Quantum Tomograms Research Articles

On quantum tomography on locally compact groups

We introduce quantum tomography on locally compact Abelian groups G. A linear map from the set of quantum states on the C⁎-algebra A(G) generated by the projective unitary representation of G to the space of characteristic functions is constructed. The dual map determining symbols of quantum observables from A(G) is derived. Given a characteristic function of a state the quantum tomogram consisting a set of probability distributions is introduced. We provide three examples in which G=R (the optical tomography), G=Zn (corresponding to measurements in mutually unbiased bases) and G=T (the tomography of the phase). As an application we have calculated the quantum tomogram for the output states of quantum Weyl channels.

Tomographic Description of a Quantum Wave Packet in an Accelerated Frame.

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.

Signatures of avoided energy-level crossings in entanglement indicators obtained from quantum tomograms

Extensive theoretical and experimental investigations on multipartite systems close to an avoided energy-level crossing reveal interesting features such as the extremisation of entanglement. Conventionally, the estimation of entanglement directly from experimental observation involves either one of two approaches: uncertainty-relation-based estimation that captures the linear correlation between relevant observables, or rigorous but error-prone quantum state reconstruction on tomograms obtained from homodyne measurements. We investigate the behaviour, close to avoided crossings, of entanglement indicators that can be calculated directly from a numerically-generated tomogram. The systems we study are two generic bipartite continuous-variable systems: a Bose–Einstein condensate trapped in a double-well potential, and a multi-level atom interacting with a radiation field. We also consider a multipartite hybrid quantum system of superconducting qubits interacting with microwave photons. We carry out a quantitative comparison of the indicators with a standard measure of entanglement, the subsystem von Neumann entropy (SVNE). It is shown that the indicators that capture the nonlinear correlation between relevant subsystem observables are in excellent agreement with the SVNE.

Emergent classical universes from initial quantum states in a tomographical description

Quantum and classical physical states are represented in a unified way when they aredescribed by symplectic tomography. Therefore this representation allows us to study directly the necessary conditions for a classical universe to emerge from a quantum state. In a previous work on the de Sitter universe this was done by comparing the classical limit of the quantum tomograms with those resulting from the classical cosmological equations. In this paper, we first review these results and extend them to all the de Sitter models. We show further that these tomograms can be obtained directly from transposing the Wheeler–De Witt equation to the tomographic variables. Subsequently, because the classic limits of the quantum tomograms are identified with their asymptotic expressions, we find the necessary conditions to extend the previous results by taking the tomograms of the WKB solutions of the Wheeler–DeWitt equation with an any potential. Furthermore, in the previous works, we found that the de Sitter models undergo the quantum-to-classical transition when the cosmological constant decays to its present value, we discuss at the end how far we can extend this result to more general models. In the conclusions, after discussing any improvements and developments of the results of this work, we sketch a phenomenological approach from which to extract information about the initial states of the universe.

Conditions for Quantum and Classical Tomogram-Like Functions to Describe System States and to Retain Normalizations During Time Evolution

It is shown that dynamical equations for quantum tomograms retain the normalizations of their solutions during time evolution only if the solutions satisfy a set of special conditions. These conditions are found explicitly. On the contrary, it is also illustrated that the classical Liouville equation, Moyal equation for Wigner function, and evolution equation for Husimi function retain normalizations of any initially normalized and quickly decaying at infinity functions on the phase space. Necessary and sufficient conditions for optical and symplectic tomogram-like functions to be tomograms of physical states are also discussed.

On Definition of Quantum Tomography via the Sobolev Embedding Theorem

We obtain sufficient conditions on kernels of quantum states under which Wigner functions, optical quantum tomograms and linking their formulas are correctly defined. Our approach is based upon the Sobolev Embedding theorem. The transition probability formula and the fractional Fourier transform are discussed in this framework.

Tomographic reconstruction of quantum metrics

In the framework of quantum information geometry we investigate the relationship between monotone metric tensors uniquely defined on the space of quantum tomograms, once the tomographic scheme is chosen, and monotone quantum metrics on the space of quantum states, classified by operator monotone functions, according to the Petz classification theorem. We show that different metrics can be related through a change in the tomographic map and prove that there exists a bijective relation between monotone quantum metrics associated with different operator monotone functions. Such a bijective relation is uniquely defined in terms of solutions of a first order second degree differential equation for the parameters of the involved tomographic maps. We first exhibit an example of a non-linear tomographic map that connects a monotone metric with a new one, which is not monotone. Then we provide a second example where two monotone metrics are uniquely related through their tomographic parameters.

Metric on the space of quantum states from relative entropy. Tomographic reconstruction

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 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.

On tomographic representation on the plane of the space of Schwartz operators and its dual

It is shown that the set of optical quantum tomograms can be provided with the topology of Frechet space. In such a case the conjugate space will consist of symbols of quantum observables including all polynomials of the position and momentum operators.

New relationship between quantum state’s tomogram and its wave function

By searching for - and -ordering form of the coordinate-momentum intermediate representation (formed by the eigenvector of with eigenvalue being x, ), and noting that is just the Radon transformation of the Wigner operator, we find new relationship between a quantum state ’s tomogram and its wave function in coordinate (or momentum) representation, i.e. the tomogram can be split into two parts, one is ’s Gaussian integration transformation with the parameter and another is ’s Gaussian integration transformation with the parameter . In this way, the quantum tomogram of number state is conveniently deduced. We also derive Radon transform of and , which can be either viewed as - and -ordered operator correspondence of the classical function .

On the relationship between Collins diffraction integration and quantum tomogram

We find a new relationship between Collins diffraction integration and quantum tomogram. We conclude that the tomogram of quantum state |ψ〉 is just the module square of the Fresnel transform function of ψ(x), and this transform is just performed through the Collins formula, which opens up a new application area of Collins formula in calculating quantum tomogram of various quantum states.

Star product, discrete Wigner functions, and spin-system tomograms

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.

Towards a tomographic representation of quantum mechanics on the plane

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.

Deformed Entropy and Information Relations for Composite and Noncomposite Systems

The notion of conditional entropy is extended to noncomposite systems. The q-deformed entropic inequalities, which usually are associated with correlations of the subsystem degrees of freedom in bipartite systems, are found for the noncomposite systems. New entropic inequalities for quantum tomograms of qudit states including the single qudit states are obtained. The Araki{Lieb inequality is found for systems without subsystems.

Wigner function and the probability representation of quantum states

The relation of theWigner 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 integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.

Feynman integral and perturbation theory in quantum tomography

Abstract We present a definition for tomographic Feynman path integral as representation for quantum tomograms via Feynman path integral in the phase space. The proposed representation is the potential basis for investigation of Path Integral Monte Carlo numerical methods with quantum tomograms. Tomographic Feynman path integral is a representation of solution of initial problem for evolution equation for tomograms. The perturbation theory for quantum tomograms is constructed.

Quantum correlations and tomographic representation

We review the probabilistic representation of quantum mechanics within which states are described by the probability distribution rather than by the wavefunction and density matrix. Uncertainty relations have been obtained in the form of integral inequalities containing measurable optical tomograms of quantum states. Formulas for the transition probabilities and purity parameter have been derived in terms of the tomographic probability distributions. Inequalities for Shannon and Renyi entropies associated with quantum tomograms have been obtained. A scheme of the star product of tomograms has been developed.

Inequalities for nonnegative numbers and information properties of qudit tomograms

We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Rényi, and Tsallis entropies of spin states.

Fidelity and purity of quantum electrical circuit states and quantum tomograms

A review of the probability representation of quantum mechanics and the symplectic tomography approach is presented. Examples of the Gaussian states of a nanoelectric circuit, a 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.

Tomographic probability representation in the problem of transitions between Landau levels

The problem of quantum motion of a charged particle in a magnetic field is considered in the framework of the tomographic probability representation. The coherent and Fock states of a charge moving in a time-dependent homogeneous magnetic field are studied in the tomographic probability representation. These states are expressed in terms of the quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. These results are then generalized and applied to calculate the transition probabilities between the Landau levels for the case of transitions between stationary states of different magnetic fields, when the initially constant field becomes time dependent and then stays at some constant value again. The problem is studied in terms of wave functions and symplectic tomograms as well.