A Werner state[1] 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
\( {\displaystyle \rho _{AB}=(U\otimes U)\rho _{AB}(U^{\dagger }\otimes U^{\dagger })} \)
for all unitary operators U acting on d-dimensional Hilbert space.
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
\( {\displaystyle \alpha =((1-2p)d+1)/(1-2p+d).} \)
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 \( T_{{A}} \) 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 \( U\otimes U\otimes ...\otimes U \) 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
Reinhard F. Werner (1989). "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A. 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/PhysRevA.40.4277. PMID 9902666.
Reinhard F. Werner and Alexander S. Holevo (2002). "Counterexample to an additivity conjecture for output purity of quantum channels". Journal of Mathematical Physics. 43 (9): 4353–4357. doi:10.1063/1.1498491.
Mark Fannes, B. Haegeman, Milan Mosonyi, and D. Vanpeteghem (2004). "Additivity of minimal entropy out- put for a class of covariant channels". arXiv:quant-ph/0410195.
Debbie Leung and William Matthews (2015). "On the power of PPT-preserving and non-signalling codes". IEEE Transactions on Information Theory. 61 (8): 4486–4499. doi:10.1109/TIT.2015.2439953.
Eggeling et al. (2008)
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