arXiv subjects
Publications and source records attributed to O. V. Man'ko.
Exact solutions of the time-dependent Schrodinger equation for a quantum oscillator subject to periodical frequency delta-kicks are obtained. We show that the oscillator occurs in the squeezed state and calculate the corresponding squeezing coefficients and the energy increase rate in terms of Chebyshev polynomials.
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 bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relation of the structure constants of the associative star-product of the operator symbols to the quantizer-dequantizer operators is reviewed.
A review of probability representation of quantum states in given for optical and photon number tomography approaches. Explicit connection of photon number tomogram with measurable by homodyne detector optical tomogram is obtained. New integral relations connecting Hermite polynomials of two variables with Laguerre polynomials are found. Examples of generic Gaussian photon states (squeezed and correlated states) are studied in detail.
The marginal distribution of squeezed, rotated and shifted quadrature for two types of nonclassical states of a trapped ion - squeezed correlated states and squeezed even and odd coherent states (squeezed Schrodinger cat states) - is studied.
Squeezed and rotated quadrature of an ion in a Paul trap is discussed in connection with reconstructing its quantum state using symplectic-tomography method. Marginal distributions of the quadrature for squeezed and correlated states and for nonlinear coherent states of a trapped ion are obtained and the density matrices in the Fock basis are expressed explicitly in terms of these marginal distributions.
The single qudit state tomograms are shown to have the no signaling property. The known and new entropic and information inequalities for Shannon, von Neumann and q-entropies of the composite and noncomposite systems characterizing correlations in these systems are discussed. The spin tomographic probability distributions determining the single qudit states are demonstrated to satisfy the strong subadditivity condition for Tsallis q-entropy. Examples of the new entropic inequalities for q-entropy are considered for qudits with j = 5=2, j = 7=2.
We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iSWAP gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verify $\log{N}$ inequalities for Shannon and Rényi entropies for the considered noncomposite quantum system.
We consider quantum correlations and quantum discord phenomena for two-qubit states with $X$-type density matrices in the tomographic framework of quantum mechanics. By introducing different measurements schemes, we establish the relation between tomographic approach to quantum discord, symmetric discord, and measurement-induced disturbance. In our consideration, $X$-states appear as approximations of ground and low temperature thermal states of two coupled harmonic oscillators realized by nanoelectric $LC$-circuits. Possibilities for control the amounts of correlations and entropic asymmetry due to variation of the frequency detuning and the coupling constant are also considered.
Using the monotonity of relative entropy of composite quantum systems we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Example of qutrit state inequalities and the "qubit portrait" bound for the distance between the qutrit states are considered in explicit form.
New kind of matrix inequality known for bipartite system density matrix is obtained for arbitrary density matrix of composite or noncomposite qudit systems including the single qudit state. The examples of two qubit system and qudit with j=3/2 are discussed.
We consider the generalized pure state density matrix which depends on different time moments. The evolution equation for this density matrix is obtained in case where the density matrix corresponds to the solutions of Gross-Pitaevskii equation.
New quantum entropic inequality for states of system of n >_ 1 qudits is obtained. The inequality has the form of quantum subadditivity condition of bipartite qudit system and coincides with this subadditivity condition for the system of two qudits. The general statement on existence of the subadditivity condition for arbitrary probability distribution and arbitrary qudit-system tomogram is formulated. The nonlinear quantum channels creating the entangled states from separable ones are discussed.
New entropic inequality for quantum and tomographic Shannon information for system of two qubits is obtained. The inequality relating quantum information and spin-tomographic information for particle with spin j = 3=2 is found. The method to extend the obtained new entropic and information inequalities for one qudit and arbitrary composite system of qudits is suggested.
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.
The scheme of photon-number tomography is discussed in the framework of star-product quantization. The connection of dual quantization scheme and observables is reviewed. The quantizer and dequantizer operators and kernels of star product of tomograms in photon-number tomography scheme and its dual one are presented in explicit form. The fidelity and state purity are discussed in photon{number tomographic scheme, and the expressions for fidelity and purity are obtained in the form of integral of the product of two photon-number tomograms with integral kernel which is presented in explicit form. The properties of quantumness are discussed in terms of inequalities on state photon{number tomograms.
Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied. It is shown that the sets have common part but there exist tomograms of classical states which are not admissible in quantum mechanics and vica versa, there exist tomograms of quantum states which are not admissible in classical mechanics. Role of different transformations of reference frames in phase space of classical and quantum systems (scaling and rotation) determining the admissibility of the tomograms as well as the role of quantum uncertainty relations is elucidated. Union of all admissible tomograms of both quantum and classical states is discussed in context of interaction of quantum and classical systems. Negative probabilities in classical mechanics and in quantum mechanics corresponding to the tomograms of classical states and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively.
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.
The process of stimulated Brillouin scattering is described by the model of two-dimensional oscillator. The phenomenon of entanglement which appears in the photon-phonon modes after the interaction with matter is discussed.