The second law of thermodynamics under unitary evolution and external operations

The von Neumann entropy cannot represent the thermodynamic entropy of equilibrium pure states in isolated quantum systems. The diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, is a natural generalization of the von Neumann entropy and applicable to equilibrium pure states. We show that the diagonal entropy is consistent with the second law of thermodynamics upon arbitrary external unitary operations. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that quantum trajectories under unitary evolution are restricted by the Hamiltonian dynamics and that the external operation is performed without reference to the microscopic state of the system.     

Adiabatic approximation in PT-symmetric quantum mechanics

In this paper, we present a quantitative sufficient condition for adiabatic approximation in PT-symmetric quantum mechanics, which yields that a state of the PT-symmetric quantum system at any time will remain approximately in the m-th eigenstate up to a multiplicative phase factor whenever it is initially in the m-th eigenstate of the Hamiltonian. In addition, we estimate the approximation errors by the distance and the fidelity between the exact solution and the adiabatic approximate solution to the time evolution equation, respectively.     

Experimental implementation of local adiabatic evolution algorithms by an NMR quantum information processor

Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N.J. Cerf for Grover's search [J. Roland, N.J. Cerf, Quantum search by local adiabatic evolution, Phys. Rev. A 65 (2002) 042308] and by Saurya Das, Randy Kobes, and Gabor Kunstatter for the Deutsch-Jozsa algorithm [S. Das, R. Kobes, G. Kunstatter, Adiabatic quantum computation and Deutsh's algorithm, Phys. Rev. A 65 (2002) 062301], yield a complexity of order N (where N=2(n) and n is the number of qubits). In this paper, we report the experimental implementation of these local adiabatic evolution algorithms on a 2-qubit quantum information processor, by Nuclear Magnetic Resonance.     

Inconsistency in the application of the adiabatic theorem

The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.     

Probability representation of the quantum evolution and energy-level equations for optical tomograms

The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.     

Entanglement of electronic subbands and coherent superposition of spin states in a Rashba nanoloop

The present work is concerned with an analysis of the entanglement between the electronic coherent superpositions of spin states and subbands in a quasi-one-dimensional Rashba nanoloop acted upon by a strong perpendicular magnetic field. We explicitly include the confining potential and the Rashba spin-orbit coupling into the Hamiltonian and then proceed to calculate the von Neumann entropy, a measure of entanglement, as a function of time. An analysis of the von Neumann entropy demonstrates that, as expected, the dynamics of entanglement strongly depends upon the initial state and electronic subband excitations. When the initial state is a pure one formed by a subband excitation and the z-component of spin states, the entanglement exhibits periodic oscillations with local minima (dips). On the other hand, when the initial state is formed by the subband states and a coherent superposition of spin states, the entanglement still periodically oscillates, exhibiting stronger correlations, along with elimination of the dips. Moreover, in the long run, the entanglement for the latter case undergoes the phenomenon of collapse-revivals. This behaviour is absent for the first case of the initial states. We also show that the degree of entanglement strongly depends upon the electronic subband excitations in both cases.     

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.     

Entropy scaling and simulability by matrix product states

We investigate the relation between the scaling of block entropies and the efficient simulability by matrix product states (MPSs) and clarify the connection both for von Neumann and Rényi entropies. Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPSs. We apply these results to illustrate that quantum computers might outperform classical computers in simulating the time evolution of quantum systems, even for completely translational invariant systems subject to a time-independent Hamiltonian.     

On precession of entangled spins in a strong laser field

A dynamics of the entanglement under an environmental influence is modelled by a bound state composed of two heavy particles interacting with a strong laser. Adopting the semiclassical attitude, a trajectory of the bound state’s center-of-mass is found from the Newton equations solved beyond the dipole approximation and taking into account the magnetic field effect. At the same time the dynamics of constituent spins under the laser coupling is studied quantum mechanically solving the nonrelativistic von Neumann equation with the effective Hamiltonian determined by the bound state’s classical trajectory. Based on the solution, the effects of an intense linearly polarized monochromatic plane wave on the precession of entangled spins are discussed for a specific kind of mixed initial states including a family of maximally entangled Werner states. The text was submitted by the authors in English.     

An Alternative Adiabatic Quantum Algorithm for the Hamiltonian Cycle Problem

We put forward an alternative quantum algorithm for finding Hamiltonian cycles in any N-vertex graph based on adiabatic quantum computing. With a von Neumann measurement on the final state, one may determine whether there is a Hamiltonian cycle in the graph and pick out a cycle if there is any. Although the proposed algorithm provides a quadratic speedup, it gives an alternative algorithm based on adiabatic quantum computation, which is of interest because of its inherent robustness.     

Entanglement dynamics of two interacting qubits under the influence of local dissipation

We investigate the dynamics of entanglement given by the concurrence of a two-qubit system in the non-Markovian setting. A quantum master equation is derived, which is solved in the eigenbasis of the system Hamiltonian for X-type initial states. A closed formula for time evolution of concurrence is presented for a pure state. It is shown that under the influence of dissipation non-zero entanglement is created in unentangled two-qubit states which decay in the same way as pure entangled states. We also show that under real circumstances, the decay rate of concurrence is strongly modified by the non-Markovianity of the evolution.     

基于超绝热捷径技术快速制备超导三量子比特Greenberger-Horne-Zeilinger态

本文提出了一个基于超绝热捷径技术快速制备超导三量子比特Greenberger-Horne-Zeilinger态的理论方案.该方案首先在量子Zeno动力学的帮助下得到系统的有效哈密顿量,之后通过引入与有效哈密顿量具有相同形式的反向导热哈密顿量来构建绝热捷径,加速了整个系统的演化过程.该方案不需要初态和目标态之间的直接耦合,在实验上也更容易实现.数值模拟结果表明该方案对超导量子比特的自发辐射、波导谐振腔的泄漏以及超导量子比特的退相位是鲁棒的.     

Unified Treatment of Geometric Phases for Statistical Ensembles of Classical, Quantum and Hybrid Systems

Geometric phases for evolution of statistical ensembles of Hamiltonian dynamical systems are introduced utilizing the fact that the Liouville equation is itself an infinite integrable Hamiltonian system. This general framework provides unified treatment of geometric phases for pure or mixed states of classical, quantum or hybrid quantum-classical systems.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

Transitionless driving on local adiabatic quantum search algorithm

We apply the transitionless driving on the local adiabatic quantum search algorithm to speed up the adiabatic process.By studying quantum dynamics of the adiabatic search algorithm with the equivalent two-level system, we derive the transitionless driving Hamiltonian for the local adiabatic quantum search algorithm. We found that when adding a transitionless quantum driving term H_D(t) on the local adiabatic quantum search algorithm, the success rate is 1 exactly with arbitrary evolution time by solving the time-dependent Schr odinger equation in eigen-picture. Moreover, we show the reason for the drastic decrease of the evolution time is that the driving Hamiltonian increases the lowest eigenvalues to a maximum of ON~(1/2).     

Measurement-induced nonlocality based on affinity

Measurement-induced nonlocality(MIN), a quantum correlation measure for bipartite systems,is an indicator of maximal global effects due to locally invariant von Neumann projective measurements. It is originally defined as the maximal square of the Hilbert-Schmidt norm of the difference between pre-and post-measurement states. In this article, we propose a new form of MIN based on affinity. This quantity satisfies all the criteria of a bona fide measure of quantum correlation measures. This quantity is evaluated for both arbitrary pure and 2×n dimensional(qubit-qudit) mixed states. The operational meaning of the proposed quantity is interpreted in terms of the interferometric power of the quantum state. We apply these results on two-qubit mixed states, such as the Werner, isotropic and Bell diagonal states.     

多粒子W态的绝热制备

通过利用时间依赖的外磁场,提出了绝热制备多粒子W态的新方法.同时给出了驱动n个自旋1/2粒子从未纠缠态到W态的相互作用哈密顿量以及绝热和非绝热演化条件,展示了能量和靶态布据随时间的演化图.     

Von Neumann entropy in a Rashba-Dresselhaus nanodot; dynamical electronic spin-orbit entanglement

The main purpose of the present article is to report the characteristics of von Neumann entropy, thereby, the electronic hybrid entanglement, in the heterojunction of two semiconductors, with due attention to the Rashba and Dresselhaus spin-orbit interactions. To this end, we cast the von Neumann entropy in terms of spin polarization and compute its time evolution; with a vast span of applications. It is assumed that gate potentials are applied to the heterojunction, providing a two dimensional parabolic confining potential (forming an isotropic nanodot at the junction), as well as means of controlling the spin-orbit couplings. The spin degeneracy is also removed, even at electronic zero momentum, by the presence of an external magnetic field which, in turn, leads to the appearance of Landau states. We then proceed by computing the time evolution of the corresponding von Neumann entropy from a separable (spin-polarized) initial state. The von Neumann entropy, as we show, indicates that electronic hybrid entanglement does occur between spin and two-dimensional Landau levels. Our results also show that von Neumann entropy, as well as the degree of spin-orbit entanglement, periodically collapses and revives. The characteristics of such behavior; period, amplitude, etc., are shown to be determined from the controllable external agents. Moreover, it is demonstrated that the phenomenon of collapse-revivals’ in the behavior of von Neumann entropy, equivalently, electronic hybrid entanglement, is accompanied by plateaus (of great importance in quantum computation schemes) whose durations are, again, controlled by the external elements. Along these lines, we also make a comparison between effects of the two spin-orbit couplings on the entanglement (von Neumann entropy) characteristics. The finer details of the electronic hybrid entanglement, which may be easily verified through spin polarization measurements, are also accreted and discussed. The novel results of the present article, with potent applications in the field of quantum information processing, provide a deeper understanding of the electronic von Neumann entropy and hybrid entanglement that occurs in two-dimensional nanodots.     

Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit

In this paper, we construct an adiabatic invariant for a large 1–d lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant.