Conditional Probability in Quantum Formalism

In the classical Kolmogorov probabilistic model (section 10.1), besides probabilities one operates with the conditional probabilities defined by the Bayes formula (see section 10.1, formula (10.9)).

Born's postulate defining quantum probability should also be completed by a definition of conditional probability. We have remarked that, for one concrete observable, the probability given by Born's rule can be treated classically. However, the definition of the conditional probability involves two observables. Such situations cannot be treated classically (at least straightforwardly, cf. section 10.14.1). Thus, conditional probability is really quantum probability.

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

Let ^ be a pure state and let kxj/ = 0. Then the probability to get the value b = pm under the condition that the value a = ak was observed in the preceding measurement of the observable a on the state xj/ is given by the formula

One can motivate this definition by appealing to the projection postulate (Liiders’ version). After the a measurement with output a = ak initially prepared state xj/ is projected onto the state

Then one applies Born's rule to the b measurement for this state.

Conditional Probability for Observables with a Nondegenerate Spectrum

Let the operator a have a nondegenerate spectrum, i.e., for any eigenvalue a the corresponding eigenspace (i.e., generated by eigenvectors with axj/ = ax//) is one-dimensional. We can write

(here aeak = akeak).

Independence of the Initial State

Thus, the conditional probability in this case does not depend on the original state f. We can say that the memory about the original state was destroyed. If also the operator b has a nondegenerate spectrum, then we have

and

By using symmetry of the scalar product we obtain the following result:

Matrix of Transition Probabilities: Symmetric

Letboth operatorsa and b have purely discrete nondegenerate spectra and let Pfxj/ Ф 0 and Pf, f Ф 0. Then conditional probability is symmetric and it does not depend on the original state f:

We remark that classical (Kolmogorov-Bayes) conditional probability is not symmetric in special situations. Thus QM is described by a very specific probabilistic model.

Matrix of Transition Probabilities: Double Stochasticity

Consider two nondegenerate observables. Set = P(b = f a = a). The matrix of transition probabilities РЬ|а = (p) is not only stochastic, i.e.,

but it is even doubly stochastic:

In Kolmogorov’s 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 specific 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 [152]. 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].

Interference of Probabilities for Incompatible Observables

We shall show that the quantum probabilistic calculus violates the conventional FTP (10.10), one of the basic laws of classical probability theory. In this section, we proceed in an abstract setting by operating with two abstract incompatible observables. The concrete realization of this setting for the two-slit experiment demonstrating the interference of probabilities in QM will be presented in section 10.14, which is closely related to Feynman's claim [79] on the nonclassical probabilistic structure of this experiment.

Let H2 = C x C be the two-dimensional complex Hilbert space and let f e H2 be a quantum state. Let us consider two dichotomous observables b = f>2 and a = ait 2 represented by Hermitian

operators b and a, respectively (one may consider simply Hermitian matrices). Let eb = (e^) and ea = (e“) be two orthonormal bases consisting of eigenvectors of the operators. The state f can be represented in the two ways:

By Postulate 4 we have

The possibility of expanding one basis with respect to another basis induces a connection between the probabilities P(a = a) and P(b = f). Let us expand the vectors e" with respect to the basis eb:

where иар = (e“, e£). Thus dx = cxun + c2u21, d2 = cxui2 + cxu22. We obtain the quantum rule for transformation of probabilities:

On the other hand, by the definition of quantum conditional probability (see (10.39)), we obtain

By combining Eqs. 10.44, 10.45 and 10.48, 10.49 we obtain the quantum formula of total probability—the formula of the interference of probabilities:

In general, cos в = 0. Thus, the quantum FTP does not coincide with the classical FTP (10.10), which is based on the Bayes' formula.

We considered only the case of observables presented by operators acting in the two-dimensional Hilbert space (see [9] for arbitrary dimensions and even representation of observables by POVMs).

Logic of Quantum Propositions

Von Neumann and Birkhoff [31, 235] suggested representing events (propositions) by orthogonal projectors in complex Hilbert space H.

For an orthogonal projector P, we set HP = Р(Я), its image, and vice versa. For subspace L of H, the corresponding orthogonal projector is denoted by the symbol PL.

The set of orthogonal projectors is a lattice with the order structure P < Qif HP c Hq or equivalently, for any )/ e H, {x//Pl/) <

We recall that the lattice of projectors is endowed with operations "and” (л) and "or” (v). For two projectors Pb P2, the projector R = Pi л P2 is defined as the projector onto the subspace PR = Hpt П Hp2 and the projector 5 = Pi v P2 is defined as the projector onto the subspace PR defined as the minimal linear subspace containing the set-theoretic union U HP.2 of subspaces HPl, HPl: this is the space of all linear combinations of vectors belonging these subspaces. The operation of negation is defined as the orthogonal complement Px = H : (y|x) = 0 for all x 6 Hp).

In the language of subspaces the operation “and” coincides with the usual set-theoretic intersection, but the operations "or” and "not” are nontrivial deformations of the corresponding set-theoretic operations. It is natural to expect that such deformations can induce deviations from classical Boolean logic.

Consider the following simple example. Let Я be a two- dimensional Hilbert space with the orthonormal basis (ei, e2) and let v = [e + e2)/s/2. Then, Pv л Pe, = 0 and Pv л Pe.2 = 0, but Pv A v Pe2) = Pv. Hence, for quantum events, in general the distributivity law is violated:

The lattice of orthogonal projectors is called quantum logic. It is considered as a (special) generalization of classical Boolean logic. Any sub-lattice consisting of commuting projectors can be treated as classical Boolean logic.

At first sight the representation of events by projectors/linear subspaces might look exotic. However, this is simply a prejudice which springs from too common usage of the set-theoretic representation of events (Boolean logic) in the modern classical probability theory. The tradition to represent events by subsets was firmly established by A. N. Kolmogorov in 1933. We remark that before him the basic classical probabilistic models were not of the set-theoretic nature. For example, the main competitor of the Kolmogorov model, the von Mises frequency model [232-234], was based on the notion of a collective.

As we have seen, quantum logic relaxes some constraints posed on the operations of classical Boolean logic, in particular the dis- tributivity constraint. This provides novel possibilities for logically consistent reasoning; cf. with the information interpretation of QM [39-43, 48-51, 58, 59, 89-97, 138, 144-149, 175, 205-209, 238] and applications of the mathematical formalism of QM (in fact, reasoning and decision making based on quantum logic) outside of physics.

Tensor Product of Hilbert Spaces and Linear Operators

Let both state spaces be L2 spaces, the spaces of complex-valued square integrable functions: H = L2(Rfc) and Z,2(Rm).

Take two functions; ф = ^(x) belongs to Hi and ф = ф[у) belongs to H2. By multiplying these functions we obtain the function of two variables Ф(х, у) = ф(х) x ф[у), where x denotes the usual pointwise product.[1] It is easy to check that this function belongs to the space H = L2{Rk+m). Take now n functions, ф[х), • • •. iAnOO- from Hi and n functions, i[y),..., Ф„(у), from H2 and consider the sum of their pairwise products:

i

This function also belongs to H.

It is possible to show that any function belonging to H can be represented as Eq. 10.52, where the sum is in general infinite. Multiplication of functions is the basic example of the operation of the tensor product. The latter is denoted by the symbol . Thus, in the example under consideration ф ® Ф{х, у) = iA(x) x Ф{у)- The tensor product structure on H = Z,2(R,(+m) is symbolically denoted as H = Hj H2.

Consider now two arbitrary orthonormal bases in spaces Hk, (ej^), к = 1,2. Then, functions (e,y = e-1^ e^) form an orthonormal basis in H : any Ф e H, can be represented as

where

Consider now two arbitrary finite dimensional Hilbert spaces, Hi, H2. For each pair of vectors фHi, ф e H2, we form a new formal entity denoted by ф ® ф. Then we consider the sums Ф = i ф, <8> Фь On the set of such formal sums we can introduce the linear space structure. (To be mathematically rigorous, we have to constrain this set by some algebraic relations to make the operations of addition and multiplication by complex numbers well defined). This construction gives us the tensor product H = Hi H2. In particular, if we take orthonormal bases in Hk, (e^), к = 1,2, then (e,y = e)1' g> eform an orthonormal basis in H, any Ф e H, can be represented as (10.53) with (10.54).

The latter representation gives the simplest possibility to define the tensor product of two arbitrary (i.e., may be infinitedimensional) Hilbert spaces as the space of formal series (10.53) satisfying the condition (10.54).

Besides the notion of the tensor product of states, we shall also use the notion of the tensor product of operators. Consider two linear operators я, : H, -> Я,, / = 1, 2. Their tensor product a = a2 : H —>■ H is defined starting with the tensor products of

two vectors:

Then it is extended by linearity. By using the coordinate representation (10.53) the tensor product of operators can be represented as

If operators S/, / = 1, 2, are represented by matrices (a^), with respect to the fixed bases, then the matrix (flw.nm) of the operator

a with respect to the tensor product of these bases can be easily calculated.

In the same way, one defines the tensor product of Hilbert spaces Hi,..., H„ denoted by the symbol H = Hi ® ... Hn. We start with forming the formal entities 1Д1 ... <8> V'n, where j/jHj, j =

1,, rt. Tensor product space is defined as the set of all sums j iAi; ® ■■■ Q'l'nj (which has to be constrained by some algebraic relations, but we omit such details). Take orthonormal bases in Hk, [ey), к = 1,..., n. Then, any Ф e H can be represented as

where 0|c„|2

  • [1] Here it is convenient to use this symbol, not just write Ф(х, у) = '1г(_х)ф(у).
 
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