2016/01/18 by Y. B. Band, Pier A. Mello, Band, Y. B. +1
Computer Science · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Quantum optics and atomic interactions
paper · pdf · doi:10.48550/arxiv.1601.04481
openalex publication_date 2016/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We first show that partial transposition for pure and mixed two-particle states in a discrete N-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter r of a Bell state |Φ+ ⟩ and the completely incoherent state, ρr = r |Φ+ ⟩ ⟨ Φ+| + (1-r)\frac\mathbbIN2: we find that for r0 < r < 1, where r0 is obtained as a function of the dimensionality N, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.