Entropy of conditional tomographic probability distributions for classical and quantum systems

Abstract

Abstract. The possibility to describe hybrid systems containing classical and quantum subsystems by means of conditional tomographic probability distributions (tomograms) is discussed. Tomographic Shannon and Rényi entropies associated with the tomograms are studied, and new tomographic uncertainty relations are obtained. IntroductionThe possibility to describe the hybrid systems containing both classical and quantum subsystems was considered, for example, in [1][2][3][4][5]. The problems presented in the process of considering combined systems are related to a difference in the description of classical and quantum states. The classical states are associated with the probability density on the phase space. The quantum states are associated either with the wave function or with the density matrix [6,7].Recently the probability representation of quantum states was introduced using the tomographic approach [8] (see also the review [9]). Within the framework of this approach, the states in quantum mechanics are identified with the probability density of the particle position (called the tomogram) measured in a rotated and scaled reference frame in the phase space. It is important that the states in classical mechanics can also be identified with tomograms [10,11].In view of the mentioned above, the difference in the concept of classical and quantum states in the tomographic approach is reduced to the difference of the sets where the classical and quantum tomograms are located. This circumstance was used to make an attempt to describe a hybrid system consisted of the classical and quantum subsystems by a joint tomographic probability distribution depending on the positions of the classical and quantum particles. Also it was introduced that the tomograms can be interpreted as conditional probability distributions [12,13]. For example, the optical tomogram w(X, θ), being dependent on a random homodyne quadrature X and the local oscillator phase θ [14,15], which is a measurable parameter (see, e.g., [16,17]), can be interpreted as the conditional probability density w(X, θ) = w(X | θ).In view of Bayesian formula, the conditional probability distribution is connected with a joint probability distribution P (X, θ) of the both random variables.

Maps of Matrices and Portrait Maps of the Density Operators of Composite and Noncomposite Systems

“…Since there exists the strong subadditivity condition for the density matrix of the three-partite system [23], we can obtain a new matrix inequality, which is an analog of this condition, for an arbitrary Hermitian N ×N matrix, including the density matrix of the single qudit state. We continue the consideration of the found matrix inequalities in the form of relations for qudit tomograms of classical and quantum system states [37][38][39][40], empolying the inequalities for the probability vectors depending on the parameters of the unitary matrix in a future publication.…”

Conditional Information and Hidden Correlations in Single-qudit States

“…As we have discussed in [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], one can consider probability distributions of composite and noncomposite systems as a set of N nonnegative numbers; in the case of N = N 1 N 2 or N = N 1 N 2 N 3 , where N 1 , N 2 , and N 3 are integers, one can obtain new entropy-information relations for noncomposite systems. Analogously, for quantum indivisible systems, one can obtain the relations for entropy and information analogous to the relations known for multiqudit systems.…”

Information Processing Using Three-Qubit and Qubit–Qutrit Encodings of Noncomposite Quantum Systems

“…Recently, multilevel (i.e., noncomposite) quantum systems attracted a significant deal of interest as a potential platform for quantum technologies [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. On the one hand, such multilevel quantum systems can be realized using a single physical system, e.g., multilevel superconducting artificial atoms [26][27][28][29][30].…”

“…Speaking in terms of the quantum Shannon theory [1], one can establish a direct correspondence between information and entropic measures for composite and noncomposite quantum systems. Comprehensive studies of information and entropic characteristics of noncomposite quantum system demonstrated their resource for information processing [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In particular, the single qudit version of the Deutsch algorithm using through anharmonic superconducting multilevel artificial atom has been suggested [10].…”