Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Kluwer Academic Publishers eBooks

The Geometry of Density States, Positive Maps and Tomograms

Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan, F. Zaccaria

Published Dec 17, 2005• 7 citations
Publisher

Abstract

The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the form of higher dimensional matrices. Probability representation of spin states (spin tomography) is reviewed and U ( N )-tomogram of spin states is presented. Properties of the tomograms as probability distribution functions are studied. Notion of tomographic purity of spin states is introduced. Entanglement and separability of density matrices are expressed in terms of properties of the tomographic joint probability distributions of random spin projections which depend also on unitary group parameters. A new positivity criterion for hermitian matrices is formulated. An entanglement criterion is given in terms of a function depending on unitary group paramete rs and semigroup of positive map parameters. The function is constructed as sum of moduli of U ( N )- tomographic symbols of the hermitian matrix obtained after action on the density matrix of composite system by a positive but not completely positive map of the subsystem density matrix. Some two-qubit and two-qutritt states are considered as examples of entangled states. The connection with the star-product quantisation is discussed. The structure of the set of density matrices and their relation to unitary group and Lie algebra of the unitary group are studied. Nonlinear quantum evolution of state vector obtained by means of applying purification rule of density matrices evolving via dynamical maps is considered. Some connection of positive maps and entanglement with random matrices is discussed and used.

DOI
10.1007/1-4020-2634-x_19

Related papers

The geometry of density states, positive maps and tomograms

Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan +1 more

The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the form of higher dimensional matrices. Probability representation of spin states (spin tomography) is reviewed and U(N)-tomogram of spin states is presented. Properties of the tomograms as probability distribution functions are studied. Notion of tomographic purity of spin states is introduced. Entanglement and separability of density matrices are expressed in terms of properties of the tomographic joint probability distributions of random spin projections which depend also on unitary group parameters. A new positivity criterion for hermitian matrices is formulated. An entanglement criterion is given in terms of a function depending on unitary group parameters and semigroup of positive map parameters. The function is constructed as sum of moduli of U(N)-tomographic symbols of the hermitian matrix obtained after action on the density matrix of composite system by a positive but not completely positive map of the subsystem density matrix. Some two-qubit and two-qutritt states are considered as examples of entangled states. The connection with the star-product quantisation is discussed. The structure of the set of density matrices and their relation to unitary group and Lie algebra of the unitary group are studied. Nonlinear quantum evolution of state vector obtained by means of applying purification rule of density matrices evolving via dynamical maps is considered. Some connection of positive maps and entanglement with random matrices is discussed and used.

arXiv:0705.3574•May 24, 2007•0 citations

Entanglement structure of adjoint representation of unitary group and tomography of quantum states

Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan +1 more

The density matrix of composite spin system is discussed in relation to the adjoint representation of unitary group U(n). The entanglement structure is introduced as an additional ingredient to the description of the linear space carrying the adjoint representation. Positive maps of density operator are related to random matrices. The tomographic probability description of quantum states is used to formulate the problem of separability and entanglement as the condition for joint probability distribution of several random variables represented as the convex sum of products of probabilities of random variables describing the subsystems. The property is discussed as a possible criterion for separability or entanglement. The convenient criterion of positivity of finite and infinite matrix is obtained. The U(n)-tomogram of a multiparticle spin state is introduced. The entanglement measure is considered in terms of this tomogram.

arXiv:quant-ph/0310022•Oct 3, 2003•1 citation

Entanglement in probability representation of quantum states and tomographic criterion of separability

Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan +1 more

Entangled and separable states of a bipartite (multipartite) system are studied in the tomographic representation of quantum states. Properties of tomograms (joint probability distributions) corresponding to entangled states are discussed. The connection with star-product quantization is presented. U(N)-tomography and spin tomography as well as the relation of the tomograms to positive and completely positive maps are considered. The tomographic criterion of separability (necessary and sufficient condition) is formulated in terms of the equality of the specific function depending on unitary group parameters and positive map semigroup parameters to unity. Generalized Werner states are used as an example.

Journal of Optics B Quantum and Semiclassical Optics•Feb 1, 2004•9 citations

Quantum states, maps, measurements and entanglement

Aik-meng Kuah

The structure of the set of density matrices, its linear transformations, generalized linear measurements, and entanglement are studied. The set of density matrices is shown to be a convex and stratified set with simplex and group symmetries. Generalized measurements for density matrices are shown to be reducible to one unitary transformation and one von Neumann measurement carried out with an ancillary system of fixed size. Linear maps of density matrices are considered and the volume of the set of maps is derived. Positive but not completely positive maps are studied in consideration of obtaining a test for entanglement in density matrices. Using the Jamiolkowski representation and Schmidt decomposition of the map eigen matrices, several properties of these maps are shown. An algebraic approach to constructing these positive but not completely positive maps is partially formulated. The positivity of the linear map describing the evolution of an open system and its dependence on the initialized to a zero-discord state, the evolution is shown to be given by a completely positive map. In quantum process tomography, the results obtained from a open system that is initially prepared using von Neumann measurements is shown to be described by a bi-linear map, not a linear map. A method for quantum process tomography is derived for qubit bi-linear maps. The difference between preparing states for an experiment by measurement and by stochastic process is analyzed, and it is shown that the two different methods will give fundamentally different outcomes.

•May 1, 2007•1 citation

Standard quantum mechanics featuring probabilities instead of wave functions

Vladimir I. Man’ko, Olga V. Man’ko

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.

Physics of Atomic Nuclei•Jun 1, 2006•6 citations

An introduction to the tomographic picture of quantum mechanics

Alberto Ibort, Vladimir I. Man’ko, Giuseppe Marmo +2 more

Starting from the famous Pauli problem on the possibility of associating quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc is clarified and a connection with abstract Hilbert space properties is established. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two-qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.

arXiv:0904.4439•Physica Scripta•Jun 1, 2009•262 citations
The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the form of higher dimensional matrices. Probability representation of spin states (spin tomography) is reviewed and U ( N )-tomogram of spin states is presented. Properties of the tomograms as probability distribution functions are studied. Notion of tomographic purity of spin states is introduced. Entanglement and separability of density matrices are expressed in terms of properties of the tomographic joint probability distributions of random spin projections which depend also on unitary group parameters. A new positivity criterion for hermitian matrices is formulated. An entanglement criterion is given in terms of a function depending on unitary group paramete rs and semigroup of positive map parameters. The function is constructed as sum of moduli of U ( N )- tomographic symbols of the hermitian matrix obtained after action on the density matrix of composite system by a positive but not completely positive map of the subsystem density matrix. Some two-qubit and two-qutritt states are considered as examples of entangled states. The connection with the star-product quantisation is discussed. The structure of the set of density matrices and their relation to unitary group and Lie algebra of the unitary group are studied. Nonlinear quantum evolution of state vector obtained by means of applying purification rule of density matrices evolving via dynamical maps is considered. Some connection of positive maps and entanglement with random matrices is discussed and used.
The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the form of higher dimensional matrices. Probability representation of spin states (spin tomography) is reviewed and U(N)-tomogram of spin states is presented. Properties of the tomograms as probability distribution functions are studied. Notion of tomographic purity of spin states is introduced. Entanglement and separability of density matrices are expressed in terms of properties of the tomographic joint probability distributions of random spin projections which depend also on unitary group parameters. A new positivity criterion for hermitian matrices is formulated. An entanglement criterion is given in terms of a function depending on unitary group parameters and semigroup of positive map parameters. The function is constructed as sum of moduli of U(N)-tomographic symbols of the hermitian matrix obtained after action on the density matrix of composite system by a positive but not completely positive map of the subsystem density matrix. Some two-qubit and two-qutritt states are considered as examples of entangled states. The connection with the star-product quantisation is discussed. The structure of the set of density matrices and their relation to unitary group and Lie algebra of the unitary group are studied. Nonlinear quantum evolution of state vector obtained by means of applying purification rule of density matrices evolving via dynamical maps is considered. Some connection of positive maps and entanglement with random matrices is discussed and used.
The density matrix of composite spin system is discussed in relation to the adjoint representation of unitary group U(n). The entanglement structure is introduced as an additional ingredient to the description of the linear space carrying the adjoint representation. Positive maps of density operator are related to random matrices. The tomographic probability description of quantum states is used to formulate the problem of separability and entanglement as the condition for joint probability distribution of several random variables represented as the convex sum of products of probabilities of random variables describing the subsystems. The property is discussed as a possible criterion for separability or entanglement. The convenient criterion of positivity of finite and infinite matrix is obtained. The U(n)-tomogram of a multiparticle spin state is introduced. The entanglement measure is considered in terms of this tomogram.
Entangled and separable states of a bipartite (multipartite) system are studied in the tomographic representation of quantum states. Properties of tomograms (joint probability distributions) corresponding to entangled states are discussed. The connection with star-product quantization is presented. U(N)-tomography and spin tomography as well as the relation of the tomograms to positive and completely positive maps are considered. The tomographic criterion of separability (necessary and sufficient condition) is formulated in terms of the equality of the specific function depending on unitary group parameters and positive map semigroup parameters to unity. Generalized Werner states are used as an example.
The structure of the set of density matrices, its linear transformations, generalized linear measurements, and entanglement are studied. The set of density matrices is shown to be a convex and stratified set with simplex and group symmetries. Generalized measurements for density matrices are shown to be reducible to one unitary transformation and one von Neumann measurement carried out with an ancillary system of fixed size. Linear maps of density matrices are considered and the volume of the set of maps is derived. Positive but not completely positive maps are studied in consideration of obtaining a test for entanglement in density matrices. Using the Jamiolkowski representation and Schmidt decomposition of the map eigen matrices, several properties of these maps are shown. An algebraic approach to constructing these positive but not completely positive maps is partially formulated. The positivity of the linear map describing the evolution of an open system and its dependence on the initialized to a zero-discord state, the evolution is shown to be given by a completely positive map. In quantum process tomography, the results obtained from a open system that is initially prepared using von Neumann measurements is shown to be described by a bi-linear map, not a linear map. A method for quantum process tomography is derived for qubit bi-linear maps. The difference between preparing states for an experiment by measurement and by stochastic process is analyzed, and it is shown that the two different methods will give fundamentally different outcomes.
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.
Starting from the famous Pauli problem on the possibility of associating quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc is clarified and a connection with abstract Hilbert space properties is established. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two-qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.