Материал: искусственный интеллект

i.e., this quantity does not depend on a basis.

Let L be a subspace of H. The orthogonal projector P : H → L onto this subspace is a Hermitian, idempotent (i.e., coinciding with its square), and positivesemideÞnite operator3:

(a)P = P ;

(b)P 2 = P ;

(c)P ≥ 0.

Here (c) is a consequence of (a) and (b). Moreover, an arbitrary linear operator satisfying (a) and (b) is an orthogonal projectorÑonto the subspace P H.

3.2Pure and Mixed States: Normalized Vectors and Density Operators

Pure quantum states are represented by normalized vectors, ψ H : ψ = 1. Two colinear vectors,

represent the same pure state. Thus, rigorously speaking, a pure state is an equivalence class of vectors having the unit norm: ψ ψ for vectors coupled by (13). The unit sphere of H is split into disjoint classesÑpure states. However, in concrete calculations one typically uses just concrete representatives of equivalent classes, i.e., one works with normalized vectors.

Each pure state can also be represented as the projection operator Pψ which projects H onto one dimensional subspace based on ψ. For a vector φ H,

The trace of the one dimensional projector Pψ equals 1:

Tr Pψ = ψ|ψ = 1.

We summarize the properties of the operator Pψ representing the pure state ψ. It is

(a)Hermitian,

(b)positive-semideÞnite,

(c)trace one,

(d)idempotent.

3To simplify formulas, we shall not put the operator-label ÒhatÓ in the symbols denoting projectors, i.e., P ≡ P .

Moreover, any operator satisfying (a)Ð(d) represents a pure state. Properties (a) and

(d) characterize orthogonal projectors, property (b) is their consequence. Property

(c) implies that the projector is one dimensional.

The next step in the development of QM was the extension of the class of quantum states, from pure states represented by one dimensional projectors to states represented by linear operators having the properties (a)Ð(c). Such operators are called density operators. (This nontrivial step of extension of the class of quantum states was based on the efforts of Landau and von Neumann.) The symbol D(H ) denotes the space of density operators in the complex Hilbert space H.

One typically distinguishes pure states, as represented by one dimensional projectors, and mixed states, the density operators which cannot be represented by one dimensional projectors. The terminology ÒmixedÓ has the following origin: any density operator can be represented as a ÒmixtureÓ of pure states (ψi ) :

(To simplify formulas, we shall not put the operator-label ÒhatÓ in the symbols denoting density operators, i.e., ρ ≡ ρ.) The state is pure if and only if such a mixture is trivial: all pi , besides one, equal zero. However, by operating with the terminology Òmixed stateÓ one has to take into account that the representation in the form (15) is not unique. The same mixed state can be presented as mixtures of different collections of pure states.

Any operator ρ satisfying (a)Ð(c) is diagonalizable (even in inÞnite-dimensional Hilbert space), i.e., in some orthonormal basis it is represented as a diagonal matrix, ρ = diag(pj ), where pj [0, 1], j pj = 1. Thus it can be represented in the form (15) with mutually orthogonal one dimensional projectors. The property (d) can be used to check whether a state is pure or not.

We point out that pure states are merely mathematical abstractions; in real experimental situations, it is possible to prepare only mixed states. The degree of purity is deÞned as

purity(ρ) = Trρ2.

Experimenters are satisÞed by getting this quantity near one.

4 Quantum Mechanics: Postulates

We state again that H denotes complex Hilbert space with the scalar product ·, · and the norm · corresponding to the scalar product.

Postulate 1 (The Mathematical Description of Quantum States) Quantum (pure) states (wave functions) are represented by normalized vectors ψ (i.e., ψ 2 =ψ, ψ = 1) of a complex Hilbert space H. Every normalized vector ψ H may

The physical meaning of Òa quantum stateÓ is not deÞned by this postulate, see Sect. 6.1.

Postulate 2 (The Mathematical Description of Physical Observables) A physical observable a is represented by a Hermitian operator a in complex Hilbert space H. Different observables are represented by different operators.

Postulate 3 (Spectral) For a physical observable a which is represented by the Hermitian operator a we can predict (together with some probabilities) values λ

Spec(a).

We restrict our considerations by simplest Hermitian operators which are analogous to discrete random variables in classical probability theory. We recall that a Hermitian operator a has purely discrete spectrum if it can be represented as

where Pαam are orthogonal projection operators related to the orthonormal eigenvectors {ekma }k of a corresponding to the eigenvalues αm by

Here k labels the eigenvectors ekma which belong to the same eigenvalue αm of a.

Postulate 4 (BornÕs Rule) Let a physical observable a be represented by a Hermitian operator a with purely discrete spectrum. The probability Pψ (a = αm) to obtain the eigenvalue αm of a for measurement of a in a state ψ is given by

If the operator a has nondegenerate (purely discrete) spectrum, then each αm is associated with one dimensional subspace. The latter can be Þxed by selecting any normalized vector, say ema . In this case orthogonal projectors act simply as

Formula (18) takes a very simple form

4Thus states are given by elements of the unit sphere of the Hilbert space H.

It is BornÕs rule in the Hilbert space formalism.

It is important to point out that if state ψ is an eigenstate of operator a representing observable a, i.e., aψ = αψ, then the outcome of observable a equals α with probability one.

We point out that, for any Þxed quantum state ψ, each quantum observable a can be represented as a classical random variable (Sect. 2). In the discrete case the corresponding probability distribution is deÞned as

where Pψ (a = αm) is given by BornÕs rule.

Thus each concrete quantum measurement can be described by the classical probability model.

Problems (including deep interpretational issues) arise only when one tries to describe classically data collected for a few incompatible observables (Sect. 5).

By using the BornÕs rule (18 ) and the classical probabilistic deÞnition of average (Sect. 2), it is easy to see that the average value of the observable a in the state ψ (belonging to the domain of deÞnition of the corresponding operator a) is given by

For example, for an observable represented by an operator with the purely discrete spectrum, we have

Postulate 5 (Time Evolution of Wave Function) Let H be the Hamiltonian of a quantum system, i.e., the Hermitian operator corresponding to the energy observable. The time evolution of the wave function ψ H is described by the Schrödinger equation

with the initial condition ψ(0) = ψ.

5 Compatible and Incompatible Observables

Two observables a and b are called compatible if a measurement procedure for their joint measurement can be designed, i.e., a measurement of the vector observable d = (a, b). In such a case their joint probability distribution is well deÞned.

62 A. Khrennikov

In the opposite case, i.e., when such a joint-measurement procedure does not exist, observables are called incompatible. The joint probability distribution of incompatible observables has no meaning.

In QM, compatible observables a and b are represented by commuting Hermitian

the QM-formalism compatibilityÐincompatibility is represented as commutativityÐ noncommutativity.

Postulate 4a (BornÕs Rule for Joint Measurements) Let observables a and b be represented by Hermitian operators a and b with purely discrete spectrum. The probability to obtain the eigenvalues αm and βk in a joint measurement of a and b in a state ψ—the joint probability distribution—is given by

It is crucial that the spectral projectors of commuting operators commute, so the probability distribution does not depend on the order of the values of observables. This is a classical probability distribution (Sect. 2). Any pair of compatible observables a and b can be represented by random variables: for example, by using the joint probability distribution as the probability measure.

A family of compatible observables a1, . . . , an is represented by commuting

where all possible permutations of projectors can be considered.

Now we point to one distinguishing feature of compatibility of quantum observables which is commonly not emphasized. The relation of commutativity of operators is the pairwise relation, it does not involve say triples of operators. Thus, for joint measurability of a group of quantum observables a1, . . . , an, their pairwise joint measurability is sufÞcient. Thus if we are able to design measurement procedures for all possible pairs, then we are always able to design a jointmeasurement procedure for the whole group of quantum observables a1, . . . , an. This is the specialty of quantum observables. In particular, if there exist all pairwise joint probability distributions Pai ,aj ;ψ , then the joint probability Pa1,...,an;ψ is deÞned as well.

The BornÕs rule can be generalized to the quantum states represented by density operators (Sect. 9.1, formula (40)).