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arXiv:quant-ph/0012041

Comments on: ``Weinberg's Nonlinear Quantum Mechanics and Einstein-Podolsky-Rosen paradox'', by Joseph Polchinski

Bogdan Mielnik

Published Dec 8, 2000• 1 citation
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Abstract

Contrary to the opinion of J. Polchinski [Phys.Rev.Lett. {\bf 66}, 397-400 (1991)], the phenomenon of superluminal messages in nonlinear versions of quantum mechanics is not a specific difficulty in a class of theories formulated by S. Weinberg [Ann.Phys. (N.Y.), {\bf 194}, 336-386 (1989)]. It appears in all schemes which try to enlarge the orthodox class of observables, while conserving the traditional structure of the pure and mixed states.

DOI
10.48550/arxiv.quant-ph/0012041

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Contrary to the opinion of J. Polchinski [Phys.Rev.Lett. {\bf 66}, 397-400 (1991)], the phenomenon of superluminal messages in nonlinear versions of quantum mechanics is not a specific difficulty in a class of theories formulated by S. Weinberg [Ann.Phys. (N.Y.), {\bf 194}, 336-386 (1989)]. It appears in all schemes which try to enlarge the orthodox class of observables, while conserving the traditional structure of the pure and mixed states.
As shown by Einstein, Podolsky, and Rosen (the EPR paradox) [A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935)], quantum mechanics is a nonlocal theory contrarily to what happens for any other modern physical theory. Alternative local theories based on superluminal communications have been also proposed in the literature. So far, no evidence for these superluminal communications has been obtained and only lower bounds for the superluminal velocities have been established. In this paper we describe an improved experiment that increases by about two orders of magnitude the maximum detectable superluminal velocities. The locality, the freedom of choice, and the detection loopholes are not addressed here. No evidence for superluminal communications has been found and a higher lower bound for their velocities has been established.
One difficulty with the correspondence principle is its vagueness. To what should the quantum theory correspond in the quasiclassical domain? Here we show that, whatever it is, it cannot be Hamilton's equations. This is done using Weinberg's generalized nonlinear quantum theory [S. Weinberg, Ann. Phys. (N.Y.) 194, 336 (1989)] by exploiting the fact that it contains an exact copy of classical dynamics [K. R. W. Jones, Phys. Rev. D 45, R2590 (1992)]. An enlarged dynamical theory incorporating mixed quantum and classical interactions is shown to have some desirable properties in relation to measurement. By studying this system, we show that the existence of observable physical domains obeying intrinsically classical laws would violate the uncertainty principle; thereby ruling out an entire class of such larger theories. We interpret this result as a demonstration that the correspondence principle is essentially approximate. Further, the given exclusion is suggestive as a guide to physical models of quasi- classical emergence in a scenario based upon environment noise and stochastic reduction.
We formally link the concept of steering (a concept created by Schr\"odinger but only recently formalized by Wiseman, Jones, and Doherty [Phys. Rev. Lett. 98, 140402 (2007)]) and the criteria for demonstrations of Einstein-Podolsky-Rosen (EPR) paradox introduced by Reid [Phys. Rev. A 40, 913 (1989)]. We develop a general theory of experimental EPR-steering criteria, derive a number of criteria applicable to discrete as well as continuous-variable observables, and study their efficacy in detecting that form of nonlocality in some classes of quantum states. We show that previous versions of EPR-type criteria can be rederived within this formalism, thus unifying these efforts from a modern quantum-information perspective and clarifying their conceptual and formal origin. The theory follows in close analogy with criteria for other forms of quantum nonlocality (Bell nonlocality and entanglement), and because it is a hybrid of those two, it may lead to insights into the relationship between the different forms of nonlocality and the criteria that are able to detect them.