Physics:Werner state

A Werner state is a [math]\displaystyle{ d^2 }[/math] × [math]\displaystyle{ d^2 }[/math]-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form [math]\displaystyle{ U \otimes U }[/math]. That is, it is a bipartite quantum state [math]\displaystyle{ \rho_{AB} }[/math] that satisfies

[math]\displaystyle{ \rho_{AB} = (U \otimes U) \rho_{AB} (U^\dagger \otimes U^\dagger) }[/math]

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

[math]\displaystyle{ 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}, }[/math]

where

[math]\displaystyle{ P^\text{sym}_{AB} = \frac{1}{2}(I_{AB}+F_{AB}), }[/math]

[math]\displaystyle{ P^\text{as}_{AB} = \frac{1}{2}(I_{AB}-F_{AB}), }[/math]

are the projectors and

[math]\displaystyle{ F_{AB} = \sum_{ij} |i\rangle \langle j|_A \otimes |j\rangle \langle i|_B }[/math]

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

[math]\displaystyle{ \rho_{AB} = \frac{1}{d^2-d \alpha}(I_{AB} - \alpha F_{AB}), }[/math]

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

[math]\displaystyle{ \alpha = ((1-2p)d+1)/(1-2p+d) . }[/math]

Two-qubit example

Two-qubit Werner states, corresponding to [math]\displaystyle{ d=2 }[/math] above, can be written explicitly in matrix form as[math]\displaystyle{ 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}. }[/math]Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state: [math]\displaystyle{ 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), }[/math] where [math]\displaystyle{ \lambda\in[-1/3,1] }[/math] (or, confining oneself to positive values, [math]\displaystyle{ \lambda\in[0,1] }[/math]) is related to [math]\displaystyle{ p }[/math] by [math]\displaystyle{ \lambda=(3-4p)/3 }[/math]. Then, two-qubit Werner states are separable for [math]\displaystyle{ \lambda \leq 1/3 }[/math] and entangled for [math]\displaystyle{ \lambda \gt 1/3 }[/math].

Werner-Holevo channels

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

[math]\displaystyle{ \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} }, }[/math]

where the quantum channels [math]\displaystyle{ \mathcal{W}_{A\rightarrow B}^{\text{sym} } }[/math] and [math]\displaystyle{ \mathcal{W}_{A\rightarrow B}^{\text{as} } }[/math] are defined as

[math]\displaystyle{ \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], }[/math]

[math]\displaystyle{ \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], }[/math]

and [math]\displaystyle{ T_{A} }[/math] denotes the partial transpose map on system A. Note that the Choi state of the Werner-Holevo channel [math]\displaystyle{ \mathcal{W}_{A\rightarrow B}^{p,d} }[/math] is a Werner state:

[math]\displaystyle{ \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}}, }[/math]

where [math]\displaystyle{ \Phi_{RA} = \frac{1}{d} \sum_{i,j} |i\rangle \langle j|_R \otimes |i\rangle \langle j|_A }[/math].

Multipartite Werner states

Werner states can be generalized to the multipartite case. An N-party Werner state is a state that is invariant under [math]\displaystyle{ U \otimes U \otimes \cdots \otimes U }[/math] 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.