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arXiv:1906.03869 · Physical Review Research

Nonlinear evolution and signaling

Jakub Rembieliński, Paweł Caban

Published Jan 24, 2020• 24 citations
arXivHTMLPDF

Abstract

This paper proposes a general rule that allows to distinguish acceptable and non-acceptable nonlinear quantum evolutions. Convex quasi-linear evolutions are free from problems with superluminal signaling. The explicit model of nonlinear but convex quasi-linear evolution of a qubit is constructed.
DOI
10.1103/physrevresearch.2.012027

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Abstract There is a growing interest, both from the theoretical as well as experimental side, to test the validity of the quantum superposition principle, and of theories which explicitly violate it by adding nonlinear terms to the Schrödinger equation. We review the original argument elaborated by Gisin (1989 Helv. Phys. Acta 62 363), which shows that the non-superluminal-signaling condition implies that the dynamics of the density matrix must be linear. This places very strong constraints on the permissible modifications of the Schrödinger equation, since they have to give rise, at the statistical level, to a linear evolution for the density matrix. The derivation is done in a heuristic way here and is appropriate for the students familiar with the textbook quantum mechanics and the language of density matrices.
Quantum theory is compatible with special relativity. In particular, though measurements on entangled systems are correlated in a way that cannot be reproduced by local hidden variables, they cannot be used for superluminal signaling. As Czachor, Gisin, and Polchinski pointed out, this is not generally true of general nonlinear modifications of the Schr\"odinger equation. Excluding superluminal signaling has thus been taken to rule out most nonlinear versions of quantum theory. The no-superluminal-signaling constraint has also been used for alternative derivations of the optimal fidelities attainable for imperfect quantum cloning and other operations. These results apply to theories satisfying the rule that their predictions for widely separated and slowly moving entangled systems can be approximated by nonrelativistic equations of motion with respect to a preferred time coordinate. This paper describes a natural way in which this rule might fail to hold. In particular, it is shown that quantum readout devices which display the values of localized pure states need not allow superluminal signaling, provided that the devices display the values of the states of entangled subsystems as defined in a nonstandard, although natural, way. It follows that any locally defined nonlinear evolution of pure states can be made consistent with Minkowski causality.
Nonlinear modifications of quantum mechanics have a troubled history. They were initially studied for many promising reasons: resolving the measurement problem, testing the limits of standard quantum mechanics, and reconciling it with gravity. Two results substantially undermined the credibility of non-linear theories. Some have been experimentally refuted, and more importantly, all deterministic non-linear theories can be used for superluminal communication. However, these results are unconvincing because they overlook the fact that the distribution of measurement results predicted by non-linear quantum mechanics depends on the interpretation of quantum mechanics that one uses. For instance, although the Everett and Copenhagen interpretations agree on the expression of Born's rule for the outcomes of multiple measurements in linear quantum mechanics, they disagree in non-linear quantum mechanics. We present the range of expressions of Born's rule that can be obtained by applying different formulations of quantum mechanics to a class of non-linear quantum theories. We then determine that many do not allow for superluminal communication but only two seem to have a reasonable justification. The first is the Everett interpretation, and the second, which we name causal-conditional, states that a measurement broadcasts its outcome to degrees of freedom in its future light-cone, who update the wavefunction that their non-linear Hamiltonian depends on according to this new information.
A possible mechanism of nonlinear quantum evolution is introduced and its implications for quantum communication are investigated. First, it is argued that an appropriate combination of wavefunction collapse and the consciousness of observer may permit the observer to distinguish nonorthogonal quantum states in principle, and thus consciousness will introduce certain nonlinearity into quantum dynamics. Next, it is shown that the distinguishability of nonorthogonal states can be used to achieve quantum superluminal communication, by which information can be transmitted nonlocally and faster than the speed of light. Finally, the issue of apparent incompatibility between superluminal communication and special relativity is briefly addressed.