Quantum signatures of chaos or quantum chaos?

A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.     

On coding of a quantum source of states with a finite frequency band: Quantum analogue of the Kotel’nikov theorem on sampling

An example of coding a source of quantum states with a finite frequency band W and finite exit power not exceeding ～(?W)W is given. The number of classical information bits that can be coded in the quantum states generated by such a source per unit time is C=W. Such a source is minimal in the sense that the filling factor for each of the orthogonal single-particle modes constituting N=WT-photon vector in time window 2T is equal to 1. This result can be treated as a quantum analogue of the Kotel’nikov theorem on sampling for classical signals     

Efficient Quantum Algorithm for the Parity Problem of a Certain Function

Based on a particular mathematical structure of a certain function f(x) under our attention, we present a novel quantum algorithm. The algorithm allows one to determine the property of a certain function. In our study, it is f(x) = f(?x). Therefore, there would be a question here, “How fast can we succeed in this?” All we need to do is only the evaluation of a single quantum state \(|\overbrace {0,0,\ldots ,0,1}^{N}\rangle \) (N ≥?2). Only using that with a little amount of information, we can derive the global property f(x) = f(?x). Our quantum algorithm overcomes a classical counterpart by a factor of the order of 2^N.     

Creating Very True Quantum Algorithms for Quantum Energy Based Computing

An interpretation of quantum mechanics is discussed. It is assumed that quantum is energy. An algorithm by means of the energy interpretation is discussed. An algorithm, based on the energy interpretation, for fast determining a homogeneous linear function f(x) := s.x = s ₁ x ₁ + s ₂ x ₂ + ? + s _N x _N is proposed. Here x = (x ₁, … , x _N ), x _j ∈ R and the coefficients s = (s ₁, … , s _N ), s _j ∈ N. Given the interpolation values \((f(1), f(2),...,f(N))=\vec {y}\), the unknown coefficients \(s = (s_{1}(\vec {y}),\dots , s_{N}(\vec {y}))\) of the linear function shall be determined, simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of N. Our method is based on the generalized Bernstein-Vazirani algorithm to qudit systems. Next, by using M parallel quantum systems, M homogeneous linear functions are determined, simultaneously. The speed of obtaining the set of M homogeneous linear functions is shown to outperform the classical case by a factor of N × M.     

Quantification of Quantum Entanglement in a Multiparticle System of Two-Level Atoms Interacting with a Squeezed Vacuum State of the Radiation Field

We quantify multiparticle quantum entanglement in a system of N two-level atoms interacting with a squeezed vacuum state of the electromagnetic field. We calculate the amount of quantum entanglement present among one hundred such two-level atoms and also show the variation of that entanglement with the radiation field parameter. We show the continuous variation of the amount of quantum entanglement as we continuously increase the number of atoms from N = 2 to N = 100. We also discuss that the multiparticle correlations among the N two-level atoms are made up of all possible bipartite correlations among the N atoms.     

All possible Cayley-Klein contractions of quantum orthogonal groups

Spaces of constant curvature and their motion groups are described most naturally in the Cartesian basis. All these motion groups, also known as CK groups, are obtained from an orthogonal group by contractions and analytical continuations. On the other hand, quantum deformation of orthogonal group SO(N) is most easily performed in the so-called symplectic basis. We reformulate its standard quantum deformation to the Cartesian basis and obtain all possible contractions of quantum orthogonal group SO _q (N) for both untouched and transformed deformation parameters. It turned out that, similar to the undeformed case, all CK contractions of SO _q (N) are realized. An algorithm for obtaining nonequivalent (as Hopf algebra) contracted quantum groups is suggested. Contractions of SO _q (N), N = 3, 4, 5, are regarded as examples.     

New insights into quantum and classical correlations in XY spin models

We compute quantum dissonance Q (non-entangled quantum correlation), entanglement E, quantum discord D (total quantum correlation) and classical correlation C for spin pairs at any distance in the infinite XY spin-1/2 chains, i.e., the anisotropic XY model and the isotropic XY model with three-spin interactions. We obtain two simple dominance relations: C ≥ E and D ≥ E + Q Except this, there are no other simple ordering relations between them. We also show that Q can detect the special points of the system where the entanglement just appears or completely disappears. In addition, it is worthwhile to mention that dissonance and classical correlation can also clearly spotlight the critical points of quantum phase transitions in XY spin-1/2 chains.     

Finite Hamiltonian systems: Linear transformations and aberrations

Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su(2), and their phase space is a sphere. Rigid motions of this phase space form the group SU(2); overall phases complete this to U(2). But since N-point states can be subject to U(N) ?U(2) transformations, the rest of the generators will provide all N ² unitary transformations of the states, which appear as nonlinear transformations—aberrations—of the system phase space. They are built through the “finite quantization” of a classical optical system.     

On the Mean Field and Classical Limits of Quantum Mechanics

The main result in this paper is a new inequality bearing on solutions of the N-body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of N identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of C^1,1 interaction potentials. The quantity measuring the approximation of the N-body quantum dynamics by its mean field limit is analogous to the Monge–Kantorovich (or Wasserstein) distance with exponent 2. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13, 115–123, (1979)]. Our approach to this problem is based on a direct analysis of the N-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.     

A Total Measure of Multi-Particle Quantum Correlations in Atomic Schrödinger Cat States

We propose a total measure of multi-particle quantum correlation in a system of N two-level atoms (N qubits). We construct a parameter that encompasses all possible quantum correlations among N two-level atoms in arbitrary symmetric pure states and define its numerical value to be the total measure of the net atom-atom correlations. We use that parameter to quantify the total quantum correlations in atomic Schrödinger cat states, which are generated by the dispersive interaction in a cavity. We study the variation of the net amount of quantum correlation as we vary the number of atoms from N=2 to N=100 and obtain some interesting results. We also study the variation of the net correlation, for fixed interaction time, as we increase the number of atoms in the excited state of the initial system, and notice some interesting features. We also observe the behaviour of the net quantum correlation as we continuously increase the interaction time, for the general state of N two-level atoms in a dispersive cavity.     

Quantum Cryptography,Quantum Communication,and Quantum Computer in a Noisy Environment

First, we study several information theories based on quantum computing in a desirable noiseless situation. (1) We present quantum key distribution based on Deutsch’s algorithm using an entangled state. (2) We discuss the fact that the Bernstein-Vazirani algorithm can be used for quantum communication including an error correction. Finally, we discuss the main result. We study the Bernstein-Vazirani algorithm in a noisy environment. The original algorithm determines a noiseless function. Here we consider the case that the function has an environmental noise. We introduce a noise term into the function f(x). So we have another noisy function g(x). The relation between them is g(x) = f(x) ± O(??). Here O(??) ? 1 is the noise term. The goal is to determine the noisy function g(x) with a success probability. The algorithm overcomes classical counterpart by a factor of N in a noisy environment.     

Collective states in highly symmetric atomic configurations and single-photon traps

We study correlated states in circular and linear-chain configurations of identical two-level atoms containing the energy of a single quasi-resonant photon in the form of a collective excitation, where the collective behavior is mediated by exchange of transverse photons between the atoms. For a circular atomic configuration containing N atoms, the collective energy eigenstates can be determined by group-theoretical means making use of the fact that the configuration possesses a cyclic symmetry group Z _N . For these circular configurations, the carrier spaces of the various irreducible representations of the symmetry group are at most two-dimensional, so that the effective Hamiltonian on the radiationless subspace of the system can be diagonalized analytically. As a consequence, the radiationless energy eigenstates carry a Z _N quantum number p = 0, 1, …, N, which is analogous to the angular momentum quantum number l = 0, 1, … carried by particles propagating in a central potential, such as a hydrogen-like system. Just as the hydrogen s states are the only electronic wave functions that can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the circular atomic sample can occupy the central atom only for vanishing Z _N quantum number p. When a central atom is present, the p = 0 state splits into two, showing level crossing at certain radii; in the regions between these radii, damped oscillations between two “ extreme” p = 0 states occur, where the excitation occupies either the outer atoms or the central atom only. For large numbers of atoms in a maximally subradiant state, a critical interatomic distance of λ/2 emerges both in the linear-chain and in the circular configuration of atoms. The spontaneous decay rate of the linear configuration exhibits a jumplike “critical” behavior for next-neighbor distances close to a half-wavelength. Furthermore, both the linear-chain and the circular configurations exhibit exponential photon trapping once the next-neighbor distance becomes less than a half-wavelength, with the suppression of spontaneous decay being particularly pronounced in the circular system. In this way, circular configurations containing sufficiently many atoms may be natural candidates for single-photon traps.     

Pairwise Quantum Discord for a Symmetric Multi-Qubit System in Different Types of Noisy Channels

We study the pairwise quantum discord (QD) for a symmetric multi-qubit system in different types of noisy channels, such as phase-flip, amplitude damping, phase-damping, and depolarizing channels. Using the QD and geometric quantum discord (GMQD) to quantify quantum correlations, some analytical and numerical results are presented. The results show that, the QD dynamics is strongly related to the number of spin particles N as well as the initial parameter ?? of the one-axis twisting collective state. With the number of spin particles N increasing, the amount of the QD increases. However, when the amount of the QD arrives at a stable maximal value, the QD is independence of the number of spin particles N increasing. The behavior of the QD is symmetrical during a period 0 ≤ ?? ≤ 2π. Moreover, we compare the QD dynamics with the GMQD for a symmetric multi-qubit system in different types of noisy channels.     

Universal Low-energy Behavior in a Quantum Lorentz Gas with Gross-Pitaevskii Potentials

We consider a quantum particle interacting with N obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form N²V (Nx) (Gross-Pitaevskii potential). We show convergence of the N dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for N large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.     

Interpreting Quantum Logic as a Pragmatic Structure

Many scholars maintain that the language of quantum mechanics introduces a quantum notion of truth which is formalized by (standard, sharp) quantum logic and is incompatible with the classical (Tarskian) notion of truth. We show that quantum logic can be identified (up to an equivalence relation) with a fragment of a pragmatic language \(\mathcal {L}_{G}^{P}\) of assertive formulas, that are justified or unjustified rather than trueor false. Quantum logic can then be interpreted as an algebraic structure that formalizes properties of the notion of empirical justification according to quantum mechanics rather than properties of a quantum notion of truth. This conclusion agrees with a general integrationist perspective that interprets nonstandard logics as theories of metalinguistic notions different from truth, thus avoiding incompatibility with classical notions and preserving the globality of logic.     

New Method of Calculating a Multiplication by using the Generalized Bernstein-Vazirani Algorithm

We present a new method of more speedily calculating a multiplication by using the generalized Bernstein-Vazirani algorithm and many parallel quantum systems. Given the set of real values \(\{a_{1},a_{2},a_{3},\ldots ,a_{N}\}\) and a function \(g:\textbf {R}\rightarrow \{0,1\}\), we shall determine the following values \(\{g(a_{1}),g(a_{2}),g(a_{3}),\ldots , g(a_{N})\}\) simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of \(N\). Next, we consider it as a number in binary representation; M₁ = (g(a₁),g(a₂),g(a₃),…,g(a _N )). By using \(M\) parallel quantum systems, we have \(M\) numbers in binary representation, simultaneously. The speed of obtaining the \(M\) numbers is shown to outperform the classical case by a factor of \(M\). Finally, we calculate the product; \( M_{1}\times M_{2}\times \cdots \times M_{M}. \) The speed of obtaining the product is shown to outperform the classical case by a factor of N × M.     

Quantum Teleportation of an Arbitrary N-qubit State via GHZ-like States

Recently Zhu (Int. J. Theor. Phys. 53, 4095, 2014) had shown that using GHZ-like states as quantum channel, it is possible to teleport an arbitrary unknown two-qubit state. We investigate this channel for the teleportation of an arbitrary N-qubit state. The strict proof through mathematical induction is presented and the rule for the receiver to reconstruct the desired state is explicitly derived in the most general case. We also discuss that if a system of quantum secret sharing of classical message is established, our protocol can be transformed to a N-qubit perfect controlled teleportation scheme from the controller’s point of view.     

Investigation of low-lying levels of the <Stack><Subscript>4</Subscript><Superscript>8</Superscript></Stack>Be nucleus in the multiquantum approximation of the orthogonal scheme

In the multiquantum approximation of the orthogonal scheme, specific calculations for the energies and radii of the ₄ ⁸ Be nucleus are performed with allowance for all states characterized by the λ=[44] Young diagram, the quantum numbers K_min and K_min+2 of the O(3(A?1)) group, and the quantum numbers E=K+2N (N≤9) of the U(3(A?1)) group. The convergence of the results with respect to the extension of the basis is studied, and the structure of relevant wave functions is revealed. The results of these calculations are compared with the results obtained in the analogous approximation of the unitary scheme.