Материал: искусственный интеллект

¥Experimental contexts (systemÕs state preparations) are represented by probabilities.

¥Observables are represented by random variables.

In principle, we can call probability a state and this is the direct analog of the quantum state (Sect. 6.1, the ensemble interpretation). However, we have to remember that the word ÒstateÓ has the meaning Òstatistical state,Ó the state of an ensemble of systems prepared for measurement.

The Kolmogorov probability space [49, 50] is any triple

( , F, P),

where is a set of any origin and F is a σ -algebra of its subsets, P is a probability measure on F. The set represents random parameters of the model. Kolmogorov called elements of elementary events. This terminology is standard in mathematical literature. Sets of elementary events are regarded as events. The key point of KolmogorovÕs axiomatization of probability theory is that not any subset of can be treated as an event. For any stochastic model, the system of events F is selected from the very beginning. The key mathematical point is that F has to be a σ -algebra. (Otherwise it would be very difÞcult to develop a proper notion of integral. And the latter is needed to deÞne average of a random variable.)

We remind that a σ -algebra is a system of sets which contains and empty set, it is closed with respect to the operations of countable union and intersection and to the operation of taking the complement of a set. For example, the collection of all subsets of is a σ -algebra. This σ -algebra is used in the case of Þnite or countable sets:

However, for Òcontinuous sets,Ó e.g., = [a, b] R, the collection of all possible subsets is too large to have applications. Typically it is impossible to describe a σ - algebra in the direct terms. To deÞne a σ -algebra, one starts with a simple system of subsets of and then consider the σ -algebra which is generated from this simple system with the aid of aforementioned operations. In particular, one of the most important for applications σ -algebras, the so-called Borel σ -algebra, is constructed in this way by staring with the system consisting of all open and closed intervals of the real line. In a metric space (in particular, in a Hilbert space), the Borel σ -algebra is constructed by starting with the system of all open and closed balls.

Finally, we remark that in American literature the term σ -Þeld is typically used, instead of σ -algebra.

The probability is deÞned as a measure, i.e., a map from F to nonnegative real numbers which is σ -additive:

j

In the case of a discrete probability space, see (1), the probability measures have the form

P(A) = pj , pj = P({ωj }).

ωj A

In fact, any Þnite measure μ, i.e., μ( ) < ∞, can be transformed into the probability measure by normalization:

A (real) random variable is a map ξ : → R which is measurable with respect to the Borel σ -algebra B of R and the σ -algebra F of . The latter means that, for any set B B, its preimage ξ −1(B) = {ω : ξ(ω) B} belongs to F. This condition provides the possibility to assign the probability to the events of the type Òvalues of ξ belong to a (Borel) subset of the real line.Ó The probability distribution of ξ is deÞned as

In the same way we deÞne the real (and complex) vector-valued random variables,

ξ : → Rn and ξ : → Cn.

Let ξ1, . . . , ξk be real-valued random variables. Their join probability distribution is deÞned as the probability distribution of the vector-valued random ξ = (ξ1, . . . , ξk ). To determine this probability measure, it is sufÞcient

to deÞne probabilities

Pξ1,...,ξk ( 1 × · · · × k ) = P(ω : ξ1(ω) 1, . . . ., ξk (ω) k ),

where j , j = 1, . . . , k, are intervals (open, closed, half-open) of the real line. Suppose now that random variables ξ1, . . . , ξk represent observables a1, . . . , ak .

For any point ω , the values of the vector ξ composed of these random variables are well deÞned ξ(ω) = (ξ1(ω), . . . , ξk (ω)). This vector represents a joint measurement of the observables and Pξ1,...,ξk represents the probability distribution for the outcomes of these jointly measured observables. Thus classical probability theory is applicable for jointly measurable observables, compatible observables in the terminology of QM (Sect. 5).

A random variable is called discrete if its image consists of Þnite or countable number of points, ξ = α1, . . . , αn, . . . . In this case its probability distribution has the form

The mean value (average) of a real-valued random variable is deÞned as its integral (the Lebesgue integral)

Eξ = ξ(ω)dP (ω). (7)

For a discrete random variable, its mean value has the simple representation:

αj B

In the Kolmogorov model the conditional probability is defined by the Bayes formula

We stress that other axioms are independent of this deÞnition.

We also present the formula of total probability (FTP) which is a simple consequence of the Bayes formula. Consider the pair, a and b, of discrete random variables. Then

Thus the b-probability distribution can be calculated from the a-probability distribution and the conditional probabilities P(b = β|a = α). These conditional probabilities are known as transition probabilities.

This formula plays the crucial role in Bayesian inference. It is applicable to the plenty of phenomena, in insurance, Þnances, economics, engineering, biology, AI, game theory, decision making, and programming. However, as was shown by the author of this review, in quantum domain FTP is violated and it is perturbed by the so-called interference term. Recently it was shown that even data collected in cognitive science, psychology, game theory, and decision making can violate classical FTP [1Ð10, 18, 38Ð41, 44Ð48].

3 Quantum Mathematics

We present the basic mathematical structures of QM and couple them to quantum physics.

3.1 Hermitian Operators in Hilbert Space

We recall the deÞnition of a complex Hilbert space. Denote it by H. This is a complex linear space endowed with a scalar product (a positive-deÞnite nondegenerate Hermitian form) which is complete with respect to the norm corresponding to the scalar product, ·|· . The norm is deÞned as

φ = φ|φ .

In the Þnite-dimensional case the norm and, hence, completeness are of no use. Thus those who have no idea about functional analysis (but know essentials of linear algebra) can treat H simply as a Þnite-dimensional complex linear space with the

For readerÕs convenience, we recall that the scalar product is a function from the Cartesian product H × H to the Þeld of complex numbers C, ψ1, ψ2 → ψ1|ψ2 , having the following properties:

1.Positive-deÞniteness: ψ|ψ ≥ 0 with ψ, ψ = 0 if and only if ψ = 0.

2.Conjugate symmetry: ψ1|ψ2 = ψ2|ψ1

3.Linearity with respect to the second argument2: φ|k1ψ1 + k2ψ2 = k1 φ|ψ1 + k2 φ|ψ2 , where k1, k2 are the complex numbers.

A reader who does not feel comfortable in the abstract framework of functional analysis can simply proceed with the Hilbert space H = Cn, where C is the set of complex numbers, and the scalar product

In this case the above properties of a scalar product can be easily derived from (11). Instead of linear operators, one can consider matrices.

2In mathematical texts one typically considers linearity with respect to the Þrst argument. Thus a mathematician has to pay attention to this difference.

We also recall a few basic notions of theory of linear operators in complex Hilbert space. A map a : H → H is called a linear operator, if it maps linear combination of vectors into linear combination of their images:

a(λ1ψ1 + λ2ψ2) = λ1aψ1 + λ2aψ2,

where λj C, ψj H, j = 1, 2.

For a linear operator a the symbol a denotes its adjoint operator which is

Let us select in H some orthonormal basis (ei ), i.e., ei |ej = δij . By denoting the matrix elements of the operators a and a as aij and aij , respectively, we rewrite the deÞnition (12) in terms of the matrix elements:

aij = a¯j i .

A linear operator a is called Hermitian if it coincides with its adjoint operator:

a = a .

If an orthonormal basis in H is Þxed, (ei ), and a is represented by its matrix, A = (aij ), where aij = aei |ej , then it is Hermitian if and only if

a¯ij = aj i .

We remark that, for a Hermitian operator, all its eigenvalues are real. In fact, this was one of the main reasons to represent quantum observables by Hermitian operators. In the quantum formalism, the spectrum of a linear operator (the set of eigenvalues while we are in the Þnite-dimensional case) coincides with the set of possibly observable values (Sect. 4, Postulate 3). We also recall that eigenvectors of Hermitian operators corresponding to different eigenvalues are orthogonal. This property of Hermitian operators plays some role in justiÞcation of the projection postulate of QM, see Sect. 5.1.

A linear operator a is positive-semidefinite if, for any φ H,

aφ|φ ≥ 0.

This is equivalent to positive-semideÞniteness of its matrix.

For a linear operator a, its trace is deÞned as the sum of diagonal elements of its matrix in any orthonormal basis:

Tr a = aii = aei |ei ,

ii