Geometry of the Schmidt decomposition and Hardy's theorem

American Journal of Physics

Volumn 64, Issue 9, 1996, Pages 1143-1150

  Aravind, P K ^a  

^a WORCESTER POLYTECHNIC INSTITUTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0346614626     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18334     Document Type: Article

References (22)

1. E. Schmidt, "Zur theorie der linearen und nichtlinearen integralgleichungen," Math. Annalen 63, 433-476 (1906). A recent graduate level quantum mechanics text that discusses the Schmidt decomposition theorem is A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993), Chap. 5.
2. E. Schmidt, "Zur theorie der linearen und nichtlinearen integralgleichungen," Math. Annalen 63, 433-476 (1906). A recent graduate level quantum mechanics text that discusses the Schmidt decomposition theorem is A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993), Chap. 5.
3. H. Everett III, "The theory of the universal wave function," in The Many-World Interpretation of Quantum Mechanics, edited by B. S. DeWitt and N. Graham (Princeton U.P., Princeton, NJ, 1973), pp. 3-140; "Relative state formulation of quantum mechanics," Rev. Mod. Phys. 29, 454-462 (1957).
4. H. Everett III, "The theory of the universal wave function," in The Many- World Interpretation of Quantum Mechanics, edited by B. S. DeWitt and N. Graham (Princeton U.P., Princeton, NJ, 1973), pp. 3-140; "Relative state formulation of quantum mechanics," Rev. Mod. Phys. 29, 454-462 (1957).
5. See, for example, N. Gisin, "Bell's inequality holds for all non-product states," Phys. Lett. A 154, 201-202 (1991); S. Popescu and D. Rohrlich, "Generic quantum nonlocality," Phys. Lett. A 166, 293-297 (1992).
6. See, for example, N. Gisin, "Bell's inequality holds for all non-product states," Phys. Lett. A 154, 201-202 (1991); S. Popescu and D. Rohrlich, "Generic quantum nonlocality," Phys. Lett. A 166, 293-297 (1992).
7. A. Ekert and P. L. Knight, "Entangled quantum systems and the Schmidt decomposition," Am. J. Phys. 63, 415-423 (1995).
8. L. Hardy, "Non-locality for two particles without inequalities for almost all entangled states," Phys. Rev. Lett. 71, 1665-1668 (1993).
9. S. Goldstein, "Nonlocality without inequalities for almost all entangled states for two particles," Phys. Rev. Lett. 72, 1951 (1994).
10. H. P. Stapp, Mind, Matter and Quantum Mechanics (Springer-Verlag, New York, 1993), pp. 5-9.
11. T. F. Jordan, "Testing Einstein-Podolsky-Rosen assumptions without inequalities with two photons or particles with spin-1/2," Phys. Rev. A 50, 62-66 (1994).
12. T. F. Jordan, "Quantum mysteries explored," Am. J. Phys. 62, 874-880 (1994).
13. N. D. Mermin, "Quantum mysteries refined," Am. J. Phys. 62, 880-887 (1994).
14. U. Fano, "Pairs of two-level systems," Rev. Mod. Phys. 55, 855-874 (1983).
15. See, for example, E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), p. 279.
16. We use the Einstein summation convention for repeated indices throughout the text.
17. 2=ρ to simplify the description of a pure state has been exploited in another context by J. Bohn, "Observable characteristics of pure quantum states," Phys. Rev. Lett. 66, 1547-1550 (1991).
18. xx′= -1,Cyy′=-1). The above states, which form an orthonormal and complete set, are often referred to as the "Bell basis;" see, for example, S. L. Braunstein, A. Mann, and M. Revzen, "Maximal violation of Bell inequalities for mixed states," Phys. Rev. Lett. 68, 3259-3261 (1994).
19. A. Einstein, B. Podolsky, and N. Rosen, "Can quantum mechanical description of physical reality be considered complete?," Phys. Rev. 47, 777-780 (1935).
20. a = 0 must be excluded because it does not lead to a distinct set of directions for a, a′, b, and b′.
21. b, and Ω. The three equations (24)-(26) suffice to determine these unknowns, but the equations are complicated enough that they can be solved only numerically.
22. J. Torgerson, D. Branning, and L. Mandel, "A method for demonstrating violation of local realism with a two-photon downconverter without use of Bell inequalities," Appl. Phys. B 60, 267-269 (1995).