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arXiv:0708.1932 · Journal of Physics A Mathematical and Theoretical

Derivation of the quantum probability law from minimal non-demolition measurement

F Herbut

Published Aug 7, 2007• 5 citations
arXivHTMLPDF

Abstract

One more derivation of the quantum probability rule is presented in order to shed more light on the versatile aspects of this fundamental law. It is shown that the change of state in minimal quantum non-demolition measurement, also known as ideal measurement, implies the probability law in a simple way. Namely, the very requirement of minimal change of state, put in proper mathematical form, gives the well-known Lüders formula, which contains the probability rule.

DOI
10.1088/1751-8113/40/34/012

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One more derivation of the quantum probability rule is presented in order to shed more light on the versatile aspects of this fundamental law. It is shown that the change of state in minimal quantum non-demolition measurement, also known as ideal measurement, implies the probability law in a simple way. Namely, the very requirement of minimal change of state, put in proper mathematical form, gives the well-known Lüders formula, which contains the probability rule.
Usual derivations of Lüders's projection rule show that Lüders's rule is the rule required by quantum statistics to calculate the final state after an ideal (minimally disturbing) measurement. These derivations are at best inconclusive, however, when it comes to interpreting Lüders's rule as a description of individual state transformations. In this paper, I show a natural way of deriving Lüders's rule from well-motivated and explicit physical assumptions referring to individual systems. This requires, however, the introduction of a concept of individual state which is not standard.
In this paper I formalize the notion of minimal disturbance, as this seems to be required by usual interpretations of the theory of quantum mechanics, and construct a quantum logical (lattice) model of the type of situation that seems to be at the root of the problem of the interpretation of Luders’ projection rule as a criterion of minimal disturbance for individual state transformations. What is particularly interesting in the situation to be depicted here is that, on the basis of a simple model, which depends only on some very general features of the lattice structure of the theory (and its semantical interpretation), usual interpretive assumptions on minimal disturbance appear to be wanting. If we restrict our attention to the statistics of measurement results, Luders’ rule can, easily be interpreted as a formula describing ‘minimal change’ for statistical states.
A fundamental prediction of quantum mechanics is that observables can be measured through ideal measurements, that is, measurements that give the same outcome when repeated and do not disturb any observable that is jointly measurable. In terms of quantum instruments, such an ideal measurement is described by projective effects together with the associated Luders state-update rule. Here, we show that quantum mechanics allows for self-testing any ideal measurement. Using solely the output statistics of bipartite Bell tests in which a non-demolition measurement is sometimes added, it is possible to certify in a device-independent manner whether the added measurement is ideal.
To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making use of the probability reproducibility condition, which is a key concept in unitary measurement theory, one obtains the relevant coher- ent distribution of the complete-measurement results in the final unitary- measurement state in agreement with the mentioned probability formula. Treating the transition from the final unitary, or premeasurement, state, where all possible results are present, to one complete-measurement result sketchily in the usual way, the well-known probability formula is derived. In conclusion it is pointed out that the entire argument is only formal unless one makes it physical assuming that the quantum probability law is valid in the extreme case of probability-one (certain) events (projectors).
Recent results obtained in quantum measurements indicate that the fundamental relations between three physical properties of a system can be represented by complex conditional probabilities. Here, it is shown that these relations provide a fully deterministic and universally valid framework on which all of quantum mechanics can be based. Specifically, quantum mechanics can be derived by combining the rules of Bayesian probability theory with only a single additional law that explains the phases of complex probabilities. This law, which I introduce here as the law of quantum ergodicity, is based on the observation that the reality of physical properties cannot be separated from the dynamics by which they emerge in measurement interactions. The complex phases are an expression of this inseparability and represent the dynamical structure of transformations between the different properties. In its quantitative form, the law of quantum ergodicity describes a fundamental relation between the ergodic probabilities obtained by dynamical averaging and the deterministic relations between three properties expressed by the complex conditional probabilities. The complete formalism of quantum mechanics can be derived from this one relation, without any axiomatic mathematical assumptions about state vectors or superpositions. It is therefore possible to explain all quantum phenomena as the consequence of a single fundamental law of physics.
According to the Heisenberg uncertainty principle, if we measure one variable describing a physical system, we inevitably introduce noise into some other variable describing the same system. In practice, the situation is worse than that: measuring one variable usually introduces random noise into the very quantity we are measuring. It is possible in principle to make measurements, known as quantum non-demolition (or QND) measurements, which do not have this latter effect. A QND measurement may approach the ideal quantum measurement commonly encountered in quantum mechanics textbooks. Philippe Grangier and his co-workers at the Université Paris-Sud, at Orsay in France, have recently attempted to measure the intensity of a beam of light in this way (Phys. Rev. Len. (1991) 66 1418). Their experiment has shown that it is indeed possible to make a measurement which at least partially preserves the quantum mechanical state of the detected beam.