Measurements Incompatible in Quantum Theory Cannot Be Measured Jointly in Any Other No-Signaling Theory

Thumbnail Image
Official URL
Full text at PDC
Publication Date
Wolf, Michael M.
Fernández, Carlos
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
American Physical Society
Google Scholar
Research Projects
Organizational Units
Journal Issue
It is well known that jointly measurable observables cannot lead to a violation of any Bell inequality-independent of the state and the measurements chosen at the other site. In this Letter we prove the converse: every pair of incompatible quantum observables enables the violation of a Bell inequality and therefore must remain incompatible within any other no-signaling theory. While in the case of von Neumann measurements it is sufficient to use the same pair of observables at both sites, general measurements can require different choices. The main result is obtained by showing that for arbitrary dimension the Clauser-Horne-Shimony-Holt inequality provides the Lagrangian dual of the characterization of joint measurability. This leads to a simple criterion for joint measurability beyond the known qubit case.
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777(1935). J. S. Bell, Physics 1, 195 (1964). A. Fine, J. Math. Phys. (N.Y.) 23, 1306 (1982). R. F. Werner and M. M. Wolf, Quantum Inf. Comput. 1, 1(2001); N. Gisin, arXiv:quant-ph/0702021. L. Masanes, A. Acin, and N. Gisin, Phys. Rev. A 73,012112 (2006). R. F. Werner, Lett. Math. Phys. 17, 359 (1989); B. M. Terhal, A. C. Doherty, and D. Schwab, Phys. Rev. Lett. 90, 157903 (2003). M. M. Wolf, F. Verstraete, and J. I. Cirac, Int. J. Quantum. Inform. 1, 465 (2003). P. Busch and H.-J. Schmidt, arXiv:0802.4167; S. Yu, N. Liu, L. Li, and C. H. Oh, arXiv:0805.1538; P. Stano, D. Reitzner, and T. Heinosaari, Phys. Rev. A 78, 012315 (2008). E. Andersson, S. M. Barnett, and A. Aspect, Phys. Rev. A 72, 042104 (2005). P. Busch, Phys. Rev. D 33, 2253 (1986). J. F. Clauser, M.A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969). L. J. Landau, Phys. Lett. A 120, 54 (1987). S. Gudder, J. Math. Phys. (N.Y.) 37, 2637 (1996); A. Gheondea, S. Gudder, and P. Jonas, J. Math. Phys. (N.Y.) 46, 062102 (2005). L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996). P. L. Halmos, Trans. Am. Math. Soc. 144, 381 (1969). A. Fine, Phys. Rev. Lett. 48, 291 (1982). T. Heinosaari, D. Reitzner, and P. Stano, Found. Phys. 38, 1133 (2008). R. F.Werner, in Quantum Information—An Introduction to Basic Theoretical Concepts and Experiments, Springer Tracts in Modern Physics (Springer, Berlin, 2001).