Werner state

A Werner state^[1] is a ${\displaystyle d^{2}}$ × ${\displaystyle d^{2}}$-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form ${\displaystyle U\otimes U}$. That is, it is a bipartite quantum state ${\displaystyle \rho _{AB}}$ that satisfies

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

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

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

where

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

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

are the projectors and

${\displaystyle F_{AB}=\sum _{ij}|i\rangle \langle j|_{A}\otimes |j\rangle \langle i|_{B}}$

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

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

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

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

Two-qubit example

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

Werner-Holevo channels

A Werner-Holevo quantum channel ${\displaystyle {\mathcal {W}}_{A\rightarrow B}^{\left(p,d\right)}}$ with parameters ${\displaystyle p\in \left[0,1\right]}$ and integer ${\displaystyle d\geq 2}$ is defined as ^{[2] [3] [4]}

${\displaystyle {\mathcal {W}}_{A\rightarrow B}^{\left(p,d\right)}=p{\mathcal {W}}_{A\rightarrow B}^{\text{sym}}+\left(1-p\right){\mathcal {W}}_{A\rightarrow B}^{\text{as}},}$

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

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

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

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

${\displaystyle {\mathcal {W}}_{A\rightarrow B}^{\left(p,d\right)}(\Phi _{RA})=p{\frac {2}{d\left(d+1\right)}}P_{RB}^{\text{sym}}+\left(1-p\right){\frac {2}{d\left(d-1\right)}}P_{RB}^{\text{as}},}$

where ${\displaystyle \Phi _{RA}={\frac {1}{d}}\sum _{i,j}|i\rangle \langle j|_{R}\otimes |i\rangle \langle j|_{A}}$.

Multipartite Werner states

Werner states can be generalized to the multipartite case.^[5] An N-party Werner state is a state that is invariant under ${\displaystyle U\otimes U\otimes \cdots \otimes U}$ 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.