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arXiv:1501.03198

Wave Function Collapse and the No-Superluminal-Signaling Principle

Edward J. Gillis

Published Jan 13, 2015• 11 citations
arXivHTMLPDF

Abstract

The assumption that wave function collapse is a real occurrence has very interesting consequences - both experimental and theoretical. Besides predicting observable deviations from linear evolution, it implies that these deviations must originate in nondeterministic effects at the elementary level in order to prevent superluminal signaling, as demonstrated by Gisin. This lack of determinism implies that information cannot be instantiated in a reproducible form in isolated microsystems (as illustrated by the No-cloning theorem). By stipulating that information is a reproducible and referential property of physical systems, one can formulate the no-signaling principle in strictly physical terms as a prohibition of the acquisition of information about spacelike-separated occurrences. This formulation provides a new perspective on the relationship between relativity and spacetime structure, and it imposes tight constraints on the way in which collapse effects are induced. These constraints indicate that wave function collapse results from (presumably small) nondeterministic deviations from linear evolution associated with nonlocally entangling interactions. This hypothesis can be formalized in a stochastic collapse equation and used to assess the feasibility of testing for collapse effects.

DOI
10.48550/arxiv.1501.03198

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The assumption that wave function collapse is a real occurrence has very interesting consequences - both experimental and theoretical. Besides predicting observable deviations from linear evolution, it implies that these deviations must originate in nondeterministic effects at the elementary level in order to prevent superluminal signaling, as demonstrated by Gisin. This lack of determinism implies that information cannot be instantiated in a reproducible form in isolated microsystems (as illustrated by the No-cloning theorem). By stipulating that information is a reproducible and referential property of physical systems, one can formulate the no-signaling principle in strictly physical terms as a prohibition of the acquisition of information about spacelike-separated occurrences. This formulation provides a new perspective on the relationship between relativity and spacetime structure, and it imposes tight constraints on the way in which collapse effects are induced. These constraints indicate that wave function collapse results from (presumably small) nondeterministic deviations from linear evolution associated with nonlocally entangling interactions. This hypothesis can be formalized in a stochastic collapse equation and used to assess the feasibility of testing for collapse effects.
The changes that quantum states undergo during measurement are both probabilistic and nonlocal. These two characteristics complement one another to insure compatibility with relativity and maintain conservation laws. The probabilistic nature of nonlocal effects prevents the superluminal transmission of information, while nonlocal entanglement relations provide a means to enforce conservation laws in a probabilistic theory. In order to explain measurement-induced changes in terms of fundamental physical processes it is important to take these two key characteristics into account. One way to do this is to modify the Schroedinger equation by adding stochastic, nonlinear terms. A number of such proposals have been made over the past few decades. A recently proposed equation based on the assumption that wave function collapse is induced by a sequence of correlating interactions of the kind that constitute measurements has been shown to maintain strict adherence to conservation laws in individual instances, and has also eliminated the need to introduce any new, ad hoc physical constants. In this work it is shone that the proposed stochastic modification to the Schroedinger equation is also Lorentz invariant, even though it is formulated in a preferred reference frame. It is further argued that the additional spacetime structure that the proposed modification requires provides a way to implement the assumption that spacelike-separated operators (and measurements) commute, and that this assumption of local commutativity should be regarded as a third postulate of relativity.
Signal causality, the prohibition of superluminal information transmission, is the fundamental property shared by quantum measurement theory and relativity, and it is the key to understanding the connection between nonlocal measurement effects and elementary interactions. To prevent those effects from transmitting information between the generating and observing process, they must be induced by the kinds of entangling interactions that constitute measurements, as implied in the Projection Postulate. They must also be nondeterministic as reflected in the Born Probability Rule. The nondeterminism of entanglement-generating processes explains why the relevant types of information cannot be instantiated in elementary systems, and why the sequencing of nonlocal effects is, in principle, unobservable. This perspective suggests a simple hypothesis about nonlocal transfers of amplitude during entangling interactions, which yields straightforward experimental consequences.
Bell [Helv. Phys.Act. 48 1975] described an idealized test of the projection postulate in order to illustrate the ambiguities in the experimental predictions of quantum theory. An approach to the implementation of such tests is outlined here, based on the interesting logical relationship between the projection postulate and the no-superluminal-signaling principle. Gisin's theorem [Helv. Phys. Act. 62 1989] shows that the no-signaling principle implies that any description of projection at the elementary level must be nondeterministic. The lack of determinism means that information cannot be instantiated in a reproducible and transmittable form in isolated microsystems. The implied limitation on information allows one to interpret the no-signaling principle in purely physical terms as a prohibition of superluminal information transmission. This prohibition, in turn, imposes tight constraints on the connection between projection and elementary processes, leading to a specific hypothesis about how projection effects originate. The hypothesis serves as the basis for the experimental proposal.
The no-signaling principle preventing superluminal communication is a limiting paradigm for physical theories. Within the information-theoretic framework it is commonly understood in terms of admissible correlations in composite systems. Here we unveil its complementary incarnation---the ''dynamical no-signaling principle''---which forbids superluminal signaling via measurements on simple physical objects (e.g., particles) evolving in time. We show that it imposes strong constraints on admissible models of dynamics. The posited principle is universal; it can be applied to any theory (classical, quantum, or postquantum) with well-defined rules for calculating detection statistics in spacetime. As an immediate application we show how one could exploit the Schr\"odinger equation to establish a fully operational superluminal protocol in the Minkowski spacetime. This example illustrates how the principle can be used to identify the limits of applicability of a given model of quantum or postquantum dynamics.
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.