PeresHorodecki

The Peres–Horodecki criterion, commonly called the PPT criterion, is a foundational test in quantum information theory for determining entanglement in bipartite quantum states. It is named after Asher Peres, who introduced the idea in 1996, and the Horodecki family (Michał, Paweł, and Ryszard Horodecki), who extended and clarified its reach.

The criterion considers a density operator ρ describing a composite system AB. One forms the partial transpose with respect to one subsystem, typically B, denoted ρ^{T_B}. A basic result is that if ρ is separable (i.e., not entangled), then ρ^{T_B} must be positive semidefinite. Conversely, in the cases of 2×2 and 2×3 dimensional systems, this condition is also sufficient: if ρ^{T_B} is positive, the state is separable.

In general, however, positivity of the partial transpose is only a necessary condition for separability and not always sufficient. In higher-dimensional systems, there exist entangled states with a positive partial transpose, known as PPT entangled or bound entangled states. Therefore, while a negative eigenvalue of ρ^{T_B} guarantees entanglement, a positive partial transpose does not guarantee separability beyond the lowest-dimensional cases.

The PPT criterion remains a widely used and practical tool for entanglement detection, providing a straightforward computational test and shaping both theoretical analyses and experimental investigations in quantum information science. It is often employed alongside other criteria to characterize the entanglement properties of complex quantum states.