A. B. Klimov

Generation of squeezed states in a resonator with a moving wall

Abstract The problem of generation of squeezed states of the electromagnetic field in a one-dimensional resonator with a moving wall is considered. The squeezing and correlation coefficients for a harmonic law of motion of the wall are calculated.

Structure of the sets of mutually unbiased bases for N qubits

For a system of N qubits, living in a Hilbert space of dimension d=2{sup N}, it is known that there exists d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of N qubits is outlined.

Quantum phenomena in resonators with moving walls

New solutions for the mode functions of the electromagnetic field in ideal cavity which boundary oscillates at a resonance frequency are obtained in the long‐time limit. The rate of photons creation from initial vacuum state is shown to be time independent and proportional to the amplitude of oscillations and the resonance frequency. Temperature corrections are evaluated. The squeezing coefficients for the quantum states of the field generated are calculated, as well as the backward reaction of the field on the vibrating wall.

Multicomplementary operators via finite Fourier transform

A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d, whenever d is a power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements. One of these classes is diagonal, and can be mapped to cyclic operators by means of the finite Fourier transform, which allows one to understand complementarity in a similar way as for the position–momentum pair in standard quantum mechanics. The relevant examples of two and three qubits and two qutrits are discussed in detail.

Tomographic representation of spin and quark states

We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU(3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.

Geometrical approach to mutually unbiased bases

We propose a unifying phase-space approach to the construction of mutually unbiased bases for a two-qubit system. It is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We also consider the feasible transformations between different kinds of curves and show that they correspond to local rotations around the Bloch-sphere principal axes. We suggest how to generalize the method to systems in dimensions that are powers of a prime.

Quantum particle in a box with moving walls

The analytical expressions for the energy gain and transition probabilities between energy levels of a nonrelativistic quantum particle confined in a box with uniformly moving walls, including the cases of adiabatic motion and a sudden change of the size of the box was obtained.

Geometrical approach to the discrete Wigner function in prime power dimensions

We analyse the Wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non-uniqueness of the representation of the Wigner function. Explicit expressions for the Wigner kernel are given in both cases.

The discrete Wigner function

Publisher Summary Until quite recently, Wigner functions were used mostly for infinite-dimensional Hilbert spaces. However, many quantum systems can be appropriately described in a finite-dimensional Hilbert space. These include, among other, spin systems, multi-level atoms, optical fields with a fixed number of photons, and electrons occupying a finite number of sites. When considering the coherent superpositions of ‘classical’ states, the discrete Wigner function behaves in a way that differs drastically from its continuous counterpart: the interference spreads over all of phase space, affecting even the regions where the original states are localized. As a consequence, the presence of interference may become hard to identify when using this class of Wigner functions. However, given two orthogonal stabilizer ‘classical’ states it is possible to define a Wigner function such that all coherent super-positions of those states have a phase-space representation in which the quantum interference is localized. This chapter discusses the continuous Wigner function, followed by discrete finite space and finite fields. It also discusses the generalized Pauli group, mutually unbiased bases, and the discrete Wigner function. This is followed by a discussion on reconstruction of the density operator from the discrete Wigner function, its applications.

Quantum phases of a qutrit

We consider various approaches to treat the phases of a qutrit. Although it is possible to represent qutrits in a convenient geometrical manner by resorting to a generalization of the Poincare sphere, we argue that the appropriate way of dealing with this problem is through phase operators associated with the algebra su(3). The rather unusual properties of these phases are caused by the small dimension of the system and are explored in detail. We also examine the positive operator-valued measures that can describe the qutrit phase properties.