2009/05/31 by Paul B. Slater · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #Combinatorics #Concurrence #Discrete mathematics #Mathematical analysis #Mathematics #Monotone polygon #Parameterized complexity #Physics #Pure mathematics #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #Quantum mechanics #Quantum state #Qubit #Random Matrices and Applications #Separable space #State (computer science) #quant-ph
paper · pdf · doi:10.1088/1751-8113/42/46/465305
published in Journal of Physics A Mathematical and Theoretical 42(46), 465305 (Institute of Physics) · 44 pages, 37 figures. Revised for J. Phys. A
arxiv created 2009/09/08 · openalex publication_date 2009/10/22 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
The probability that a generic real, complex or quaternionic two-qubit state is separable can be considered to be the sum of three contributions. One is from those states that are absolutely separable, that is those (which can not be entangled by unitary transformations) for which the maximal concurrence over spectral orbits ( C max ) is zero. The other two contributions are from the states for which , and for which . We have previously (arXiv:0805.0267) found exact formulas for the absolutely separable contributions in terms of the Hilbert–Schmidt metric over the quantum states, and here advance hypotheses as to the exact contributions for . A crucial element in understanding the two contributions for C max > 0 is the nature of the ratio ( R ) of the C max -parameterized separability function for the complex states to the square of the comparable function for the real states–both such functions having clearly displayed jump discontinuities at . For , the ratio R appears to be of the form 1 + kC max , except near , while for , there is strong numerical evidence that it equals 2 (thus, according to the Dyson-index pattern of random matrix theory). Related phenomena also occur for the minimally degenerate two-qubit states and the qubit–qutrit states. Our results have immediate application to the computation of separability probabilities in terms of other metrics, such as the Bures (minimal monotone) metric. The paper begins with continuous embeddings of the separability probability question in terms of four metrics of interest, using 'generalized Peres–Horodecki conditions'.