Exact evolution of time-reversible symplectic integrators and their phase error for the harmonic oscillator

The evolution of any factorized time-reversible symplectic integrators, when applied to the harmonic oscillator, can be exactly solved in a closed form. The resulting modified Hamiltonians demonstrate the convergence of the Lie series expansions. They are also less distorted than modified Hamiltonian of non-reversible algorithms. The analytical form for the modified angular frequency can be used to assess the phase error of any time-reversible algorithm.Comment: Submitted to Phys. Lett. A, Six Pages two Column

Unitary S Matrices With Long-Range Correlations and the Quantum Black Hole

We propose an S matrix approach to the quantum black hole in which causality, unitarity and their interrelation play a prominent role. Assuming the 't Hooft S matrix ansatz for a gravitating region surrounded by an asymptotically flat space-time we find a non-local transformation which changes the standard causality requirement but is a symmetry of the unitarity condition of the S matrix. This new S matrix then implies correlations between the in and out states of the theory with the involvement of a third entity which in the case of a quantum black hole, we argue is the horizon S matrix. Such correlations are thus linked to preserving the unitarity of the S matrix and to the fact that entangling unitary operators are nonlocal. The analysis is performed within the Bogoliubov S matrix framework by considering a spacetime consisting of causal complements with a boundary in between. No particular metric or lagrangian dynamics need be invoked even to obtain an evolution equation for the full S matrix. Constraints imposed by the new causality requirement and implications for the effectiveness of field theoretical descriptions and for complementarity are also discussed. We find that the tension between information preservation and complementarity may be resolved provided the full quantum gravity theory either through symmetries or fine tuning forbids the occurrence of closed time like curves of information flow. Then, even if causality is violated near the horizon at any intermediate stage, a standard causal ordering may be preserved for the observer away from the horizon. In the context of the black hole, the novelty of our formulation is that it appears well suited to understand unitarity at any intermediate stage of black hole evaporation. Moreover, it is applicable generally to all theories with long range correlations including the final state projection models.Comment: 47 pages Latex, 1 figure.Corrected typos. Some section titles changed. Minor clarifying additions to all sections. Conclusions unchanged. Accepted for publication in JHE

Recursive Representations of Arbitrary Virasoro Conformal Blocks

We derive recursive representations in the internal weights of N-point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.Comment: 39 pages, 8 figures, v2: comments on references added, reference added, typos corrected, v3: comments on the relation between the plumbing and the Schottky parameters added, v4: typos correcte

Yang-Baxter maps: dynamical point of view

A review of some recent results on the dynamical theory of the Yang-Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric crystals and tropical combinatorics are discussed and demonstrated on several concrete examples.Comment: 24 pages. Extended version of lectures given at the meeting "Combinatorial Aspect of Integrable Systems" (RIMS, Kyoto, July 2004

Concepts of quantum non-Markovianity: a hierarchy

Markovian approximation is a widely-employed idea in descriptions of the dynamics of open quantum systems (OQSs). Although it is usually claimed to be a concept inspired by classical Markovianity, the term quantum Markovianity is used inconsistently and often unrigorously in the literature. In this report we compare the descriptions of classical stochastic processes and quantum stochastic processes (as arising in OQSs), and show that there are inherent differences that lead to the non-trivial problem of characterizing quantum non-Markovianity. Rather than proposing a single definition of quantum Markovianity, we study a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, such as quantum white noise, factorization approximation, divisibility, Lindblad master equation, etc.. Others are first proposed in this report, including those we call past-future independence, no (quantum) information backflow, and composability. All of these concepts are defined under a unified framework, which allows us to rigorously build hierarchy relations among them. With various examples, we argue that the current most often used definitions of quantum Markovianity in the literature do not fully capture the memoryless property of OQSs. In fact, quantum non-Markovianity is highly context-dependent. The results in this report, summarized as a hierarchy figure, bring clarity to the nature of quantum non-Markovianity.Comment: Clarifications and references added; discussion of the related classical hierarchy significantly improved. To appear in Physics Report

Continued-fraction representation of the Kraus map for non-Markovian reservoir damping

Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir. By performing factorizations in the Kraus map a perturbation theory is set up that conserves in arbitrary perturbative order both positivity and probability of the density matrix. The latter is determined by an integral equation for a bitemporal matrix and a finite hierarchy for Kraus matrices. In lowest perturbative order this hierarchy reduces to one equation for one Kraus matrix. Its solution is given by a continued fraction of a much simpler structure as compared to the non-perturbative case. In lowest perturbative order our non-Markovian evolution equations are applied to the damped Jaynes-Cummings model. From the solution for the atomic density matrix it is found that the atom may remain in the state of maximum entropy for a significant time span that depends on the initial energy of the radiation field

Equivalent qubit dynamics under classical and quantum noise

We study the dynamics of quantum systems under classical and quantum noise, focusing on decoherence in qubit systems. Classical noise is described by a random process leading to a stochastic temporal evolution of a closed quantum system, whereas quantum noise originates from the coupling of the microscopic quantum system to its macroscopic environment. We derive deterministic master equations describing the average evolution of the quantum system under classical continuous-time Markovian noise and two sets of master equations under quantum noise. Strikingly, these three equations of motion are shown to be equivalent in the case of classical random telegraph noise and proper quantum environments. Hence fully quantum-mechanical models within the Born approximation can be mapped to a quantum system under classical noise. Furthermore, we apply the derived equations together with pulse optimization techniques to achieve high-fidelity one-qubit operations under random telegraph noise, and hence fight decoherence in these systems of great practical interest.Comment: 5 pages, 2 figures; converted to PRA format, added Fig. 2, corrected typo