Showing papers by "Vladimir I. Man’ko published in 2017"
TL;DR: In this article, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus is derived, from quantum relative Tsallis entropy.
Abstract: In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2, N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states. By using the tomographic picture of quantum mechanics we have obtained the Fisher- Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.
74 citations
01 Jan 2017
70 citations
TL;DR: In this article, the authors introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distribution describing quantum states. And they present quantum channels for qubits in the probability representation.
Abstract: We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.
49 citations
TL;DR: In this article, the density matrix of the qubit (spin-1/2) state associated with the Bloch sphere was given in the tomographic probability representation onto vertices of a triangle determining Triada of Malevich's squares.
Abstract: We map the density matrix of the qubit (spin-1/2) state associated with the Bloch sphere and given in the tomographic probability representation onto vertices of a triangle determining Triada of Malevich’s squares. The three triangle vertices are located on three sides of another equilateral triangle with the sides equal to
$$ \sqrt{2} $$
. We demonstrate that the triangle vertices are in one-to-one correspondence with the points inside the Bloch sphere and show that the uncertainty relation for the three probabilities of spin projections +1/2 onto three orthogonal directions has the bound determined by the triangle area introduced. This bound is related to the sum of three Malevich’s square areas where the squares have sides coinciding with the sides of the triangle. We express any evolution of the qubit state as the motion of the three vertices of the triangle introduced and interpret the gates of qubit states as the semigroup symmetry of the Triada of Malevich’s squares. In view of the dynamical semigroup of the qubit-state evolution, we constructed nonlinear representation of the group U(2).
49 citations
TL;DR: In this paper, a review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin-systems with discrete variables, and new entropic information inequalities are obtained for Franck-Condon factors.
Abstract: Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin--systems with discrete variables. New entropic--information inequalities are obtained for Franck--Condon factors. Density matrices of qudit states are expressed in terms of probabilities of artificial qubits as well as the quantum suprematism approach to geometry of these states using the triadas of Malevich squares is developed. Examples of qubits, qutrits and ququarts are considered.
16 citations
TL;DR: In this article, the authors discuss the procedure of different partitions in the finite set of N integer numbers and construct generic formulas for a bijective map of real numbers sy, where y = 1, 2,…, N, N = \underset{k=1}{\overset{n}{\varPi}}{X}_k, \) and Xk are positive integers.
Abstract: We discuss the procedure of different partitions in the finite set of N integer numbers and construct generic formulas for a bijective map of real numbers sy, where y = 1, 2,…, N, N = \( \underset{k=1}{\overset{n}{\varPi}}{X}_k, \) and Xk are positive integers, onto the set of numbers s(y(x1, x2,…, xn)). We give the functions used to present the bijective map, namely, y(x1, x2, …, xn) and xk(y) in an explicit form and call them the functions detecting the hidden correlations in the system. The idea to introduce and employ the notion of “hidden gates” for a single qudit is proposed. We obtain the entropic-information inequalities for an arbitrary finite set of real numbers and consider the inequalities for arbitrary Clebsch–Gordan coefficients as an example of the found relations for real numbers.
10 citations
TL;DR: An explicit description of all the minimum self-dual sets of dequantizers and quantizers for a qubit system is derived.
9 citations
TL;DR: In this paper, a generalization of the quantizer-dequantizer scheme on spin-1/2 states is discussed. But the results are restricted to spin tomograms of the state density matrix.
Abstract: We review the quantizer–dequantizer formalism of constructing symbols of the density operators and quantum observables, such as Wigner functions and tomographic-probability distributions. We present a tutorial consideration of the technique of obtaining minimal sets of dequantizers (quorum) related to the observable eigenvalues for one-qubit states. We discuss a generalization of the quantizer–dequantizer scheme on the example of spin-1/2 states. We consider the possibilities of extending the results to two-qubit systems using spin tomograms of the state density matrix.
9 citations
TL;DR: In this article, the gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated, and explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained.
Abstract: The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations have the Liouville equation in corresponding representations as the classical limits are found.
7 citations
TL;DR: In this paper, the evolution of qubit states for the Demkov problem in the presence of dephasing processes in the spin tomographic-probability representation is considered.
Abstract: We consider the evolution of qubit states for the Demkov problem in the presence of dephasing processes in the spin tomographic-probability representation. We present an explicit solution of the spin tomogram in terms of the 1
F
2 hypergeometric function. We calculate the tomographic Shannon and q entropies through the solution of the master equation in the form of tomographic-probability distribution of the qubit states obtained.
5 citations
TL;DR: In this article, the authors consider tomograms and quasidistribution functions like the Wigner functions that violate the standard normalization condition, and obtain the conditions under which a reconstruction of the density matrix using these tomograms, and then they study an example of the de Broglie plane wave.
Abstract: We consider tomograms and quasidistribution functions like the Wigner functions that violate the standard normalization condition and obtain the conditions under which a reconstruction of the density matrix using these tomograms and quasidistribution functions is possible. Then we study an example of the de Broglie plane wave.
TL;DR: In this paper, the entanglement of even and odd f-coherent states is evaluated by the linear entropy, and the concept of nonlinear f-oscillators and their properties are recalled.
Abstract: Symplectic tomographies of classical and quantum states are shortly reviewed. The concept of nonlinear f-oscillators and their properties are recalled. The tomographic probability representations of oscillator coherent states and the problem of entanglement are then discussed. The entanglement of even and odd f-coherent states is evaluated by the linear entropy.
TL;DR: In this article, the authors used properties of the Shannon and Tsallis entropies to obtain new inequalities for the Clebsch-Gordan coefficients of SU(2).
Abstract: Using properties of the Shannon and Tsallis entropies, we obtain new inequalities for the Clebsch–Gordan coefficients of the group SU(2). For this, we use squares of the Clebsch–Gordan coefficients as probability distributions. The obtained relations are new characteristics of correlations in a quantum system of two spins. We also find new inequalities for Hahn polynomials and the hypergeometric functions 3F2.
TL;DR: In this paper, the entanglement of even and odd f-coherent states is evaluated by the linear entropy, and the concept of nonlinear f-oscillators and their properties are recalled.
Abstract: Symplectic tomographies of classical and quantum states are shortly reviewed. The concept of nonlinear f-oscillators and their properties are recalled. The tomographic probability representations of oscillator coherent states and the problem of entanglement are then discussed. The entanglement of even and odd f-coherent states is evaluated by the linear entropy.
TL;DR: New entropic and information inequalities for density matrices and vector tomographic portraits of spin-1 quantum particle states were obtained in this article, where the authors also considered the problem of spin 1 particle states.
Abstract: New entropic and information inequalities for density matrices and vector tomographic portraits of spin-1 quantum particle states are obtained
TL;DR: The aim of this paper is to give an interpretation of the linear or nonlinear filtration problem as a quantization problem, and extends the optimal filtering equation known from the Stratonovich filtering theory on the quantum process case.
Abstract: We extend the optimal filtering equation known from the Stratonovich filtering theory on the quantum process case. The used observation model is based on an indirect measurement method, where the measurement is performed on an ancilla system that is interacted with an unknown one. Observation model for single qudit system is proposed.
TL;DR: In this article, the authors investigated quantum correlations in the state of a four-level atom by using generic unitary transforms of the classical density matrix and derived properties of entanglement in terms of concurrence, entropy and negativity.
Abstract: Quantum correlations in the state of four-level atom are investigated by using generic unitary transforms of the classical (diagonal) density matrix. Partial cases of pure state, $X$-state, Werner state are studied in details. The geometrical meaning of unitary Hilbert reference-frame rotations generating entanglement in the initially separable state is discussed. Characteristics of the entanglement in terms of concurrence, entropy and negativity are obtained as functions of the unitary matrix rotating the reference frame.
TL;DR: In this article, a general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations is given, and the correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators in the form of singular and regular generalized functions are derived.
Abstract: Symplectic and optical joint probability representations of quantum mechanics are considered, in which the functions describing the states are the probability distributions with all random arguments (except the argument of time). The general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations is given. Taking the Gaussian functions as the distributions of the tomographic parameters the correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators in the form of singular and regular generalized functions are derived. Evolution equations and stationary states equations for symplectic and optical joint probability distributions are obtained.
TL;DR: In this article, a simple system of two particles in a bidimensional configurational space S is studied, and the possibility of breaking in S the time-independent Schrodinger equation of the system into two separated one-dimensional one-body Schodoringer equations is assumed, where the latter property is countered by imposing such boundary conditions as confinement to a limited region of S and/or restrictions on the joint coordinate probability density stemming from the sign-invariance condition of the relative coordinate.
TL;DR: In this article, an invertible mapping for the irreducible unitary representation of groups SU(2) and SU(1, 1) like Jacoby polynomials and Gauss hypergeometric functions, respectively, are used.
Abstract: Using the entropic inequalities for Shannon and Tsallis entropies new inequalities for some classical polynomials are obtained. To this end, an invertible mapping for the irreducible unitary representation of groups SU(2) and SU(1, 1) like Jacoby polynomials and Gauss’ hypergeometric functions, respectively, are used.
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TL;DR: In this article, the notion of quantum channels was formulated in the framework of quantum tomography and the issue of whether such channels can be regarded as classical stochastic maps was addressed.
Abstract: We formulate the notion of quantum channels in the framework of quantum tomography and address there the issue of whether such maps can be regarded as classical stochastic maps. In particular kernels of maps acting on probability representation of quantum states are derived for qubit and bosonic systems. In the latter case it results that a single mode Gaussian quantum channel corresponds to non-Gaussian classical channels.
TL;DR: In this article, the entropy-energy inequality for a three-level atom implemented on superconducting circuits with the Josephson junction was investigated and the positivity of the relative entropy of the qutritquantum system was used for verification of tomography of quantum states of qudits.
Abstract: We consider the entropy-energy inequality for a three-level atom implemented on superconducting circuits with the Josephson junction. It is suggested to use the positivity of the relative entropy of the qutritquantum system for verification of tomography of quantum states of qudits. The relations obtained are considered in detail on the example of the temperature density matrix.
TL;DR: In this article, the authors consider quadratic tomography in star product formalism and consider the behavior of the associative algebra of quad ratic tomographic symbols in the ℓ-rightarrow 0$ limit.
Abstract: We consider quadratic tomography in star product formalism. The contraction and the behavior of the associative algebra of quadratic tomographic symbols in $\hbar\rightarrow 0$ limit are discussed. A simple $k$-deformation example is illustrated.