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Separability probability formulas and their proofs for generalized two-qubit X-matrices endowed with Hilbert–Schmidt and induced measures

Charles F. Dunkl

Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA

E-mail Address: cfd5z@virginia.edu

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and

Paul B. Slater

University of California, Santa Barbara, CA 93106-4030, USA

E-mail Address: slater@kitp.ucsb.edu

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https://doi.org/10.1142/S2010326315500185Cited by:5 (Source: Crossref)

Abstract

Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert–Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general two-qubit states — deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter $α$ employed.

Keywords:

AMSC: 15B52, 81P40, 33B15

References

1. M. Ali, A. R. P. Rau and G. Alber, Quantum discord for two-qubit X-states, Phys. Rev. A 81(4) (2010) 042105. Crossref, Google Scholar
2. G. Aubrun, S. J. Szarek and D. Ye, Entanglement thresholds for random induced states, Commun. Pure Appl. Math. 67(1) (2014) 129–171. Crossref, Google Scholar
3. I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, 2006). Crossref, Google Scholar
4. I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 47 (2006) 5830–5847. Crossref, Google Scholar
5. C. F. Dunkl and G. Gasper, The sums of a double hypergeometric series and of the first $m + 1$ terms of $_{3} F_{2} (a, b, c; (a + b + 1) / 2, 2 c; 1)$ when $c = - m$ is a negative integer, preprint (2014), arXiv:1412.4022. Google Scholar
6. J. Fei and R. Joynt, Numerical computations of separability probabilities, preprint (2014), arXiv:1409.1993. Google Scholar
7. M. Horodecki, P. Horodecki and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223(1) (1996) 1–8. Crossref, Google Scholar
8. A. Khvedelidze and I. Rogojina, On the geometric probability of entangled mixed states, Zap. Nauch. Sem. POMI 432 (2015) 274–296. Google Scholar
9. P. Mendonça, M. A. Marchiolli and D. Galetti, Entanglement universality of two-qubit X-states, Ann. Phys. 351 (2014) 79–103. Crossref, Google Scholar
10. S. Milz and W. T. Strunz, Volumes of conditioned bipartite state spaces, preprint (2014), arXiv:1408.3666v2. Google Scholar
11. F. Olver, D. Lozier, R. Boisvert and C. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). Google Scholar
12. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77(8) (1996) 1413. Crossref, Google Scholar
13. S. B. Provost, Moment-based density approximants, Math. J. 9(4) (2005) 727–756. Google Scholar
14. P. B. Slater, A concise formula for generalized two-qubit Hilbert–Schmidt separability probabilities, J. Phys. A: Math. Theor. 46(44) (2013) 445302. Crossref, Google Scholar
15. P. B. Slater and C. F. Dunkl, Moment-based evidence for simple rational-valued Hilbert–Schmidt generic $2 \times 2$ separability probabilities, J. Phys. A: Math. Theor. 45(9) (2012) 095305. Crossref, Google Scholar
16. P. B. Slater and C. F. Dunkl, Formulas for rational-valued separability probabilities of random induced generalized two-qubit states, Advances in Mathematical Physics (2015) Article ID: 62153 (9 pages). Google Scholar
17. P. B. Slater and C. F. Dunkl, Generalized two-qubit whole and half Hilbert–Schmidt separability probabilities, J. Geometry and Physics 90 (2015) 42–54. Crossref, Google Scholar
18. K. Życzkowski, P. Horodecki, A. Sanpera and M. Lewenstein, Volume of the set of separable states, Phys. Rev. A 58(2) (1998) 883. Crossref, Google Scholar
19. K. Życzkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, J. Phys. A: Math. Gen. 34(35) (2001) 7111. Crossref, Google Scholar
20. K. Życzkowski and H.-J. Sommers, Hilbert–Schmidt volume of the set of mixed quantum states, J. Phys. A: Math. Gen. 36(39) (2003) 10115. Crossref, Google Scholar