Ket and Bra Vectors: Dirac's Symbolism

Dirac’s notations [65] are widely used in quantum information theory. Vectors of H are called ket vectors and are denoted as xj/}. The elements of the dual space H' of H, the space of linear continuous functionals on H, are called bra vectors and are denoted as (ф.

Originally, the expression (ф ф) was used by Dirac for the duality form between H' and H, i.e., {фФ) is the result of application of the linear functional to the vector |). In mathematical notation it can be written as follows. Denote the functional (x// by / and the vector Iф) by simply ф. Then, {фф) = /(). To simplify the model, Dirac later took the assumption that H is a Hilbert space, i.e., H1 can be identified with H. We remark that this assumption is an axiom simplifying the mathematical model of QM. However, in principle, Dirac’s formalism [65] is applicable for any topological linear space H and its dual space H', so it is more general than von Neumann's formalism [235] rigidly based on Hilbert space.

Consider an observable a given by the Hermitian operator a with a nondegenerate spectrum and restrict our consideration to the case of finite-dimensional H. Thus, the normalized eigenvectors e, of A form the orthonormal basis in H. Let oe, = a,e,. In Dirac's notation e, is written as |a, ) and, hence, any pure state can be written as

Since the projector onto |a,-> is denoted as Pai = |a,•>{<*,|, the operator a can be written as

Now consider two Hilbert spaces Hi and H2 and their tensor product H = Hi H2. Let (|<*,)) and be orthonormal bases in Hi and H2 corresponding to the eigenvalues of two observables A and B. Then, vectors |a, ) H. Typically in physics the sign of the tensor product is omitted and these vectors are written as |a/)|/S;) or even as |or,-/3y). Thus, any vector (/H = Hi H2 can be represented as

where ctj e C (in the infinite-dimensional case these coefficients are constrained by the condition y|cy|2 < oo).

Quantum Bit: Using State Superposition for Information Encoding

In particular, in quantum information theory, typically qubit states are represented with the aid of observables having the eigenvalues 0, 1. Each qubit space is two-dimensional:

A pair of qubits is represented in the tensor product of single qubit spaces. Here, pure states can be represented as superpositions:

where ijctj2 = 1. In the same way the n-qubit state is represented in the tensor product of n one-qubit state spaces (it has the dimension 2"):

where x _0,i |cXl...Xn|2 = 1. We remark that the dimension of the n-qubit state space grows exponentially with the growth of n. The natural question about possible physical realizations of such multidimensional state spaces arises. The answer to it is not completely clear; it depends very much on the used interpretation of the wave function.

Entanglement of Pure and Mixed Quantum States

Consider the tensor product Я = Hi H2 .-®Hn of Hilbert spaces

Hk, к = 1,2,, n. The states of the space H can be separable and nonseparable (entangled). We start by considering pure states. The states from the first class, separable pure states, can be represented in the form

where x//k) e Hk. The states that cannot be represented in this way are called nonseparable, or entangled. Thus, mathematically, the notion of entanglement is very simple: it means impossibility of tensor factorization.

For example, let us consider the tensor product of two one- qubit spaces. Select in each of them an orthonormal basis denoted as |0), |1). The corresponding orthonormal basis in the tensor product has the form 100), 101), 110), 111). Here, we used Dirac's notations (see section 10.11) near the end. Then, so-called Bell's states

are entangled.

Although the notion of entanglement is mathematically simple, its physical interpretation is one of the main problems of modern quantum foundations. The common interpretation is that entanglement encodes quantum nonlocality, the possibility of action at a distance (between parts of a system in an entangled state). Such an interpretation implies drastic change in all classical physical presentations about nature, at least about the microworld. In probabilistic terms, entanglement induces correlations which are too strong to be described by classical probability theory. (At least this is the common opinion of experts in quantum information theory and quantum foundations.) Such correlations violate the famous Bell inequality, which can be derived only in a classical probability framework. The latter is based on the use of a single probability space covering probabilistic data collected in a few incompatible measurement contexts.

Now consider a quantum state given by density operator p in H. This state is called separable if it can be factorized in the product of density operators in spaces Hk:

Otherwise the state p is called entangled. We remark that an interpretation of entanglement for mixed states is even more complicated than for pure states.

Two-Slit Experiment and Violation of the Classical Law of Total Probability

On the Possibility of Classical Probabilistic Description of Quantum Experiments

In this section we discuss the old foundational problem of QM: whether it is possible to represent quantum states by classical probability (CP) distributions and quantum observables by random variables [197]. Since the first days of QM, it was commonly believed that classical probability theory (based on Kolmogorov's axiomatics [191]) cannot serve to represent incompatible quantum observables. At the early stage of development of QM this belief was firmly based on the Heisenberg uncertainty principle. By a straightforward interpretation of the Heisenberg uncertainty relation, position and momentum cannot be jointly assigned to an individual quantum system. Under such an interpretation, it was meaningless to even speak about the joint probability distribution

(jpd) for position and momentum. For example, for Wigner [237], it was clear that

In quantum theory there does not exist any similar simple expression for the probability, because one cannot ask for the simultaneous probability for the coordinates and momenta.

(p. 749)

Here, "similar” is related to the case of classical statistical mechanics and the Gibbs-Boltzmann formula for statistical equilibrium.

The two-slit experiment is the basic example demonstrating that QM describes statistical properties in microscopic phenomena, to which the classical probability theory seems to be not applicable (see Feynman [78]) (italic shrift was added by the authors of this paper):

From about the beginning of the twentieth century experimental physics amassed an impressive array of strange phenomena which demonstrated the inadequacy of classical physics. The attempts to discover a theoretical structure for the new phenomena led at first to a confusion in which it appeared that light,and electrons, sometimes behaved like waves and sometimes like particles. This apparent inconsistency was completely resolved in 1926 and 1927 in the theory called quantum mechanics. The new theory asserts that there are experiments for which the exact outcome is fundamentally unpredictable, and that in these cases one has to be satisfied with computing probabilities of various outcomes. But far more fundamental was the discovery that in nature the laws of combining probabilities were not those of the classical probability theory of Laplace.

The latter statement of Feynman is based on his analysis of the probabilistic structure of the two-slit experiment [78, 79]. This analysis demonstrated (at least as Feynman believed) that probabilities associated with this experiment violate the law of additivity of probability, the basic law of classical probability theory. In a series of authors' works [133, 136, 141, 145, 153, 154], Feynman's analysis [79] was embedded into the contextual framework, i.e., treatment of the two-slit experiment as a bunch of experiments related to different experimental contexts (see Figs. 10.2, 10.3, 10.4). This contextual probabilistic analysis led to the conclusion that the probabilities for the two-slit experiment (in fact, a bunch of experiments) collected experimentally and predicted by QM violate FTP (section 10.1, Eq. 10.10). In the abstract framework for incompatible observables, this statement was proven in section 10.8. QM predicts a modified formula of total probability, FTP with the interference term.

We remark that Feynman’s no-go conclusion [79] based on consideration of the two-slit experiment is justified better than the original claims based on the Heisenberg uncertainty relation. The role of Heisenberg’s uncertainty relation for quantum foundations is a complicated topic. On the one hand, this relation played a crucial role in establishing the complementarity principle by Bohr. On the other, it is in general interpreted too straightforwardly as carrying the message about the impossibility of the joint determination of the position and momentum for an individual quantum system. As was rightly pointed by Margenau and Ballentine [17-19] (see, e.g., [153] for discussion, Heisenberg's uncertainty relation involves the standard deviations for the position and momentum observables. These deviations are measured in two separate experiments, for the position and momentum observables, respectively. It seems that such separate measurements have nothing to do with joint measurability of these observables. Therefore, this relation cannot be used to justify the claim that the classical probability distribution for these observables does not exist.

In any event the problem of the interrelation of classical and quantum probabilities is very complex. Of course, as was already remarked, for one concrete quantum observable, the statistics of its outcomes is well described by classical probability theory. The main problem is whether it is possible to use classical probability to describe the statistics of outcomes of incompatible quantum observables. On the one hand, the contextual structure of measurements and the impossibility to combine the contexts of incompatible measurements lead to the contextual probabilistic description [133, 136, 141, 148, 151, 153, 172], which is not covered straightforwardly by the Kolmogorov probability model. For example, as was shown in section 10.1, classical FTP is violated in QM formalism. In section 10.14.2, we shall illustrate this situation by the two-slit experiment. On the other hand, the contextual decomposition of probability is not totally incompatible with Kolmogorov’s theory. In principle, incompatible contexts can be embedded into the common Kolmogorov probability space, but not straightforwardly.

The first classical probabilistic representation of QM based on the symplectic tomogram was constructed [66, 195, 196]. Another construction of the classical probabilistic representation of QM is based on the so-called prequantum classical statistical field theory [157-160, 162, 174]. Representation of the Bohm-Bell- type experiments based on classical conditional probabilities was constructed [13, 165, 179]; see [164] for application of such a technique to the two-slit experiment. Recently, Dzhafarov et al. [70- 72] [and especially article [73] and the comment on the latter by Khrennikov [183]) proposed a classical probabilistic description of the basic QM experiments based on the coupling technique which widely used in classical probability.

Interference of Wave Functions

In this section, we consider the experiment with the symmetric setting: the source of photons is located symmetrically with respect to two slits (Fig. 10.1).

Consider the pair of observables a and b. We select a as the "slit passing observable," i.e., a = 0, 1 (Fig. 10.1) (we use indexes 0, 1 to be close to qubit notation) and observable b as the position on the photosensitive plate (Fig. 10.2). We remark that the b- observable has a continuous range of values, the position x on the photosensitive plate. We denote P(a = /) by P(/) (/ = 0, 1), and P(b = x) by P(x). Physically, the я-observable corresponds to measurement of position (coarse-grained to "which slit?”) and the Ь-observable represents the measurement of momentum.

In quantum foundational studies, various versions of the two-slit experiment have been successfully performed not only with photons but also with electrons and even with macroscopic molecules (by

Experimental setup

Figure 10.1 Experimental setup.

Context with both slits open

Figure 10.2 Context with both slits open.

Zeilinger’s group). All those experiment demonstrated matching with predictions of QM. Experimenters reproduce the interference patterns predicted by QM and calculated by using wave functions.

Context with one slit closed-I

Figure 10.3 Context with one slit closed-I.

The probability that a photon is detected at position x on the photosensitive plate is represented as

where j/0 and i/q are two wave functions, whose squared absolute values |^,(x)|2 g*ve the distributions of photons passing through the slit / = 0, 1, see Figs. 10.3 and 10.4. Here, we explored the rule of addition of complex probability amplitudes, a quantum analogue of the rule of addition of probabilities. This rule is the direct consequence of the linear space structure of quantum state spaces.

The term

implies the interference effect of two wave functions. Let us denote |2 by P(x|i). Then, Eq. 10.67 is represented as

Context with one slit is closed-11

Figure 10.4 Context with one slit is closed-11.

Here, the values of probabilities P(0) and P(l) are equal to 1/2 since we consider the symmetric settings. For general experimental settings, P(0) and P(l) can be taken as the arbitrary nonnegative values satisfying P(0) + P(l) = 1. In the above form, the classical probability law (FTP)

is violated, and the term of interference

specifies the violation.

The crucial point is that the two-slit experiment has a multicon- textual structure with three different contexts:

  • (1) C0: only the / = Oth slit is open (Fig. 10.3)
  • (2) Cx only the / = 1th slit is open (Fig. 10.4)
  • (3) C01: both slits are open (Fig. 10.2)

Comparison of possibilities is represented as comparison of the corresponding probability distributions P(x|/), P(x). In the contextual notations they can be written as

Here, conditioning is not classical probabilistic event conditioning, but context conditioning [153]: different contexts are mathematically represented by different Kolmogorov probability spaces. The general contextual probability theory including its representation in a complex Hilbert space is presented in detail in my monograph [153].

 
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