Introduction to Quantum Theory

This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [191], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Luders's forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (Liiders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac's notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.

We hope that this chapter would be interesting for newcomers to quantum-like modeling. Maybe even experts can find something useful, say the treatment of interference of probabilities as a violation of the law of total probability.

Classical Probability Theory: Kolmogorov's Measure-Theoretic Axiomatics

The main aim of QM is to provide probabilistic predictions on the results of measurements. Moreover, statistics of the results of measurements of a single quantum observable can be described by classical probability theory. We now present an elementary introduction to this theory.

The Kolmogorov probability space [191] is any triple

where £2 is a set of any origin and T is a о-algebra of its subsets and P is a probability measure on T. The set £1 represents random parameters of the model. Kolmogorov called elements of £1 elementary events. This terminology is standard in mathematical literature. Sets of elementary events are regarded as events. The key point of Kolmogorov’s axiomatization of probability theory is that not any subset of £2 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 a algebra. (Otherwise it would be very difficult to develop a proper notion of integral. And the latter is needed to define average of a random variable.)

We remind that a a algebra is a system of sets which contains £2 and an empty set, and 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 £2 is a algebra. This a algebra is used in the case of finite or countable sets:

However, for "continuous sets," e.g., £2 = [a, b] c R, the collection of all possible subsets is too large to have applications. Typically it is impossible to describe a a algebra in the direct terms. To define a a algebra, one starts with a simple system of subsets of £2 and then considers the a algebra which is generated from this simple system with the aid of aforementioned operations. In particular, one of the most important for applications a algebras, the so-called Borel a Igebra, 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, a Hilbert space), the Borel a algebra is constructed by starting with the system of all open and closed balls.

Finally, we remark that in American literature the term a field is typically used, instead of a algebra.

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

where Aj e F and А, П Aj = 0, / = j. The probability measure is always normalized by one:

In the case of a discrete probability space (see (10.1)), the probability measures have the form

In fact, any finite measure ц, i.e., ц (£2) < oo, can be transformed into the probability measure by normalization:

A (real) random variable is a map £ : £2 -> R which is measurable with respect to the Borel a algebra В of R and the a algebra J" of £2. The latter means that, for any set В eB, its preimage £_1 (£?) = {w e £2 : £(a>В} belongs to F. This condition provides the possibility of assigning the probability to the events of the type "values of £ belong to a (Borel) subset of the real line." The probability distribution of § is defined as

In the same way we define the real (and complex) vector valued random variables, £ : £2 -> R" (£ : £2 -* C").

A random variable is called discrete if its image consists of finite

or countable number of points, f = alf..., a„,____In this case its

probability distribution has the form

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

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

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

We stress that other axioms are independent of this definition.

We also present th e 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 = pa = a). These conditional probabilities are known as transition probabilities.

Mathematical Structure of Quantum Theory

Complex Hilbert Space

We recall the definition of a complex Hilbert space. Denote it by H. This is a complex linear space endowed with a scalar product (a positive-definite nondegenerate Hermitian form) which is complete with respect to the norm corresponding to the scalar product (ф). The norm is defined as

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

For a complex number z = x + iy, x, у e R, its conjugate is denoted by z, where z = x — iy. The absolute value of z is given by z2 = zz = x2 + y2.

For the reader's convenience, we recall that the scalar product is a function from the Cartesian product H x H to the field of complex numbers C, i/'i. ф2 -*■ (ФЛФг), having the following properties:

  • (1) Positive-definiteness: (фф) > 0 with (1//, ф) = 0 if and only if ф = 0.
  • (2) Conjugate symmetry: (V'i I<Аг) = (MVd)
  • (3) Linearity with respect to the second argument[1]: {фк1/ + кгфг) = кх{фф{) + к2(фф2), where к and к2 are complex numbers.

It generates the norm on H : ||^|| = у/{фФ)-

A reader who does not feel comfortable in the abstract framework of functional analysis can simply proceed with the Hilbert space H = C", 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 Eq. 10.11. Instead of linear operators, one can consider matrices.

Linear Operators

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

where Aj e C, j/j e H, j = 1, 2.

For a linear operator a the symbol a* denotes its adjoint operator which is defined by the equality

Let us select in H some orthonormal basis (e,), i.e., (e, |ey) = <$y. By denoting the matrix elements of the operators a and a* as ay and o?, respectively, we rewrite the definition (10.12) in terms of the matrix elements:

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

If an orthonormal basis in H is fixed, (e, ), and a is represented by its matrix, A = (ay), where ay = ;), then it is Hermitian if and only if

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 quantum formalism, the spectrum of a linear operator (the set of eigenvalues while we are in the finite dimensional case) coincides with the set of possibly observable values (section 10.3, Postulate 3). We also recall that the eigenvectors of Hermitian operators corresponding to different eigenvalues are orthogonal. This property of Hermitian operators plays some role in the justification of the projection postulate of QM (see section 10.3.1).

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

This is equivalent to the positive-semidefiniteness of its matrix.

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

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 positive-semidefinite operator[2]:

  • (1) P* = P;
  • (2) P[2] = P;
  • (3) P > 0.

Here (3) is a consequence of (1) and (2). Moreover, an arbitrary linear operator satisfying (1) and (2) is an orthogonal projector— onto the subspace P H.

Representation of (Pure) States by Normalized Vectors

Pure quantum states are represented by normalized vectors, ф e H :ф = 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 ф' ~ ф for vectors coupled by (10.13). The unit sphere of H is split into disjoint classes: pure 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 Рф which projects H onto a one-dimensional subspace based on ф. For a vector ф e H,

The trace of the one-dimensional projector Рф equals 1:

We summarize the properties of the operator Рф representing the pure state ф. It is

  • (1) Hermitian,
  • (2) positive-semidefinite,
  • (3) trace one,
  • (4) idempotent.

Moreover, any operator satisfying (l)-(4) represents a pure state. Properties (1) and (4) characterize orthogonal projectors, and property (2) is their consequence. Property (3) implies that the projector is one-dimensional.

Representation of Mixed States by Density Operators

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 properties (l)-(3). 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 onedimensional projectors, and mixed states, the density operators which cannot be represented by one-dimensional projectors. The term mixed has the following origin: any density operator can be represented as a mixture of pure states (i/',):

(To simplify formulas, we shall not put the operator-label "hat” in the symbols denoting density operators, i.e., p = p.) The state is pure if and only if such a mixture is trivial: all ph besides one, equal zero. However, by operating with the term mixed state one has to take into account that the representation in the form (10.15) is not unique. The same mixed state can be presented as mixtures of different collections of pure states.

Any operator p satisfying (l)-(3) is diagonalizable (even in infinite-dimensional Hilbert space), i.e., in some orthonormal basis it is represented as a diagonal matrix, p = diag(py), where pj e [0, 1], j pj = 1. Thus it can be represented in the form (10.15) with mutually orthogonal one-dimensional projectors. Property (4) 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. One defines the degree of purity as Trp2. Experimenters are satisfied by getting this quantity near one.

Hilbert Space of Square Integrable Functions

Although we generally proceed with finite-dimensional Hilbert spaces, it is useful to mention the most important example of infinite-dimensional Hilbert space used in QM. Consider the space of complex-valued functions, ^ : Rm -*■ C, which are square integrable with respect to the Lebesgue measure on Rm:

It is denoted by the symbol L2(Rm). Here the scalar product is given by

A delicate point is that, for some measurable functions, j/ : Rm C, which are not identically zero, the integral

We remark that the latter equality implies that = Oa.e. (almost everywhere). Thus the quantity defined by Eq. 10.16 is, in fact, not the norm: ||Vf|| = 0 does not imply that j/ = 0. To define a proper Hilbert space, one has to consider as its elements not simply functions, but classes of equivalent functions, where the equivalence relation is defined as (/ ~ ф if and only if j/(x) = (x) a.e. In particular, all functions satisfying Eq. 10.17 are equivalent to the zero function.

  • [1] 4n mathematical texts one typically considers linearity with respect to the firstargument. Thus a mathematician has to pay attention to this difference.
  • [2] To simplify formulas, we shall not put the operator label "hat" in the symbolsdenoting projectors, i.e., P = P.
  • [3] To simplify formulas, we shall not put the operator label "hat" in the symbolsdenoting projectors, i.e., P = P.
 
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