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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,

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

1 Parametrization
2 Properties
3 See also
4 References

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.