Материал: искусственный интеллект

5.1 Post-Measurement State From the Projection Postulate

The projection postulate is one of the most questionable and debatable postulates of QM. We present it in the separate section to distinguish it from other postulates of QM, Postulates 1Ð5, which are commonly accepted.

Consider pure state ψ and quantum observable (Hermitian operator) a representing some physical observable a. Suppose that a has nondegenerate spectrum; denote its eigenvalues by α1, .., αm, . . . and the corresponding eigenvectors by e1a , . . . , ema , . . . (here αi = αj , i = j.) This is an orthonormal basis in H. We expand the vector ψ with respect to this basis:

where (kj ) are complex numbers such that

By using the terminology of linear algebra we say that the pure state ψ is a superposition of the pure states ej . The von Neumann projection postulate describes the post-measurement state and it can be formulated as follows:

Postulate 6VN (Projection Postulate, von Neumann) Measurement of observable

a resulting in output αi induces reduction of superposition (25) to the basis vector eia .

The procedure described by the projection postulate can be interpreted in the following way:

Superposition (25) reßects uncertainty in the results of measurements for an observable a. Before measurement a quantum system Òdoes not know how it will answer to the question a.Ó The BornÕs rule presents potentialities for different answers. Thus a quantum system in the superposition state ψ does not have propensity to any value of a as its objective property. After the measurement the superposition is reduced to the single term in the expansion (25) corresponding to the value of a obtained in the process of measurement.

Consider now an arbitrary quantum observable a with purely discrete spectrum, i.e., a = α1Pαa1 + · · · + αmPαam + · · · . The LŸders projection postulate describes the post-measurement state and it can be formulated as follows:

Postulate 6L (Projection Postulate, LŸders) Measurement of observable a resulting in output αm induces projection of state ψ on state

P a ψ

ψαm = αm . Pαam ψ

In contrast to the majority of books on quantum theory, we sharply distinguish the cases of quantum observables with nondegenerate and degenerate spectra. von

Neumann formulated Postulate 6VN only for observables with nondegenerate spectra. LŸders ÒgeneralizedÓ von NeumannÕs postulate to the case of observables with degenerate spectra. However, for such observables, von Neumann formulated [60] a postulate which is different from Postulate 6L. The post-measurement state need not be again a pure state.

We remark that Postulate 6L can be applied even to quantum states which are represented by density operators (Sect. 9.1, formula (41)).

6 Interpretations of Quantum Mechanics

Now we are going to discuss one of the most important and complicated issues of quantum foundations, the problem of an interpretation of a quantum state. There were elaborated numerous interpretations which can differ crucially from each other. This huge diversity of interpretations is a sign of the deep crises in quantum foundations.

In this section, we brießy discuss a few basic interpretations. Then in Sect. 9.1 we shall consider the VŠxjš (realist ensemble contextual) interpretation. Its presentation needs additional mathematical formulas. Therefore we placed it into a separate section.

6.1 Ensemble and Individual Interpretations

The Ensemble Interpretation A (pure) quantum state provides a description of certain statistical properties of an ensemble of similarly prepared quantum systems.

This interpretation is upheld, for example, by Einstein, Popper, Blokhintsev, Margenau, Ballentine, Klyshko, and recent years by, e.g., De Muynck, De Baere, Holevo, Santos, Khrennikov, Nieuwenhuizen, Adenier, Groessing, and many others.

The Copenhagen Interpretation A (pure) quantum state provides the complete description of an individual quantum system.

This interpretation was supported by a great variety of members, from SchršdingerÕs original attempt to identify the electron with a wave function solution of his equation to the several versions of the Copenhagen interpretation [12Ð14, 53, 54] (for example, Heisenberg, Bohr, Pauli, Dirac, von Neumann, Landau, Fock, and recent years by, e.g., Greenberger, Mermin, Lahti, Peres, Summhammer). Nowadays the individual interpretation is extremely popular, especially in quantum information and computing.

Instead of EinsteinÕs terminology Òensemble interpretation,Ó Ballentine [7Ð9] used the terminology “statistical interpretation.” However, BallentineÕs terminology is rather misleading, because the term Òstatistical interpretationÓ was also used

by von Neumann [60] for individual randomness! For him Òstatistical interpretationÓ had the meaning which is totally different from the BallentineÕs Òensemblestatistical interpretation.Ó J. von Neumann wanted to emphasize the difference between deterministic (Newtonian) classical mechanics in that the state of a system is determined by values of two observables (position and momentum) and quantum mechanics in that the state is determined not by values of observables, but by probabilities. We shall follow Albert Einstein and use the terminology ensemble interpretation.

We remark that following von Neumann [60] the supporters of the individual interpretation believe in irreducible quantum randomness, i.e., that the behavior of an individual quantum system is irreducibly random. Why does it behave in such a way? Because it is quantum, so it can behave so unusually. Nowadays this von NeumannÕs claim is used to justify superiority of the quantum technology over the classical technology. For example, superiority of quantum random generators.

6.2 Information Interpretations

The quantum information revolution generated a variety of information interpretations of QM (see, for example, [16, 17, 19, 20]). By these interpretations the quantum formalism describes special way of information processing, more general than the classical information processing. Roughly speaking one can forget about physics and work solely with probability, entropy, and information. Quantum Bayesianism (QBism) [25, 26] can be considered as one of such information, in its extreme form: the quantum formalism describes very general scheme of assignments of subjective probabilities to possible outcomes of experiments, assignment by human agents.

7 Quantum Conditional (Transition) Probability

In the classical Kolmogorov probabilistic model (Sect. 2), besides probabilities one operates with the conditional probabilities deÞned by the Bayes formula (see Sect. 2, formula (9)). The BornÕs postulate deÞning quantum probability should also be completed by a deÞnition of the conditional probability. We have remarked that, for one concrete observable, the probability given by BornÕs rule can be treated classically. However, the deÞnition of the conditional probability involves two observables. Such situations cannot be treated classically (at least straightforwardly, cf. Sect. 2). Thus conditional probability is really a quantum probability.

Let physical observables a and b be represented by Hermitian operators with purely discrete (may be degenerate) spectra:

66 A. Khrennikov

Let ψ be a pure state and let Pαak ψ = 0. Then the probability to get the value b = βm under the condition that the value a = αk was observed in the preceding measurement of the observable a on the state ψ is given by the formula

Then one applies BornÕs rule to the b-measurement for this state.

Let the operator a has nondegenerate spectrum, i.e., for any eigenvalue α the corresponding eigenspace (i.e., generated by eigenvectors with aψ = αψ) is one dimensional. We can write

(here aeka = αk eka ). Thus the conditional probability in this case does not depend on the original state ψ. We can say that the memory about the original state was

destroyed. If also the operator b has nondegenerate spectrum, then we have: Pψ (b =

βm|a = αk ) = | emb , eka |2 and Pψ (a = αk |b = βm) = | eka , emb |2. By using symmetry of the scalar product we obtain the following result:

Let both operators a and b have purely discrete nondegenerate spectra and let

8Observables with Nondegenerate Spectra: Double-Stochasticity of the Matrix of Transition Probabilities

We remark that classical (KolmogorovÐBayes) conditional probability is not symmetric, besides very special situations. Thus QM is described by a very specific probabilistic model.

Consider two nondegenerate observables. Set pβ|α = P(b = β|a = α). The matrix of transition probabilities Pb|a = (pβ|α ) is not only stochastic, i.e.,

pβ|α = 1

β

but it is even doubly stochastic:

pβ|α = | eβb , eαa |2 = eβb , eβb = 1.

αα

In KolmogorovÕs model, stochasticity is the general property of matrices of transition probabilities. However, in general classical matrices of transition probabilities are not doubly stochastic. Hence, double stochasticity is a very speciÞc property of quantum probability.

We remark that statistical data collected outside quantum physics, e.g., in decision making by humans and psychology, violates the quantum law of double stochasticity [38]. Such data cannot be mathematically represented with the aid of Hermitian operators with nondegenerate spectra. One has to consider either Hermitian operators with degenerate spectra or positive operator valued measures (POVMs).

9 Formula of Total Probability with the Interference Term

We shall show that the quantum probabilistic calculus violates the conventional FTP (10), one of the basic laws of classical probability theory. In this section, we proceed in the abstract setting by operating with two abstract incompatible observables. The concrete realization of this setting for the two-slit experiment demonstrating interference of probabilities in QM will be presented in Sect. 16 which is closely related to FeynmanÕs claim [22, 23] on the nonclassical probabilistic structure of this experiment.

Let H2 = C×C be the two dimensional complex Hilbert space and let ψ H2 be a quantum state. Let us consider two dichotomous observables b = β1, β2 and a = α1, α2 represented by Hermitian operators b and a, respectively (one may consider simply Hermitian matrices). Let eb = {eβb } and ea = {eαa } be two orthonormal bases consisting of eigenvectors of the operators. The state ψ can be represented in the two ways