A Werner state is a d 2 {\displaystyle d^{2}} × d 2 {\displaystyle d^{2}} -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form U ⊗ U {\displaystyle U\otimes U} . That is, it is a bipartite quantum state ρ A B {\displaystyle \rho _{AB}} that satisfies
ρ A B = ( U ⊗ U ) ρ A B ( U † ⊗ U † ) {\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 W A B ( p , d ) {\displaystyle W_{AB}^{(p,d)}} is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} being the main parameter that defines the state, in addition to the dimension d ≥ 2 {\displaystyle d\geq 2} :
W A B ( p , d ) = p 2 d ( d + 1 ) P A B sym + ( 1 − p ) 2 d ( d − 1 ) P A B as , {\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
P A B sym = 1 2 ( I A B + F A B ) , {\displaystyle P_{AB}^{\text{sym}}={\frac {1}{2}}(I_{AB}+F_{AB}),}
P A B as = 1 2 ( I A B − F A B ) , {\displaystyle P_{AB}^{\text{as}}={\frac {1}{2}}(I_{AB}-F_{AB}),}
are the projectors and
F A B = ∑ i j | i ⟩ ⟨ j | A ⊗ | j ⟩ ⟨ i | B {\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
ρ A B = 1 d 2 − d α ( I A B − α F A B ) , {\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
α = ( ( 1 − 2 p ) d + 1 ) / ( 1 − 2 p + d ) . {\displaystyle \alpha =((1-2p)d+1)/(1-2p+d).}
Two-qubit example Two-qubit Werner states, corresponding to d = 2 {\displaystyle d=2} above, can be written explicitly in matrix form as W A B ( p , 2 ) = p 6 ( 2 0 0 0 0 1 1 0 0 1 1 0 0 0 0 2 ) + ( 1 − p ) 2 ( 0 0 0 0 0 1 − 1 0 0 − 1 1 0 0 0 0 0 ) = ( p 3 0 0 0 0 3 − 2 p 6 − 3 + 4 p 6 0 0 − 3 + 4 p 6 3 − 2 p 6 0 0 0 0 p 3 ) . {\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: W A B ( λ , 2 ) = λ | Ψ − ⟩ ⟨ Ψ − | + 1 − λ 4 I A B , | Ψ − ⟩ ≡ 1 2 ( | 01 ⟩ − | 10 ⟩ ) , {\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 λ ∈ [ − 1 / 3 , 1 ] {\displaystyle \lambda \in [-1/3,1]} (or, confining oneself to positive values, λ ∈ [ 0 , 1 ] {\displaystyle \lambda \in [0,1]} ) is related to p {\displaystyle p} by λ = ( 3 − 4 p ) / 3 {\displaystyle \lambda =(3-4p)/3} . Then, two-qubit Werner states are separable for λ ≤ 1 / 3 {\displaystyle \lambda \leq 1/3} and entangled for λ > 1 / 3 {\displaystyle \lambda >1/3} .
Werner-Holevo channels A Werner-Holevo quantum channel W A → B ( p , d ) {\displaystyle {\mathcal {W}}_{A\rightarrow B}^{\left(p,d\right)}} with parameters p ∈ [ 0 , 1 ] {\displaystyle p\in \left[0,1\right]} and integer d ≥ 2 {\displaystyle d\geq 2}
is defined as
W A → B ( p , d ) = p W A → B sym + ( 1 − p ) W A → B as , {\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 W A → B sym {\displaystyle {\mathcal {W}}_{A\rightarrow B}^{\text{sym}}} and
W A → B as {\displaystyle {\mathcal {W}}_{A\rightarrow B}^{\text{as}}} are defined as
W A → B sym ( X A ) = 1 d + 1 [ Tr [ X A ] I B + id A → B ( T A ( X A ) ) ] , {\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],}
W A → B as ( X A ) = 1 d − 1 [ Tr [ X A ] I B − id A → B ( T A ( X A ) ) ] , {\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 T A {\displaystyle T_{A}} denotes the partial transpose map on system A. Note that the Choi state of the Werner-Holevo channel W A → B p , d {\displaystyle {\mathcal {W}}_{A\rightarrow B}^{p,d}}
is a Werner state:
W A → B ( p , d ) ( Φ R A ) = p 2 d ( d + 1 ) P R B sym + ( 1 − p ) 2 d ( d − 1 ) P R B as , {\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 Φ R A = 1 d ∑ i , j | i ⟩ ⟨ j | R ⊗ | i ⟩ ⟨ j | A {\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. An N-party Werner state is a state that is invariant under U ⊗ U ⊗ ⋯ ⊗ U {\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.
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