Muon spin rotation/relaxation/resonance (MuSR) technique for studying matter structures is considered by means of a recently introduced probability representation of quantum spin states. A relation between experimental MuSR histograms and muon spin tomograms is established. Time evolution of muonium, anomalous muonium, and a muonium-like system is studied in the tomographic representation. Entanglement phenomenon of a bipartite muon-electron system is investigated via tomographic analogues of Bell number and positive partial transpose (PPT) criterion. Reconstruction of the muon-electron spin state as well as the total spin tomography of composed system is discussed.
Description of system containing classical and quantum subsystems by means of tomographic probability distributions is considered. Evolution equation of the system states is studied.
Distances between quantum states are reviewed within the framework of the tomographic-probability representation. Tomographic approach is based on observed probabilities and is straightforward for data processing. Different states are distinguished by comparing corresponding probability-distribution functions. Fidelity as well as other distance measures are expressed in terms of tomograms.
Tomographic probability representation is introduced for fermion fields. The states of the fermions are mapped onto probability distribution of discrete random variables (spin projections). The operators acting on the fermion states are described by fermionic tomographic symbols. The product of the operators acting on the fermion states is mapped onto star-product of the fermionic symbols. The kernel of the star-product is obtained. The antisymmetry of the fermion states is formulated as the specific symmetry property of the tomographic joint probability distribution associated with the states…
The steering property known for two qubit-state in terms of specific inequalities for 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 two qubit system and the single qudit is presented in an integral form with a intertwining kernel calculated explicitly in tomographic probability terms.
Introduced recently approach based on tomographic probability distribution of quantum states is shown to be closely related with the known notion of the quantum probability measures discussed in quantum information theory and positive operator valued measures approach. Partial derivative of the distribution function of quantum probability measure associated with the homodyne quadrature (symplectic quantum measure) is shown to be equal the tomogram of the quantum state. Analogous relation of the spin tomogram to quantum probability measure associated with spin state is obtained. Star-product of…
In view of the tomographic-probability representation of quantum states, we reconsider the approach to quantumness tests of a single system developed in [Alicki and Van Ryn 2008 J. Phys. A: Math. Theor. 41 062001]. For qubits we introduce a general family of quantumness witnesses which are operators depending on an extra parameter. Spin tomogram and dual spin tomographic symbols are used to study qubit examples and the test inequalities which are shown to satisfy simple relations within the framework of the standard probability theory.
The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The coherent states tomograms are shown to be described by normal distributions with varying dispersions and means. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments.
We introduce a new method to calculate Franck-Condon factors in polyatomic molecules that is based on the tomographic probability approach to quantum mechanics. This approach is implemented to calculate transition probabilities in various systems under an instantaneous change of frequency and equilibrium position of nuclei in a molecule by an external force. The problem is considered for different types of the Dushinsky matrix and for any quantity of atoms in a molecule.
A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomo…
Some inequalities for probability vector are discussed. The probability representation of quantum mechanics where the states are mapped onto probability vectors (either finite or infinite dimensional) called the state tomograms is used. Examples of inequalities for qudit tomograms and a state extended uncertainly relation are considered. Tomographic cumulant related to photon state tomographic probability distributions is introduced and it is used as parameter of the state nongaussianity.
The symmetry properties under permutation of tomograms representing the states of a system of identical particles are studied. Starting from the action of the permutation group on the density matrix we define its action on the tomographic probability distribution. Explicit calculations are performed in the case of the two-dimensional harmonic oscillator.
The problem of moving of a charged particle in electromagnetic field is considered in terms of tomographic probability representation. The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. The obtained results are generalized in the present work and are applied to determining t…
States of nonlinear quantum oscillators (f-oscillators) are considered in the Weyl-Wigner-Moyal representation and the tomographic probability representation, where the states are described by standard probability distributions instead of wave functions or density matrices. The evolving integrals of motion for classical and quantum f-oscillators are found and the solution for the Liouville equation associated with the probability distribution on the phase space for this oscillator is obtained along with the solution of Moyal equation for quantum f-oscillator, which provide the solutions for pa…
The entanglement phenomenon on example of Smolin state of four qubits is discussed. This state is known as bound entangled state and the spin tomogram of the state is found in explicit form. The qubit portrait method is used the Bell inequality violation which provides another tool to prove the property of entanglement of the four-qubit state under consideration.
Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups $IGL(n, \mathbb{R})$ and $GL(n, \mathbb{R})$ respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities ar…
Probability representation entropy (tomographic entropy) of arbitrary quantum state is introduced. Using the properties of spin tomogram to be standard probability distribution function the tomographic entropy notion is discussed. Relation of the tomographic entropy to Shannon entropy and von Neumann entropy is elucidated.
Statistical properties of classical random process are considered in tomographic representation. The Radon integral transform is used to construct the tomographic form of kinetic equations. Relation of probability density on phase space for classical systems with tomographic probability distributions is elucidated. Examples of simple kinetic equations like Liouville equations for one and many particles are studied in detail.
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.
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.
Within the framework of the probability representation of quantum mechanics, we study a superposition of generic Gaussian states associated to symmetries of a regular polygon of n sides; in other words, the cyclic groups (containing the rotational symmetries) and dihedral groups (containing the rotational and inversion symmetries). We obtain the Wigner functions and tomographic probability distributions (symplectic and optical tomograms) determining the density matrices of the states explicitly as the sums of Gaussian terms. The obtained Wigner functions demonstrate nonclassical behavior, i.e.…
Based on a geometric picture, the example of free particle motion for both classical and quantum domains is considered in the tomographic probability representation. Wave functions and density operators as well as optical and symplectic tomograms are obtained as solutions of kinetic classical and quantum equations for the state tomograms. The difference of tomograms of free particle for classical and quantum states is discussed.
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. 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 arbitr…
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 having the Liouville equation in corresponding representations as the classical limit are fo…
The problem of quantum particle moving in Dirac delta potential with instant changing well depth is studied by using formalism of tomographic representation of quantum mechanics.The bound state tomogram is given in terms of error function.The ionisation probability due to instant change of the potential parameter is calculated in terms of integral containing the state tomograms.The probability of ionisation is also expressed in terms of the Wigner function.
The relation of the Wigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner--Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of…
Starting from the famous Pauli problem on the possibility to associate 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 of those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to nonnegativity of probability density on phase sp…
Glauber coherent states of quantum systems are reviewed. We construct the tomographic probability distributions of the oscillator states. The possibility to describe quantum states by tomographic probability distributions (tomograms) is presented on an example of coherent states of parametric oscillator. The integrals of motion linear in the position and momentum are used to explicitly obtain the tomogram evolution expressed in terms of trajectories of classical parametric oscillator
Tomograms introduced for the description of quantum states in terms of probability distributions are shown to be related to a standard star-product quantization with appropriate kernels. Examples of symplectic tomograms and spin tomograms are presented.
The positive vector optical tomogram fully describing the quantum state of spin 1/2 particle without any redundancy is introduced. Reciprocally the vector symplectic tomogram and vector quasidistributions $\vec W({\mathbf q},{\mathbf p})$, $\vec Q({\mathbf q},{\mathbf p})$, $\vec P(\vecα)$ are introduced. The evolution equations for proposed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are obtained. The quantum system of charged spin 1/2 particle in arbitrary electro-magnetic field is considered in proposed representations and evolution equati…
We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system's phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic …
On the base of symplectic quantum tomogram we define a probability distribution on the plane. The dual map transfers all observables which are polynomials of the position and momentum operators to the set of polynomials of two variables. In this representation the average values of observables can be calculated by means of integration over all the plane.
Multi-qudit systems are studied in tomographic probability representations of quantum qudit states. Results of calculations for Bell-type numbers within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states.
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 $Ω$-operator ordering theory is made.
From Discrete to Continuous-Variable Systems via Jordan-Schwinger Tomographic Transformation
Universiteit Leiden（莱顿大学）；Applied Quantum Algorithms Leiden（莱顿应用量子算法）；Russian Quantum Center（俄罗斯量子中心）；Lomonosov Moscow State University（莫斯科罗蒙诺索夫国立大学）；Lebedev Physical Institute, Russian Academy of Sciences（俄罗斯科学院列别杰夫物理研究所）
The tomographic probability distribution is used to decribe the kinetic equations for open quantum systems. Damped oscillator is studied. Purity parameter evolution for different damping regime is considered.
The contextuality and noncontextuality notions are considered in framework of probability representation of quantum states. Example of qutrit states and violation of the noncontextuality inequalities are presented by using the spin tomogram and tomographic symbols of the observables.
We develop a Radon like transformation, in which $P$ quasiprobability distribution for spin 1/2 states is written in terms the tomographic probability distribution $w$.
New type of tomographic probability distribution, which contains complete information on the density matrix (wave function) related to the Fresnel transform of the complex wave function, is introduced. Relation to symplectic tomographic probability distribution is elucidated. Multimode generalization of the Fresnel tomography is presented. Examples of applications of the present approach are given.
The new inequality recently found by Trifonov and called the state-extended inequality is considered in the tomographic-probability representation of quantum mechanics. The Trifonov uncertainty relations are expressed in terms of optical tomograms and can be checked in experiments on homodyne detection of the photon states.
We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schrödinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position a…
Green function and linear integrals of motion for a charged particle moving in electric field are discussed. Wigner function and tomogram of the ststionary states of the charge are obtained. Connection of quantum propagators for Schrodinger evolution equation, Moyal evolution equation and evolution equation in tomographic probability representation for charge moving in electric field is discussed.
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.
The linear canonical transforms of position and momentum are used to construct the tomographic probability representation of quantum states where the fair probability distribution determines the quantum state instead of the wave function or density matrix. The example of Moshinsky shutter problem is considered.
We give a review of the tomographic probability representation of quantum mechanics. We present the formalism of quantum states and quantum observables using the formalism of standard probability distributions and classical-like random variables. We study the coherent and number states of photons in the probability representation and obtain the evolution equation and energy spectra in the form of equations for probability distributions.
The probability representation of quantum and classical statistical mechanics is discussed. Symplectic tomography, center-of-mass tomography, and spin tomography are studied. The connection of tomographic probabilities with dynamic symmetries like symplectic group is considered. Entropic uncertainty relations and inequalities for spin tomograms are reviewed.
The quasidistributions corresponding to the diagonal representation of quantum states are discussed within the framework of operator-symbol construction. The tomographic-probability distribution describing the quantum state in the probability representation of quantum mechanics is reviewed. The connection of the diagonal and probability representations is discussed. The superposition rule is considered in terms of the density-operator symbols. The separability and entanglement properties of multipartite quantum systems are formulated as the properties of the density-operator symbols of the sys…
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…
Using tomographic-probability representation of spin states, quantum behavior of qudits is examined. For a general j-qudit state we propose an explicit formula of quantumness witnetness whose negative average value is incompatible with classical statistical model. Probability representations of quantum and classical (2j+1)-level systems are compared within the framework of quantumness tests. Trough employing Jordan-Schwinger map the method is extended to check quantumness of two-mode light states.
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.
In this work, the operator-sum representation of a quantum process is extended to the probability representation of quantum mechanics. It is shown that each process admitting the operator-sum representation is assigned a kernel, convolving of which with the initial tomogram set characterizing the system state gives the tomographic state of the transformed system. This kernel, in turn, is broken into the kernels of partial operations, each of them incorporating the symbol of the evolution operator related to the joint evolution of the system and an ancillary environment. Such a kernel decomposi…
Review of the probability representation of quantum mechanics and the symplectic tomography approach are presented. The examples of Gaussian states of nanoelectric circuit, Josephson junction, and two interacting high-quality resonant circuits are considered. The Shannon entropy, quantum information, fidelity, and purity of quantum states in the tomographic representation of quantum mechanics are studied.
The review of star-product formalism providing the possibility to describe quantum states and quantum observables by means of the functions called symbols of operators which are obtained by means of bijective maps of the operators acting in Hilbert space onto these functions is presented. Examples of the Wigner-Weyl symbols (like Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified with the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators which are needed to construct the bij…
Using the tomographic probability representation of qudit states and the inverse spin-portrait method, we suggest a bijective map of the qudit density operator onto a single probability distribution. Within the framework of the approach proposed, any quantum spin-j state is associated with the (2j+1)(4j+1)-dimensional probability vector whose components are labeled by spin projections and points on the sphere. Such a vector has a clear physical meaning and can be relatively easily measured. Quantum states form a convex subset of the 2j(4j+3) simplex, with the boundary being illustrated for qub…
New inequalities for tomographic probability distributions and density matrices of qutrit states are obtained by means of generalization of qubit portrait method. The approach based on the qudit portrait method to get new entropic inequalities is proposed. It can be applied to the case of arbutrary nonnegative hermitian matrices including the density matrices of multipartite qudit states.
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 determine…
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 r…
