A semidefinite programming approach to optimal unambiguous discrimination of quantum states

IEEE Transactions on Information Theory

Volumn 49, Issue 2, 2003, Pages 446-456

  Eldar, Yonina C ^a,b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Compound geometrically uniform (CGU) quantum states; Equal probability measurement (EPM); Geometrically uniform (GU) quantum states; Quantum detection; Semidefinite programming; Unambiguous discrimination

Indexed keywords

ALGORITHMS; POLYNOMIALS; QUANTUM THEORY; THEOREM PROVING;

SEMIDEFINITE PROGRAMMING;

INFORMATION THEORY;

EID: 0037321338     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2002.807291     Document Type: Article

References (33)

1. C. H. Bennett and P. W. Shor, "Quantum information theory," IEEE Trans. Inform. Theory, vol. 44, pp. 2724-2742, Oct. 1998.
2. A. Peres, Quantum Theory: Concepts and Methods. Boston, MA: Kluwer, 1995.
3. C. W. Helstrorn, Quantum Detection and Estimation Theory. New York: Academic, 1976.
4. A. S. Holevo, "Statistical decisions in quantum theory," J. Multivar. Anal., vol. 3, pp. 337-394, Dec. 1973.
5. H. P. Yuen, R. S. Kennedy, and M. Lax, "Optimum testing of multiple hypotheses in quantum detection theory," IEEE Trans. Inform. Theory, vol. IT-21, pp. 125-134, Mar. 1975.
6. Y. C. Eldar, A. Megretski, and G. C. Verghese, "Designing optimal quantum detectors via semidefinite programming," IEEE Trans. Inform. Theory. [Online]. Available: http://www.arXiv.org/abs/quant-ph/0205178, to be published.
7. I. D. Ivanovic, "How to differentiate between nonorthogonal states," Phys. Lett. A, vol. 123, pp. 257-259, Aug. 1987.
8. D. Dieks, "Overlap and distinguishability of quantum states," Phys. Lett. A, vol. 126, pp. 303-307, 1988.
9. A. Peres, "How to differentiate between nonorthogonal states," Phys. Lett. A, vol. 128, p. 19, Mar. 1988.
10. G. Jaeger and A. Shimony, "Optimal distinction between two nonorthogonal quantum states," Phys. Lett. A, vol. 197, pp. 83-87, 1995.
11. A. Peres and D. R. Terno, "Optimal distinction between nonorthogonal quantum states," J. Phys. A, vol. 31, pp. 7105-7111, 1998.
12. A. Chefles, "Unambiguous discrimination between linearly independent quantum states," Phys. Lett. A, vol. 239, pp. 339-347, Apr. 1998.
13. A. Chefles and S. M. Barnett, "Optimum unambiguous discrimination between linearly independent symmetric states," Phys. Lett. A, vol. 250, pp. 223-229, 1998.
14. A. Peres, "Neumark's theorem and quantum inseparability," Found. Phys., vol. 20, no. 12, pp. 1441-1453, 1990.
15. L. Vandenberghe and S. Boyd, "Semidefinite programming," SIAM Rev., vol. 38, no. 1, pp. 40-95, Mar. 1996.
16. Y. Nesterov and A. Nemirovski, Interior-Point Polynomial Algorithms in Convex Programming. Philadelphia, PA: SIAM, 1994.
17. F. Alizadeh, "Combinatorial optimization with interior point methods and semi-definite matrices," Ph.D. dissertation, Univ. Minnesota, Minneapolis, Oct. 1991.
18. G.D. Forney, Jr., "Geometrically uniform codes," IEEE Trans. Inform. Theory, vol. 37, pp. 1241-1260, Sept. 1991.
19. Y. C. Eldar and G. D. Forney, Jr., "On quantum detection and the square-root measurement," IEEE Trans. Inform. Theory, vol. 47, pp. 858-872, Mar. 2001.
20. Y.C. Eldar, A. Megretski, and G. C. Verghese. Optimal detection of symmetric mixed quantum states. [Online] http://www.arXiv.org/abs/quant-ph/0211111.
21. Y. C. Eldar and H. Bölcskei, "Geometrically uniform frames," IEEE Trans. Inform. Theory. [Online]. Available: http://arXiv.org/abs/math.FA/0108096, to be published.
22. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1996.
23. M. Ježek, J. Řeháček, and J. Fiurášek. Finding optimal strategies for minimum-error quantum-state discrimination. [Online]. Available: http://www.arXiv.org/abs/quant-ph/0201109.
24. A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. Distinguishing separable and entangled states. [Online]. Available: http://www.arXiv.org/abs/quant-ph/0112007.
25. E. M. Rains, "A semidefinite program for distillable entanglement," IEEE Trans. Inform. Theory, vol. 47, pp. 2921-2933, Nov. 2001.
26. K. Audenaert and B. De Moor. Optimizing completely positive maps using semidefinite programming. [Online]. Available: http://www.arXiv.org/abs/quant-ph/0109155.
27. X. Sun, S. Zhang, Y. Feng, and M. Ying, "Mathematical nature of and a family of lower bounds for the success probability of unambiguous discrimination," Phys. Rev. A, vol. 65, Sept. 2002.
28. M. X. Goemans and D. P. Williamson, "Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming," in Proc. ACM Symp. Theory of Computing, 2001, pp. 443-452.
29. J. F. Strum, "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones," Optimiz. Methods and Software, vol. 11-12, pp. 625-653, 1999.
30. D. Peaucelle, D. Henrion, and Y. Labit. Users guide for SeDuMi interface 1.03. [Online]. Available: http://www.laas.fr/peaucell/SeDuMiInt.html.
31. D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Belmont, MA: Athena Scientific, 1997.
32. A. Megretski, C.-Y. Kan, U. Jönsson, and A. Rantzer. A guide to IQCß: Software for robustness analysis. [Online]. Available: http://web.mit.edu/cykao/www.
33. M. A. Armstrong, Groups and Symmetry. New York: Springer-Verlag, 1988.