2009/02/28 by Salman Beigi, Peter W. Shor
Computer Science · Mathematics · Physics and Astronomy · #Combinatorics #Computer science #Dimension (graph theory) #Eigenvalues and eigenvectors #Mathematical analysis #Mathematics #Peres–Horodecki criterion #Physics #Positive-definite matrix #Pure mathematics #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #Quantum mechanics #Separable space #Separable state #Set (abstract data type) #TRACE (psycholinguistics) #Transpose #quant-ph
paper · pdf · doi:10.1063/1.3364793
published as J. Math. Phys. 51, 042202 (2010) · 12 pages, published version
openalex publication_date 2010/04/01 · arxiv created 2011/06/21 · arxiv updated 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, is efficient. In this paper, we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states. More precisely, we prove that, as the dimension of the space tends to infinity, the maximum trace distance of a positive partial transpose state from separable states tends to 1. Using similar techniques, we show that the same result holds for other well-known separability criteria such as reduction criterion, majorization criterion, and symmetric extension criterion. We also bring in evidence that the sets of positive partial transpose states and separable states have totally different shapes.