Showing papers by "Vladimir I. Man’ko published in 2016"
Abstract: We extend the Bell inequality known for two qubits to the four-level atom, including an artificial atom realized by the superconducting circuit, and qudit with j = 3/2. We formulate the extended inequality as the inequality valid for an arbitrary Hermitian nonnegative 4×4 matrix with unit trace for both separable and entangled matrices.
18 citations
TL;DR: In this article, the quantum-to-classical transition from the point of view of contractions of associative algebras is considered from the perspective of the quantum case.
Abstract: The quantum-to-classical transition is considered from the point of view of contractions of associative algebras. Various methods and ideas to deal with contractions of associative algebras are discussed that account for a large family of examples. As an instance of them, the commutative algebra of functions in phase space, corresponding to classical physical observables, is obtained as a contraction of the Moyal star-product which characterizes the quantum case. Contractions of associative algebras associated to Lie algebras are discussed, in particular the Weyl–Heisenberg and SU(2) groups are considered.
14 citations
TL;DR: In this article, the von Neumann and linear entropies were calculated to measure entanglement properties between the modes of a parametric amplifier and used to determine the evolution of a general two-mode Gaussian state in the tomographic probability representation.
Abstract: We obtain the linear time-dependent constants of motion of the parametric amplifier and use them to determine the evolution of a general two-mode Gaussian state in the tomographic-probability representation. By means of the discretization of the continuous variable density matrix, we calculate the von Neumann and linear entropies to measure the entanglement properties between the modes of the amplifier. We compare the obtained results for the nonlocal correlations with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. We use this qubit portrait procedure to establish Bell-type inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. We define the other no-signaling nonlocal correlations through the portrait procedure for noncomposite systems.
12 citations
TL;DR: In this paper, the star-product formalism for spin states was developed and different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them.
Abstract: We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.
9 citations
TL;DR: In this article, a nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed.
Abstract: The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions like Wigner function, Husimi Q−function, and Glauber-Sudarshan P−function. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitrary electro-magnetic field, which are analogs of non-relativistic Proca equation in appropriate representations. The generalization of proposed approach to the cases of arbitrary spin is discussed. The possibility of formulation of quantum mechanics of the systems with spins in terms of joint probability distributions without the use of wave functions or density matrices is explicitly demonstrated.
9 citations
TL;DR: In this article, the quantum-to-classical transition is considered from the point of view of contractions of associative algebras, in particular the Weyl-Heisenberg and $SU(2)$ groups, and the commutative algebra of functions in phase space, corresponding to classical physical observables, is obtained as a contraction of the Moyal star product which characterizes the quantum case.
Abstract: The quantum-to-classical transition is considered from the point of view of contractions of associative algebras. Various methods and ideas to deal with contractions of associative algebras are discussed that account for a large family of examples. As an instance of them, the commutative algebra of functions in phase space, corresponding to classical physical observables, is obtained as a contraction of the Moyal star-product which characterizes the quantum case. Contractions of associative algebras associated to Lie algebras are discussed, in particular the Weyl-Heisenberg and $SU(2)$ groups are considered.
7 citations
TL;DR: In this article, the steering property known for two-qubit state in terms of specific inequalities for the correlation function is translated for the state of qudit with the spin j = 3 / 2.
Abstract: The steering property known for two-qubit state in terms of specific inequalities for the correlation function is translated for the state of qudit with the spin j = 3 / 2 . Since most steering detection inequalities are based on the correlation functions we introduce analogs of such functions for the single qudit systems. The tomographic probability representation for the qudit states is applied. The connection between the correlation function in the two-qubit system and the single qudit is presented in an integral form with an intertwining kernel calculated explicitly in tomographic probability terms.
5 citations
TL;DR: In this paper, the star-product quantization procedure for spin-1/2 particles (qubits) employing the construction of a pair of operators (dequantizers and quantizers) is described.
Abstract: We show the star-product quantization procedure for spin-1/2 particles (qubits) employing the construction of a pair of operators – dequantizers and quantizers. We present an explicit description of all minimal systems of such dequantizers and quantizers and discuss their relation to the probability representation of spin states where the fair probability distribution is identified with the spin states. We give some examples and discuss the possibility of constructing a symplectic structure in the finite-dimensional phase space.
4 citations
TL;DR: In this article, it was shown that the one-qubit teleportation can be considered as a state transfer between subspaces of the whole Hilbert space of an indivisible eight-dimensional system.
Abstract: Teleportation protocol is conventionally treated as a method for quantum state transfer between two spatially separated physical carriers. Recent experimental progress in manipulation with high-dimensional quantum systems opens a new framework for implementation of teleportation protocols. We show that the one-qubit teleportation can be considered as a state transfer between subspaces of the whole Hilbert space of an indivisible eight-dimensional system. We explicitly show all corresponding operations and discuss an alternative way of implementation of similar tasks.
4 citations
TL;DR: In this article, the Wigner 3-j symbol was shown to satisfy new inequalities for Hahn polynomials and hypergeometric functions, using the properties of Shannon and Tsallis entropies.
Abstract: The Clebsch-Gordan coefficients of the group SU(2) are shown to satisfy new inequalities. They are obtained using the properties of Shannon and Tsallis entropies. The inequalities associated with the Wigner 3-j symbols are obtained using the relation of Clebsch-Gordan coefficients with probability distributions interpreted either as distributions for composite systems or distributions for noncomposite systems. The new inequalities were found for Hahn polynomials and hypergeometric functions
TL;DR: In this article, the notion of weighted quantum entropy was extended to the case of indivisible qudit systems, such as a qutrit, and the role of weighted entropy with respect to nonlinear quantum channels was discussed.
Abstract: We review the notion of weighted quantum entropy and consider the weighted quantum entropy for bipartite and noncomposite quantum systems. We extend the subadditivity condition, the inequality known for the weighted entropy information, to the case of indivisible qudit system, such as a qutrit. We discuss the new inequality for the qutrit density matrix for different weights and states, as well as the role of weighted entropy with respect to nonlinear quantum channels.
TL;DR: In this article, the authors studied the quantum information properties of a seven-level system realized by a particle in a onedimensional square-well trap and discussed the features of encodings of seven-layer systems in a form of three-qubit or qubit-qutrit systems.
Abstract: We study quantum information properties of a seven-level system realized by a particle in a onedimensional square-well trap and discuss the features of encodings of seven-level systems in a form of three-qubit or qubit–qutrit systems. We use the three-qubit encoding of the system in order to investigate the subadditivity and strong subadditivity conditions for the particle’s thermal state. We employ the qubit–qutrit encoding to suggest a single qudit algorithm for calculating the parity of a bit string. The results obtained indicate on the potential resource of multilevel systems for realization of quantum information processing.
TL;DR: In this paper, two interacting superconducting circuits based on Josephson junctions, which can be precisely engineered and easily controlled, are discussed. And the authors use the parametric excitation of two circuits realized by an instant change of the qubit coupling to study entropic and information properties of a composite system.
Abstract: We discuss the known construction of two interacting superconducting circuits based on Josephson junctions, which can be precisely engineered and easily controlled. In particular, we use the parametric excitation of two circuits realized by an instant change of the qubit coupling to study entropic and information properties of the density matrix of a composite system. We obtain the density matrix from the initial thermal state and perform its analysis in the approximation of small perturbation parameter and sufficiently low temperature. We also check the subadditivity condition for this system both for the von Neumann entropy and deformed entropies and check the dependence of mutual information on the system temperature. Finally, we discuss the applicability of this approach to describe the two coupled superconducting qubits as harmonic oscillators with limited Hilbert space.
01 Mar 2016
TL;DR: In this article, it was shown that the density-matrix states of non-composite qudit systems satisfy entropic and information relations like the subadditivity condition, strong subadditive condition, and Araki-Lieb inequality, which characterize hidden quantum correlations associated with these indivisible systems.
Abstract: We show that the density-matrix states of noncomposite qudit systems satisfy entropic and information relations like the subadditivity condition, strong subadditivity condition, and Araki-Lieb inequality, which characterize hidden quantum correlations of observables associated with these indivisible systems. We derive these relations employing a specific map of the entropic inequalities known for density matrices of multiqudit systems to the inequalities for density matrices of single-qudit systems. We present the obtained relations in the form of mathematical inequalities for arbitrary Hermitian N × N-matrices. We consider examples of superconducting qubits and qudits. We discuss the hidden correlations in single- qudit states as a new resource for quantum technologies analogous to the known resource in correlations associated with the entanglement in multiqudit systems.
TL;DR: This work uses the discretization of the density matrix as a nonlinear positive map for systems with continuous variables for calculating the entanglement between two modes through different criteria, such as Tsallis entropy, von Neumann entropy and linear entropy, and the logarithmic negativity.
Abstract: The discretization of the density matrix is proposed as a nonlinear positive map for systems with continuous variables. This procedure is used to calculate the entanglement between two modes through different criteria, such as Tsallis entropy, von Neumann entropy and linear entropy and the logarithmic negativity. As an example, we study the dynamics of entanglement for the two-mode squeezed vacuum state in the parametric amplifier and show good agreement with the analytic results. The loss of information on the system state due to the discretization of the density matrix is also addressed.
TL;DR: In this paper, 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 paper, the authors propose the discretization of the density matrix as a nonlinear positive map for systems with continuous variables and use this procedure for calculating the entanglement between two modes through different criteria, such as Tsallis entropy, von Neumann entropy and linear entropy, and the logarithmic negativity.
Abstract: We propose the discretization of the density matrix as a nonlinear positive map for systems with continuous variables. We use this procedure for calculating the entanglement between two modes through different criteria, such as Tsallis entropy, von Neumann entropy and linear entropy, and the logarithmic negativity. As an example, we study the dynamics of entanglement for the two-mode squeezed-vacuum state in the parametric amplifier and show good agreement with the analytic results. Also we address the loss of information on the system state due to the discretization of the density matrix.
TL;DR: In this paper, 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.
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TL;DR: In this paper, the authors extended the Bell inequality known for two qubits to the four-level atom, including an artificial atom realized by the superconducting circuit and qudit with j = 3/2.
Abstract: We extend the Bell inequality known for two qubits to the four-level atom, including an artificial atom realized by the superconducting circuit, and qudit with j=3/2. We formulate the extended inequality as the inequality valid for an arbitrary Hermitian nonnegative 4x4-matrix with unit trace for both separable and entangled matrices.
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TL;DR: In this article, the entropy-energy inequality and the von Neumann entropic inequality for three level atom implemented on superconducting circuits with Josephson junction were compared for the qutrit system.
Abstract: We compare the entropy-energy inequality and the von Neumann entropic inequality for three level atom implemented on superconducting circuits with Josephson junction. The positivity of entropy and energy relations for the qutrit system are used for verification of state tomography of qudit systems. The results obtained are valid for generic quantum states (qudits) and are illustrated on the example of the temperature density matrix of the single qutrit state.
TL;DR: In this article, the Minkowski-type trace inequality for the density matrices of the qudit states in terms of the purity parameters was analyzed, paying special attention to the X-state of two qubits and a single qudit.
Abstract: We analyze the recently found inequality for eigenvalues of the density matrix and purity parameters describing either a bipartite-system state or a single-qudit state. We rewrite the Minkowski-type trace inequality for the density matrices of the qudit states in terms of the purity parameters and discuss the properties of the inequality obtained, paying special attention to the X-states of two qubits and a single qudit. Also we study the relation of the purity inequalities obtained with the entanglement.
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TL;DR: In this article, photon distribution functions for one-, two-and multi-mode squeezed states in terms of Hermite, Laguerre, Legendre polynomials and Gauss' hypergeometric functions are used.
Abstract: Using entropic inequalities for Shannon entropies new inequalities for some classical polynomials are obtained. To this end, photon distribution functions for one-, two- and multi-mode squeezed states in terms of Hermite, Laguerre, Legendre polynomials and Gauss' hypergeometric functions are used. The dependence between the violation of the quadrature uncertainty relation, the sign and the existence of the distribution function of such states is considered.
TL;DR: In this paper, the authors obtained new quantum inequalities for von Neumann entropy of the five-level atom, which are analogs of the subadditivity condition known for bipartite quantum systems and the strong subadditive condition for tripartite quantum system.
Abstract: We obtain new quantum inequalities for von Neumann entropy of the five-level atom, which are analogs of the subadditivity condition known for bipartite quantum systems and the strong subadditivity condition known for tripartite quantum systems. We discuss the possibility to check the inequalities for the single qudit with j = 2, which can be realized as a five-level atom in the experiments with superconducting circuits. We present the strong subadditivity conditions for the finite-level atomic populations.
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TL;DR: In this article, the positive invertible map of the mixed states of a single qutrit onto the states of two identical fermions is constructed, and it is shown that using this one-to-one correspondence between qutris states and states of the two identical three-dimensional fermians, one may attribute hidden entanglement to a single mixed state of qutres.
Abstract: We construct the positive invertible map of the mixed states of a single qutrit onto the states of two identical fermions. It is shown that using this one-to-one correspondence between qutrit states and states of two identical three-dimensional fermions (the unphysical odd dimension is picked for simplicity) one may attribute hidden entanglement to a single mixed state of qutrit.
01 Jan 2016
TL;DR: In this paper, the von Neumann and linear entropies are calculated to measure the entanglement properties between the modes of the amplifier and other no-signaling nonlocal correlations are defined through the portrait procedure for noncomposite systems.
Abstract: The linear time-dependent constants of motion of the parametric amplifier are obtained and used to determine in the tomographic-probability representation the evolution of a general two-mode Gaussian state. By means of the discretization of the continuous variable density matrix, the von Neumann and linear entropies are calculated to measure the entanglement properties between the modes of the amplifier. The obtained results for the nonlocal correlations are compared with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. This qubit portrait procedure is used to establish Bell-type's inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. Other no-signaling nonlocal correlations are defined through the portrait procedure for noncomposite systems.
TL;DR: The positive invertible map of the mixed states of a single qutrit onto the antisymmetrized bipartite quasifermions was constructed in this article.
Abstract: We construct the positive invertible map of the mixed states of a single qutrit onto the antisymmetrized bipartite qutrit states (quasifermions). It is shown that using this one-to-one correspondence between qutrit states and states of two three-dimensional quasifermions one may attribute hidden entanglement to a single mixed state of qutrit.
TL;DR: The qubit–qutrit encoding is employed to suggest a single qudit algorithm for calculating the parity of a bit string and the results obtained indicate on the potential resource of multilevel systems for realization of quantum information processing.
Abstract: We study quantum information properties of a seven-level system realized by a particle in an one-dimensional square-well trap. Features of encodings of seven-level systems in a form of three-qubit or qubit-qutrit systems are discussed. We use the three-qubit encoding of the system in order to investigate subadditivity and strong subadditivity conditions for the thermal state of the particle. The qubit-qutrit encoding is employed to suggest a single qudit algorithm for calculation of parity of a bit string. Obtained results indicate on the potential resource of multilevel systems for realization of quantum information processing.
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TL;DR: In this paper, the authors discuss the procedure of different partitions in the finite set of integer numbers and construct generic formulas for a bijective map of real numbers, where the functions used to present the map are the functions detecting the hidden correlations in the system.
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 $s_y$, where $y=1,2,\ldots,N$, $N=\prod \limits_{k=1}^{n} X_k$, and $X_k$ are positive integers, onto the set of numbers $s(y(x_1,x_2,\ldots,x_n))$. We give the functions used to present the bijective map, namely, $y(x_1,x_2,...,x_n)$ and $x_k(y)$ 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.