Werner state - Alchetron, The Free Social Encyclopedia

A Werner state is a d × d-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form U ⊗ U . That is, it is a quantum state ρ that satisfies

ρ = ( U ⊗ U ) ρ ( U † ⊗ U † )

for all unitary operators U acting on d-dimensional Hilbert space.

Every Werner state is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight p_sym being the only parameter that defines the state.

ρ = p sym 2 d 2 + d P sym + ( 1 − p sym ) 2 d 2 − d P as ,

where

P sym = 1 2 ( 1 + P ) , P as = 1 2 ( 1 − P ) ,

are the projectors and

P = ∑ i j | i ⟩ ⟨ j | ⊗ | j ⟩ ⟨ i |

is the permutation operator that exchanges the two subsystems.

Werner states are separable for p_sym ≥ ¹⁄₂ and entangled for p_sym < ¹⁄₂. All entangled Werner states violate the 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

ρ = 1 d 2 − d α ( 1 − α P ) ,

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

α = ( ( 1 − 2 p sym ) d + 1 ) / ( 1 − 2 p sym + d ) .

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 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

Werner state Wikipedia

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