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Consistency and Linearity in Quantum Theory

Ariel Caticha

Published Jan 1, 1998• 20 citations
OpenAlex

Abstract

Quantum theory is formulated as the uniquely consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if the amplitude of a quantum process can be computed in two different ways, the two answers must agree. The constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schrödinger equation must be linear: non-linear variants of quantum mechanics violate the requirement of consistency. PACS: 03.65.Bz, 03.65.Ca. The formalism of quantum mechanics is very robust; modifications are invariably unsuccessful. A number of non-linear alternative theories have been proposed, [1] experimentally tested, [2] and promptly discarded, but the question of whether new non-linear variants might prove successful is not settled. In fact, interest in the subject remains in spite of, or perhaps, partly because of

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It is argued (Part A) that quantum mechanics can be derived as a principle-based dynamical framework, the basic equation of which is an alternative form of the Hamilton-Jacobi equation. Schrodinger's equation obtains as a result of linearising that equation, and so-called wave functions can be given no straightforward physical interpretation. It is suggested, partly in relation to a theorem by Gromov, that a finite action quantum would make it practically inevitable, for purposes of prediction, to resort to a probabilistic formulation. The structure of the space of square-integrable solutions of the Schrodinger equation happens to lend itself to the introduction of the appropriate kind of predictive scheme. Investigating the nature and scope of such a scheme is the subject of Part B. It is shown that basic features of the formalism of quantum theory, like composition rules for 'amplitudes' or the 'Born' probability rule, can be derived independently of any physical assumptions. A generalisation of the basic formalism using tensor product composition appears to be required if all correlations are to be extracted from locally accessed data. A detailed discussion of quantum teleportation leads to the conclusion that a 'one-shot' account leads to a distorted picture of what is actually achieved. An analogy with classical cryptography is made and the statistical significance of the 'transfer', which does not require introducing any novel form of 'quantum information', is emphasised. Results obtained over the last decade using the extended formalism of positive operator-valued measures are reviewed and discussed. These lend further support to the idea that the set of basic 'quantum' rules functions as a general kind of probabilistic scheme for prediction, the structural features of which are not constrained in any direct way by the underlying physics. On the other hand, the very existence of such a predictive framework hinges on selecting a particular class of solutions of the Schrodinger equation, which selection has been incorrectly interpreted as reflecting a physical necessity.

•Jan 1, 2004•0 citations
Quantum theory is formulated as the uniquely consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if the amplitude of a quantum process can be computed in two different ways, the two answers must agree. The constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schrödinger equation must be linear: non-linear variants of quantum mechanics violate the requirement of consistency. PACS: 03.65.Bz, 03.65.Ca. The formalism of quantum mechanics is very robust; modifications are invariably unsuccessful. A number of non-linear alternative theories have been proposed, [1] experimentally tested, [2] and promptly discarded, but the question of whether new non-linear variants might prove successful is not settled. In fact, interest in the subject remains in spite of, or perhaps, partly because of
Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a very natural consistency constraint: if there are two different ways to compute the amplitude for a given process, the two answers must agree. This constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schr\"odinger equation must be linear; i.e., nonlinear variants of quantum mechanics are inconsistent. The physical interpretation of the theory is given in terms of a single natural rule. This rule, which does not itself involve any probabilities, is used to obtain a proof of Born's statistical postulate: we show that the probability of a certain outcome in an experiment is given by the square of the modulus of the corresponding amplitude. Thus, consistency leads to indeterminism.
A model of correlated particles described by a generalized probability theory is suggested whose dynamics is subject to a non-linear version of Schrödinger equation. Such equations arise in many different contexts, most notably in the proposals for the gravitationally induced collapse of wave function. Here, it is shown that the consequence of the connection demonstrates a possible deviation of the theory from the standard formulation of quantum mechanics in the probability prediction of experiments. The links are identified from the fact that the analytic solution of the equation is given by Dirichlet eigenvalues which can be expressed by generalized trigonometric function. Consequently, modified formulation of Born's rule is obtained by relating the event probability of the measuement to an arbitrary exponent of the modulus of the eigenvalue solution. Such system, which is subject to the non-linear dynamic equation, illustrates the violation of the Clauser-Hore-Shimony-Holt inequality proportional to the degree of the non-linearity as it can be tested by a real experiment. Depending upon the degree, it is found that the violation can go beyond Tsirelson bound 222\sqrt{2}22​ and reaches to the value of nonlocal box.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
It is argued (Part A) that quantum mechanics can be derived as a principle-based dynamical framework, the basic equation of which is an alternative form of the Hamilton-Jacobi equation. Schrodinger's equation obtains as a result of linearising that equation, and so-called wave functions can be given no straightforward physical interpretation. It is suggested, partly in relation to a theorem by Gromov, that a finite action quantum would make it practically inevitable, for purposes of prediction, to resort to a probabilistic formulation. The structure of the space of square-integrable solutions of the Schrodinger equation happens to lend itself to the introduction of the appropriate kind of predictive scheme. Investigating the nature and scope of such a scheme is the subject of Part B. It is shown that basic features of the formalism of quantum theory, like composition rules for 'amplitudes' or the 'Born' probability rule, can be derived independently of any physical assumptions. A generalisation of the basic formalism using tensor product composition appears to be required if all correlations are to be extracted from locally accessed data. A detailed discussion of quantum teleportation leads to the conclusion that a 'one-shot' account leads to a distorted picture of what is actually achieved. An analogy with classical cryptography is made and the statistical significance of the 'transfer', which does not require introducing any novel form of 'quantum information', is emphasised. Results obtained over the last decade using the extended formalism of positive operator-valued measures are reviewed and discussed. These lend further support to the idea that the set of basic 'quantum' rules functions as a general kind of probabilistic scheme for prediction, the structural features of which are not constrained in any direct way by the underlying physics. On the other hand, the very existence of such a predictive framework hinges on selecting a particular class of solutions of the Schrodinger equation, which selection has been incorrectly interpreted as reflecting a physical necessity.