Isotropic state

An isotropic state[1] is a d \times d dimensional bipartite quantum state that is invariant under any unitary of the form U \otimes U^*, where * denotes complex conjugate. That is, any state with the property that for any unitary U on one part of the system,

\rho = (U \otimes U^*) \rho (U^\dagger \otimes (U^*)^\dagger).

Parametrization

The isotropic states is a one-parameter family of states and can be written as

(1-\alpha) I/d^2 + \alpha |\phi^+\rangle \langle \phi^+|,

where -1/(d^2-1) \leq \alpha \leq 1 and |\phi^+\rangle = \frac{1}{\sqrt{d}} \sum_j |j\rangle \otimes |j \rangle i.e. a mixture (or pseudomixture for α < 0) of the maximally mixed state and the maximally entangled state.

In terms of the singlet fraction F, the fidelity to the maximally entangled state, the isotropic states can be parametrized as

\rho = \frac{d^2}{d^2-1}\left[ (1-F) I/d^2 + (F-1/d^2) |\phi^+\rangle \langle \phi^+| \right]

where 0 ≤ F ≤ 1.

Properties

Isotropic states are separable for F ≤ 1/d or equivalently α ≤ 1/(d+1), and entangled otherwise. All entangled isotropic states violate the reduction separability criterion, and are therefore also distillable.

See also

References

[1] Reduction criterion of separability and limits for a class of distillation protocols, Michał Horodecki, Paweł Horodecki, Publication Phys. Rev. A 59, 4206 - 4216 (1999) pre-print

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