\right],

and $T_$ denotes the partial transpose map on system ''A''. Note that the
Channel-state duality, Choi state of the Werner-Holevo channel $\mathcal_^$
is a Werner state:
:$\mathcal_^(\Phi_)=p \fracP_^+ \left( 1-p\right)\fracP_^,$
where $\Phi_ = \frac \sum_ , i\rangle \langle j, _R \otimes , i\rangle \langle j, _A$.

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.

References

Quantum states
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