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A '''Werner state'''<ref>
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{{cite  journal
|author=Reinhard F. Werner
|year=1989
|title=Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
|journal=[[Physical Review A]]
|volume=40 |issue=8 |pages=4277–4281
|doi=10.1103/PhysRevA.40.4277
|bibcode=1989PhRvA..40.4277W
|pmid=9902666
}}</ref> is a {{nowrap|''d'' × ''d''}}-dimensional bipartite [[quantum state]] density matrix that is [[Invariant (mathematics)|invariant]] under all [[unitary operator]]s of the form <math>U \otimes U</math>. That is, it is a [[quantum state]] ''ρ'' that satisfies
:<math>\rho = (U \otimes U) \rho (U^\dagger \otimes U^\dagger)</math>
for all unitary operators ''U'' acting on ''d''-dimensional Hilbert space.
 
Every Werner state is a mixture of [[projection (linear algebra)|projectors]] onto the [[symmetric space|symmetric]] and [[antisymmetric space|antisymmetric]] [[subspace topology|subspaces]], with the relative weight ''p''<sub>sym</sub> being the only parameter that defines the state.  
:<math>\rho = p_\text{sym} \frac{2}{d^2 + d} P_\text{sym} + (1-p_\text{sym}) \frac{2}{d^2 - d} P_\text{as},</math>
where
:<math>P_\text{sym} = \frac{1}{2}(1+P),</math> <math>P_\text{as} = \frac{1}{2}(1-P),</math>
are the projectors and
:<math>P = \sum_{ij} |i\rangle \langle j| \otimes |j\rangle \langle i|</math>
is the permutation operator that exchanges the two subsystems.
 
Werner states are [[Separable states|separable]] for ''p''<sub>sym</sub> ≥ {{frac|1|2}} and entangled for ''p''<sub>sym</sub> < {{frac|1|2}}. All entangled Werner states violate the [[Peres-Horodecki criterion|PPT separability criterion]], but for ''d'' ≥ 3 no Werner states violate the weaker [[reduction criterion]]. Werner states can be parametrized in different ways. One way of writing them is
:<math>\rho = \frac{1}{d^2-d \alpha}(1 - \alpha P),</math>
where the new parameter ''α'' varies between −1 and 1 and relates to ''p''<sub>sym</sub> as
:<math>\alpha = ((1-2p_\text{sym})d+1)/(1-2p_\text{sym}+d) .</math>
 
== Multipartite Werner states ==
 
Werner states can be generalized to the [[multipartite]] case.<ref>Eggeling ''et al.'' (2008)</ref> An ''N''-party Werner state is a state that is invariant under <math>U \otimes U \otimes ... \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.
 
== References ==
<references/>
 
{{DEFAULTSORT:Werner State}}
[[Category:Quantum mechanics]]

Latest revision as of 01:04, 9 November 2014

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