Peres–Horodecki criterion

The Peres–Horodecki criterion is a necessary condition, for the joint density matrix ρ {displaystyle ho } of two quantum mechanical systems A {displaystyle A} and B {displaystyle B} , to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2x2 and 2x3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. In higher dimensions, the test is inconclusive, and one should supplement it with more advanced tests, such as those based on entanglement witnesses. If we have a general state ρ {displaystyle ho } which acts on H A ⊗ H B {displaystyle {mathcal {H}}_{A}otimes {mathcal {H}}_{B}} Its partial transpose (with respect to the B party) is defined as Note that the partial in the name implies that only part of the state is transposed. More precisely, ( I ⊗ T ) ( ρ ) {displaystyle (Iotimes T)( ho )} is the identity map applied to the A party and the transposition map applied to the B party.