Yu. I. Bogdanov

Qutrit state engineering with biphotons.

The novel experimental realization of three-level optical quantum systems is presented. We use the polarization state of biphotons to generate a specific sequence of states that are used in the extended version of four-state QKD protocol quantum key distribution protocol. We experimentally verify the orthogonality of the basic states and demonstrate the ability to easily switch between them. The tomography procedure is employed to reconstruct the density matrices of generated states.

Statistical reconstruction of qutrits

We discuss a procedure of measurement followed by the reproduction of the quantum state of a three-level optical system - a frequency- and spatially degenerate two-photon field. The method of statistical estimation of the quantum state based on solving the likelihood equation and analyzing the statistical properties of the obtained estimates is developed. Using the root approach of estimating quantum states, the initial two-photon state vector is reproduced from the measured fourth moments in the field . The developed approach applied to quantum states reconstruction is based on the amplitudes of mutually complementary processes. Classical algorithm of statistical estimation based on the Fisher information matrix is generalized to the case of quantum systems obeying Bohrs complementarity principle. It has been experimentally proved that biphoton-qutrit states can be reconstructed with the fidelity of 0.995-0.999 and higher.

Statistical estimation of the efficiency of quantum state tomography protocols.

Y u. I. Bogdanov, G.Brida,M.Genovese, S. P. Kulik, E. V. Moreva, A. P. Shurupov 1 Institute of Physics and Technology, Russian Academy of Sciences, Moscow, Russia 2 INRIM, I-10135, Torino, Italy 3 Faculty of Physics, Moscow State University, 119992, Moscow, Russia 4 Moscow National Research Nuclear University ”MEPHI”, Russia and 5 Dipartimento di Fisica, Politecnico di Torino, I-10129, Torino, Italy (Dated: February 18, 2010)

Polarization states of four-dimensional systems based on biphotons

We discuss the concept of polarization states of four-dimensional quantum systems based on frequency non-degenerate biphoton field. Several quantum tomography protocols were developed and implemented for measurement of an arbitrary state of ququart. A simple method that does not rely on interferometric technique is used to generate and measure the sequence of states that can be used for quantum communication purposes.

Unified statistical method for reconstructing quantum states by purification

Mixed-state purification is used as a basis to formulate a general statistical method for reconstructing the density matrix of an arbitrary quantum state. A universal statistical distribution is obtained for the fidelity of the reconstructed quantum state. The proposed theory is supported by results of numerical simulations.

Fundamental notions of classical and quantum statistics: A root approach

A new method of solving classical and quantum problems of statistical data analysis based on the symbiosis of notions of quantum theory and mathematical statistics is considered. Particular attention is given to the specificity of quantum problems, determined by mutually complementary measurements (according to the Bohr complementarity principle), when, for example, a spatial-temporal picture is complemented by a momentum-energy one. The possibility of construction of multiparametric statistical models admitting a stable reconstruction of the parameters from observations (the inverse statistical problem) is studied. In this case, the only universal model of such a kind is the root model, based on the representation of the probability density as the square of the modulus of some function (called the psi function by analogy with quantum mechanics). The psi function is represented as an expansion in terms of an orthonormal basis, with the expansion coefficients being estimated by the maximum likelihood technique. The root approach makes it possible to represent the Fisher information matrix, covariance matrix, and statistical properties of the estimates of the reconstructed states in the simplest and a universal form. Being asymptotically efficient, the method allows one to reconstruct the states with an accuracy close to the theoretically attainable accuracy. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition, which makes it possible to choose, from among all the statistical models, which, on the average, are consistent with the laws of classical mechanics, those systems that are described by quantum mechanics.

Statistical reconstruction of the quantum states of three-level optical systems

The procedure of measurement followed by the reconstruction of the quantum state of a three-level optical system is implemented for a frequency-and spatially degenerate two-photon field. The method of statistical estimation of the quantum state from a solution to the likelihood equation and the analysis of the statistical properties of the obtained estimators is developed. Using the root method of estimating quantum states, the initial two-photon (qutrit) wave function is reconstructed from the measured fourth-order field moments.

Optimization of a quantum tomography protocol for polarization qubits

An operational method has been proposed for estimating the efficiency of quantum cryptography protocols for quantum states with discrete variables. The method is based on the estimate of the quality of the protocol by means of a universal asymptotic distribution of the characteristic of the accuracy of the reconstruction of the quantum state, which is called fidelity. For a specially designed measurement matrix, the condition number, which is minimal for the optimal protocol, is introduced. The method has been tested in a numerical simulation and real experiments with polarization qubits. It has been shown that the optimal choice of a polarization transformer makes it possible to strongly improve the quality of the reconstruction of states.

Statistical reconstruction of mixed states of polarization qubits

The method of estimating the adequacy, completeness, and accuracy of quantum tomography protocols is generalized to the case of mixed states of polarization qubits. The efficiency of the method is illustrated based on mathematical modeling and experimental investigation of some practically important quantum tomography protocols.

Schmidt modes and entanglement in continuous-variable quantum systems

The extraction of Schmidt modes for continuous-variable systems is considered. An algorithm based on the singular-value decomposition of a matrix is proposed. It is applied to the entanglement in (i) an atom—photon system with spontaneous emission and (ii) a system of biphotons with spontaneous parametric downconversion (SPDC) of type II. For the atom—photon system, the evolution of entangled states is found to be governed by a parameter approximately equal to the fine-structure constant times the atom-to-electron mass ratio. An analysis is made of the dynamics of atom—photon entanglement on the assumption that the system’s evolution is determined by the superposition of an initial and a final state. It is shown that in the course of emission the entanglement entropy first rises on a timescale of order the excited-state lifetime and then falls, approaching asymptotically a residual level due to the initial energy spread of the atomic packet (momentum spread squared). SPDC of type II is analyzed by means of the polarization density matrix and a newly introduced coherence parameter for two spatially separated modes. The loss of intermodal coherence is addressed that results from the difference in behavior between ordinary-and extraordinary-ray photons in a nonlinear crystal. The degree of intermodal coherence is investigated as a function of the product of crystal length and pump bandwidth.