Showing papers by "Vladimir I. Man’ko published in 2006"
TL;DR: In this paper, Hilbert space operators are mapped onto a space of ordinary functions (operator symbols) equipped with an associative (but noncommutative) star-product, and a unified framework for such maps is reviewed.
Abstract: Hilbert space operators may be mapped onto a space of ordinary functions (operator symbols) equipped with an associative (but noncommutative) star-product. A unified framework for such maps is reviewed. Because of its clear probabilistic interpretation, a particular class of operator symbols (tomograms) is proposed as a framework for quantum information problems. Qudit states are identified with maps of the unitary group into the simplex. The image of the unitary group on the simplex provides a geometrical characterization of the nature of the quantum states. Generalized measurements, typical quantum channels, entropies, and entropy inequalities are discussed in this setting.
95 citations
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TL;DR: Generalized measurements, typical quantum channels, entropies, and entropy inequalities are discussed in this setting and a particular class of operator symbols (tomograms) is proposed as a framework for quantum information problems.
Abstract: Hilbert space operators may be mapped onto a space of ordinary functions (operator symbols) equipped with an associative (but noncommutative) star-product. A unified framework for such maps is reviewed. Because of its clear probabilistic interpretation, a particular class of operator symbols (tomograms) is proposed as a framework for quantum information problems. Qudit states are identified with maps of the unitary group into the simplex. The image of the unitary group on the simplex provides a geometrical characterization of the nature of the quantum states. Generalized measurements, typical quantum channels, entropies and entropy inequalities are discussed in this setting.
86 citations
TL;DR: Using the tomographic probability distribution (symplectic tomogram) describing the quantum state (instead of the wave function or density matrix) and properties of recently introduced tomographic entropy associated with the probability distribution, the new uncertainty relation for the Tomographic entropy is obtained as discussed by the authors.
Abstract: Using the tomographic probability distribution (symplectic tomogram) describing the quantum state (instead of the wave function or density matrix) and properties of recently introduced tomographic entropy associated with the probability distribution, the new uncertainty relation for the tomographic entropy is obtained. Examples of the entropic uncertainty relation for squeezed states and solitons of the Bose-Einstein condensate are considered.
53 citations
TL;DR: Using the tomographic probability distribution (symplectic tomogram) describing the quantum state (instead of the wave function or density matrix) and properties of recently introduced tomographic entropy associated with the probability distribution, the new uncertainty relation for the Tomographic entropy is obtained as mentioned in this paper.
Abstract: Using the tomographic probability distribution (symplectic tomogram) describing the quantum state (instead of the wave function or density matrix) and properties of recently introduced tomographic entropy associated with the probability distribution, the new uncertainty relation for the tomographic entropy is obtained. Examples of the entropic uncertainty relation for squeezed states and solitons of the Bose--Einstein condensate are considered.
46 citations
TL;DR: In this paper, the authors show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins, and compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics.
Abstract: We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics.
40 citations
TL;DR: In this article, a tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product.
Abstract: The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.
33 citations
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TL;DR: In this article, a linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state, and the Bell inequalities for two qubits and two qutrits are discussed in framework of probability representation of quantum mechanics.
Abstract: A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. Using the qubit portrait method the Bell inequalities for two qubits and two qutrits are discussed in framework of probability representation of quantum mechanics. Semigroup of stochastic matrices is associated with tomographic probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as ansatz to provide a necessary condition for separability of quantum states.
30 citations
TL;DR: In this article, the partial scaling criterion of separability of multimode continuous variable system is discussed in the context of using nonpositive maps of density matrices and the example of nonpositive trace-class Hermitian operator for which Robertson-Schrodinger uncertainty relation is fulfilled is presented.
28 citations
TL;DR: In this article, entropic uncertainty relations for Shannon entropies associated with tomographic probability distributions of continuous quadratures are reviewed in the form of inequalities for integrals containing the tomograms of quantum states and deformation parameter.
Abstract: Entropic uncertainty relations for Shannon entropies associated with tomographic probability distributions of continuous quadratures are reviewed. New entropie uncertainty relations in the form of inequalities for integrals containing the tomograms of quantum states and deformation parameter are obtained.
23 citations
TL;DR: In this paper, the authors revisited the tomographic approach to quantum mechanics as a direct tool to investigate the violation of Bell-like inequalities and emphasized that the tomogram approach is the most natural one to compare the predictions of classical and quantum theory.
Abstract: The tomographic approach to quantum mechanics is revisited as a direct tool to investigate the violation of Bell-like inequalities. Since quantum tomograms are well defined probability distributions, the tomographic approach is emphasized to be the most natural one to compare the predictions of classical and quantum theory. Examples of inequalities for two qubits and two qutrits are considered in the tomographic probability representation of spin states.
20 citations
TL;DR: In this article, a 1D stability analysis of a recently proposed method to filter and control localized states of the Bose-Einstein condensate (BEC) is presented.
Abstract: We present one-dimensional (1D) stability analysis of a recently
proposed method to filter and control localized states of the
Bose–Einstein condensate (BEC), based on novel trapping
techniques that allow one to conceive methods to select a
particular BEC shape by controlling and manipulating the external
potential well in the three-dimensional (3D)
Gross–Pitaevskii equation (GPE). Within the framework of this
method, under suitable conditions, the GPE can be exactly
decomposed into a pair of coupled equations: a transverse
two-dimensional (2D) linear Schrodinger equation and a
one-dimensional (1D) longitudinal nonlinear Schrodinger
equation (NLSE) with, in a general case, a time-dependent
nonlinear coupling coefficient. We review the general idea how
to filter and control localized solutions of the GPE. Then,
the 1D longitudinal NLSE is numerically solved
with suitable non-ideal controlling potentials that differ from
the ideal one so as to introduce relatively small errors
in the designed spatial profile. It is shown that a BEC with an
asymmetric initial position in the confining potential exhibits
breather-like oscillations in the longitudinal direction but,
nevertheless, the BEC state remains confined within the potential
well for a long time. In particular, while the condensate remains
essentially stable, preserving its longitudinal soliton-like
shape, only a small part is lost into “radiation”.
TL;DR: In this paper, the probability representation of quantum mechanics including propagators and tomograms of quantum states of the universe and its application to quantum gravity and cosmology are reviewed and the notion of tomographic entropy and its properties are used to find some inequalities for the tomographic probability determining the quantum state.
Abstract: The probability representation of quantum mechanics including propagators and tomograms of quantum states of the universe and its application to quantum gravity and cosmology are reviewed. The minisuperspaces modeled by oscillator, free pointlike particle and repulsive oscillator are considered. The notion of tomographic entropy and its properties are used to find some inequalities for the tomographic probability determining the quantum state of the universe. The sense of the inequality as a lower bound for the entropy is clarified.
TL;DR: In this article, probability distributions associated to classical and quantum optical signals are introduced. And nonlinear evolution equations are discussed in the probability representation for spin states, entropy and information corresponding to spin tomograms are studied.
Abstract: Probability distributions (tomograms) associated to classical and quantum optical signals are introduced. Nonlinear evolution equations are discussed in the probability representation. For spin states, entropy and information corresponding to spin tomograms are studied.
TL;DR: A tomographic approach to quantum states that leads to a probability representation of quantum states is discussed, which can be regarded as a classical-like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator.
Abstract: It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical-like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.
01 Apr 2006
TL;DR: In this article, the properties of spin tomograms and tomographic entropy and information were studied and the specific structure of joint probability distributions of standard probability theory were compared with the spin-tomogram properties for two-qubits.
Abstract: We study the properties of spin tomograms and tomographic entropy and information. The specific structure of joint probability distributions of standard probability theory are compared with the spin-tomogram properties for two-qubits.
TL;DR: In this paper, a new formulation of quantum mechanics (probability representation) is discussed, in which a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function.
Abstract: A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.
01 Jan 2006
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TL;DR: The probability representation of states in standard quantum mechanics where the quantum states are associated with fair probability distributions (instead of wave function or density matrix) is shortly commented and bibliography related to the probability representation is given as mentioned in this paper.
Abstract: The probability representation of states in standard quantum mechanics where the quantum states are associated with fair probability distributions (instead of wave function or density matrix) is shortly commented and bibliography related to the probability representation is given.
TL;DR: In this article, the problem of transmitting states belonging to finite dimensional Hilbert space through a quantum channel associated with a larger (even infinite dimensional) Hilbert space is addressed. But it is not addressed in this paper.
Abstract: We address the problem of transmitting states belonging to finite dimensional Hilbert space through a quantum channel associated with a larger (even infinite dimensional) Hilbert space.
TL;DR: In this article, a method of constructing the tomographic probability distributions describing quantum states in parallel with density operators is presented, and known examples of Husimi-Kano quasi-distribution and photon number tomography are reconsidered in the new setting.
Abstract: The method of constructing the tomographic probability distributions describing quantum states in parallel with density operators is presented. Known examples of Husimi-Kano quasi-distribution and photon number tomography are reconsidered in the new setting. New tomographic schemes based on coherent states and nonlinear coherent states of deformed oscillators, including q-oscillators, are suggested. The associated identity decompositions providing Gram-Schmidt operators are explicitly given, and contact with the Agarwal-Wolf $\Omega$-operator ordering theory is made.
TL;DR: In this paper, the tomographic entropy of the state of two particles with total spin j is expressed in terms of 3j-symbols and matrix elements of irreducible representation of the SU(2) group.
Abstract: Tomographic probability distributions (tomograms) for spin states of two particles with spin j
1 and j
2 are explicitly calculated. For a system of two spins, the relation of the state tomogram with total spin j and the spin projection m to tomograms of spin states with spin j
1 and j
2 and spin projections m
1 and m
2 is found. The tomographic probability distribution of the state of two particles with total spin j is calculated using the Clebsch-Gordan coefficients. The tomographic entropy of the state with total spin j is expressed in terms of 3j-symbols and matrix elements of irreducible representation of the SU(2) group.
TL;DR: In this paper, a two-mode symmetric oscillator is used to construct the qubit model as the superposition of the first excited degenerate level states of the oscillator.
Abstract: The two-mode symmetric oscillator is used to construct the qubit model as the superposition of the first excited degenerate level states of the oscillator. The entanglement properties of the oscillator states are studied using the known criterion of separability. Application to the quantum computing model based on light modes propagating in optical waveguides is briefly discussed.
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TL;DR: In this article, new inequalities for symplectic tomograms of quantum states and their connection with entropic uncertainty relations are discussed within the framework of the probability representation of quantum mechanics.
Abstract: New inequalities for symplectic tomograms of quantum states and their connection with entropic uncertainty relations are discussed within the framework of the probability representation of quantum mechanics.
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TL;DR: The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product.
Abstract: The tomographic description of a quantum state is formulated in an abstract infinite dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.
TL;DR: In this article, the authors revisited the tomographic approach to quantum mechanics as a direct tool to investigate violation of Bell-like inequalities, and emphasized that the tomogram approach is the most natural one to compare the predictions of classical and quantum theory.
Abstract: The tomographic approach to quantum mechanics is revisited as a direct tool to investigate violation of Bell-like inequalities. Since quantum tomograms are well defined probability distributions, the tomographic approach is emphasized to be the most natural one to compare the predictions of classical and quantum theory. Examples of inequalities for two qubits an two qutrits are considered in the tomographic probability representation of spin states.
TL;DR: In this article, a tomographic approach to quantum states that leads to a probability representation of quantum states is discussed, which can be regarded as a classical-like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator.
Abstract: It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical-like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.