The Peres–Horodecki criterion is a necessary condition, for the joint density matrix of two quantum mechanical systems and , to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2×2 and 2×3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. The theorem was discovered in 1996 by Asher Peres and the Horodecki family (Michał, Paweł, and Ryszard) In higher dimensions, the test is inconclusive, and one should supplement it with more advanced tests, such as those based on entanglement witnesses. Definition If we have a general state which acts on 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, is the identity map applied to the A party and the transposition map applied to the B party. This definition can be seen more clearly if we write the state as a block matrix: Where , and each block is a square matrix of dimension . Then the partial transpose is The criterion states that if is separable then all the eigenvalues of are non-negative. In other words, if has a negative eigenvalue, is guaranteed to be entangled. The converse of these statements is true if and only if the dimension of the product space is or . The result is independent of the party that was transposed, because . Example Consider this 2-qubit family of Werner states: It can be regarded as the convex combination of , a maximally entangled state, and the identity element, a maximally mixed state. Its density matrix is and the partial transpose Its least eigenvalue is . Therefore, the state is entangled for . Demonstration If ρ is separable, it can be written as In this case, the effect of the partial transposition is trivial: As the transposition map preserves eigenvalues, the spectrum of is the same as the spectrum of , and in particular must still be positive semidefinite. Thus must also be positive semidefinite. This proves the necessity of the PPT criterion. Showing that being PPT is also sufficient for the 2 X 2 and 3 X 2 (equivalently 2 X 3) cases is more involved. It was shown by the Horodeckis that for every entangled state there exists an