Isotropic state

An '''isotropic state'''horodecki97reduction is a $d\; \backslash times\; d$ dimensional [[bipartite]] quantum state that is invariant under any unitary of the form $U\; \backslash otimes\; U^*$, where * denotes complex conjugate. That is, any state with the property that for any unitary U on one part of the system, :$\backslash rho\; =\; (U\; \backslash otimes\; U^*)\; \backslash rho\; (U^\backslash dagger\; \backslash otimes\; (U^*)^\backslash dagger).$ == Parametrization == The isotropic states is a one-parameter family of states and can be written as :$(1-\backslash alpha)\; I/d^2\; +\; \backslash alpha\; |\backslash phi^+\backslash rangle\; \backslash langle\; \backslash phi^+|,$ where $-1/(d^2-1)\; \backslash leq\; \backslash alpha\; \backslash leq\; 1$ and $|\backslash phi^+\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{d\}\}\; \backslash sum\_j\; |j\backslash rangle\; \backslash otimes\; |j\; \backslash rangle$ i.e. a [[mixture]] (or [[pseudomixture]] for ) 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 :$\backslash rho\; =\; \backslash frac\{d^2\}\{d^2-1\}\backslash left[\; (1-F)\; I/d^2\; +\; (F-1/d^2)\; |\backslash phi^+\backslash rangle\; \backslash langle\; \backslash phi^+|\; \backslash right]$ 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 criterion|reduction separability criterion]], and are therefore also [[Entanglement distillation|distillable]]. == See also == * [[Werner state]] == References == [[Category: Quantum States]]