An **isotropic state**horodecki97reduction is a *d* × *d* dimensional bipartite quantum state that is invariant under any unitary of the form *U* ⊗ *U*^{ * }, where * denotes complex conjugate. That is, any state with the property that for any unitary U on one part of the system,

*ρ* = (*U* ⊗ *U*^{ * })*ρ*(*U*^{ † } ⊗ (*U*^{ * })^{ † }).

### Parametrization

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

(1 − *α*)*I*/*d*^{2} + *α*∣*ϕ*^{ + }⟩⟨*ϕ*^{ + }∣,

where − 1/(*d*^{2} − 1) ≤ *α* ≤ 1 and $|\phi^+\rangle = \frac{1}{\sqrt{d}} \sum_j |j\rangle \otimes |j \rangle$ i.e. a mixture (or pseudomixture for $\alpha < 0$) 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.