Table of contents

The spectral density of the Rabi model is calculated exactly within a continued-fraction approach. It is shown that the method provides a simple algorithm for the spectral density with convergent solutions. We compare these recursive solutions with the solutions of the Jaynes–Cummings model and discuss the effect of approximations on the spectral properties.

Winter's measurement compression theorem stands as one of the most penetrating insights of quantum information theory. In addition to making an original and profound statement about measurement in quantum theory, it also underlies several other general protocols used for entanglement distillation and local purity distillation. The theorem provides for an asymptotic decomposition of any quantum measurement into noise and information. This decomposition leads to an optimal protocol for having a sender simulate many independent instances of a quantum measurement and send the measurement outcomes to a receiver, using as little communication as possible. The protocol assumes that the parties have access to some amount of common randomness, which is a strictly weaker resource than classical communication. In this review, we provide a second look at Winter's measurement compression theorem, detailing the information processing task, giving examples for understanding it, reviewing Winter's achievability proof, and detailing a new approach to its single-letter converse theorem. We prove an extension of the theorem to the case in which the sender is not required to receive the outcomes of the simulated measurement. The total cost of common randomness and classical communication can be lower for such a 'non-feedback' simulation, and we prove a single-letter converse theorem demonstrating optimality. We then review the Devetak–Winter theorem on classical data compression with quantum side information, providing new proofs of its achievability and converse parts. From there, we outline a new protocol that we call 'measurement compression with quantum side information,' announced previously by two of us in our work on triple trade-offs in quantum Shannon theory. This protocol has several applications, including its part in the 'classically-assisted state redistribution' protocol, which is the most general protocol on the static side of the quantum information theory tree, and its role in reducing the classical communication cost in a task known as local purity distillation. We also outline a connection between measurement compression with quantum side information and recent work on entropic uncertainty relations in the presence of quantum memory. Finally, we prove a single-letter theorem characterizing measurement compression with quantum side information when the sender is not required to obtain the measurement outcome.

The Donsker–Varadhan rate function for occupation-time fluctuations has been seen numerically to exhibit monotone return to stationary non-equilibrium (Maes et al 2011 Phys. Rev. Lett.107 010601). That rate function is related to dynamical activity and, except under detailed balance, it does not derive from the relative entropy for which the monotonicity in time is well understood. We give a rigorous argument that the Donsker–Varadhan function is indeed monotone under the Markov evolution at large enough times with respect to the relaxation time, provided that a 'normal linear-response' condition is satisfied.

We study a class of directed random graphs. In these graphs, the interval [0, x] is the vertex set, and from each y ∈ [0, x], directed links are drawn to points in the interval (y, x] which are chosen uniformly with density 1. We analyze the length of the longest directed path starting from the origin. In the x → ∞ limit, we employ traveling wave techniques to extract the asymptotic behavior of this quantity. We also study the size of a cascade tree composed of vertices which can be reached via directed paths starting at the origin.

We propose the set of coupled ordinary differential equations dn_j/dt = n²_{j − 1} − n²_j as a discrete analogue of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration.

This paper examines the complex trajectories of a classical particle in the potential V(x) = −cos (x). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there are two special classes of trajectories x(t) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position x(0) after some real time T. The second class consists of trajectories for which there exists a real time T such that x(t + T) = x(t) ± 2π. These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.

By extending the concept of generalized bi-Schrödinger maps to the case that the target manifold is a para-Kähler manifold, we show that the third-order timelike and spacelike correction models of the vortex filament in the Minkowski 3-space are equivalent to the generalized bi-Schrödinger maps from $\mathbb R$ to the hyperbolic 2-space $\mathbb H^2$ and the de Sitter 2-space $\mathbb S^{1,1}$, respectively. As a consequence, all three typical second to fourth order integrable systems of the AKNS hierarchy are interpreted in a unified way in terms of generalized bi-Schrödinger maps. Based on this exploitation and a general discussion of the generalized bi-Schödinger maps from $\mathbb R$ into an arbitrary Riemannian surface, we reveal a property of generalized bi-Schrödinger maps that is not admitted for Schrödinger maps.

Obtaining eigenvalues of permutations acting on the product space of N representations of SU(n) usually involves either diagonalizing their representation matrices on total-weight subspaces or decomposing their characters, which can be obtained from Frobenius' formula or via graphical methods using Young tableaux (YT). For products of fundamental representations of SU(n), Schuricht and one of us proposed the method of extended YT (eYT), which allows reading the eigenvalues of the cyclic permutation C_N directly off the, slightly modified, standard YT labelling an irreducible SU(n) representation. Here we generalize the method to all symmetric representations of SU(n), and show that C_N eigenvalue computation based on eYT is at least linearly faster than the standard methods mentioned.

We consider Frenkel–Kontorova models corresponding to one-dimensional quasi-crystal with non-nearest neighbor and many-body interactions. We formulate and prove a KAM type theorem which establishes the existence of quasi-periodic solutions. The interactions we consider do not need to be of finite range or involve finitely many particles, but have to decay sufficiently fast with respect to the distance of the position of the atoms. The KAM theorem we present has an a posteriori format. We do not need to assume that the system is close to integrable. We just assume that there is an approximate solution for the functional equation which satisfies some non-degeneracy conditions.

It is shown that the Poisson–Nernst–Planck equations for a single ion species can be formulated as one equation in terms of the electric field. This previously not analyzed equation shows similarities to the vector Burgers equation and is identical with it in the one dimensional case. Several unsteady exact solutions for one and multidimensional cases are presented. Besides new mathematical insights which these first known unsteady solutions give, they can serve as test cases in computer simulations to analyze numerical algorithms and to verify code.

This paper demonstrates the quantization of a spatial Cournot duopoly model with product choice, a two stage game focusing on non-cooperation in locations and quantities. With quantization, the players can access a continuous set of strategies, using a continuous variable quantum mechanical approach. The presence of quantum entanglement in the initial state identifies a quantity equilibrium for each location pair choice with any transport cost. Also higher profit is obtained by the firms at Nash equilibrium. Adoption of quantum strategies rewards us by the existence of a larger quantum strategic space at equilibrium.

In order to demonstrate a generalized Bloch-sphere approach for the treatment of noise in coupled qubits, we perform a case study of the decoherence of a system composed of two interacting qubits in a noisy environment. In particular, we investigate the effects of correlations in the noise acting on distinct qubits. Our treatment of the two-qubit system by use of the generalized Bloch vector leads to tractable analytic equations for the dynamics of the four-level Bloch vector and allows for the application of geometrical concepts from the well-known two-level Bloch sphere. We find that in the presence of correlated or anticorrelated noise, the rate of decoherence is very sensitive to the initial two-qubit state, as well as to the symmetry of the Hamiltonian. In the absence of symmetry in the Hamiltonian, correlations only weakly impact the decoherence rate.

We seek to gain insight into the nature of the determinantal moments 〈|ρ^PT|ⁿ|ρ|^k〉^Bures of generic (nine-dimensional) two-rebit and (fifteen-dimensional) two-qubit systems (ρ), PT denoting partial transpose. Such information—as it has proved to be in the Hilbert–Schmidt counterpart—should be useful, employing probability-distribution reconstruction (inverse) procedures, in obtaining Bures 2 × 2 separability probabilities. The (regularizing) strategy we first adopt is to plot the ratio of numerically-generated (Ginibre ensemble) estimates of the Bures moments to the corresponding (apparently) exactly-known Hilbert–Schmidt moments (Slater and Dunkl 2012 J. Phys. A: Math. Theor.45 095305). Then, through a combination of symbolic and numerical computations, we obtain strong evidence as to the exact values (and underlying patterns) of certain Bures determinantal moments. In particular, the first moment (average) of |ρ^PT| (where $|\rho ^{\rm PT}| \in [-\frac{1}{16},\frac{1}{256}]$) for the two-qubit systems is, remarkably, $-\frac{1}{256} = -2^{-8} \approx -0.003\,906\,25$. The analogous value for the two-rebit systems is $-\frac{2663}{860\,160} = -\frac{2663}{2^{13} \times 3 \times 5 \times 7} \approx -0.003\,095\,94$. While $\frac{\langle |\rho ^{\rm PT}|^{n}|\rho |^{k}\rangle }{\langle |\rho |^{k}\rangle }$ in the Hilbert–Schmidt case is the ratio of 3n-degree polynomials in k, it appears to be the ratio of 5n-degree polynomials in k in the Bures case.

In this paper, we introduce a general framework to study the concept of robust self-testing which can be used to self-test maximally entangled pairs of qubits (EPR pairs) and local measurement operators. The result is based only on probabilities obtained from the experiment, with tolerance to experimental errors. In particular, we show that if the results of an experiment approach the Cirel'son bound, or approximate the Mayers–Yao-type correlations, then the experiment must contain an approximate EPR pair. More specifically, there exist local bases in which the physical state is close to an EPR pair, possibly encoded in a larger environment or ancilla. Moreover, in these bases the measurements are close to the qubit operators used to achieve the Cirel'son bound or the Mayers–Yao results.

We expand the solutions of linearly coupled Mathieu equations in terms of infinite-continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the (Floquet–Lyapunov) transformation to coordinates in which the motion is that of decoupled linear oscillators. We use this transformation to solve the Heisenberg equations of the corresponding quantum-mechanical problem, and find the quantum wavefunctions for stable oscillations, expressed in configuration space. The obtained transformation and quantum solutions can be applied to more general linear systems with periodic coefficients (coupled Hill equations, periodically driven parametric oscillators), and to nonlinear systems as a starting point for convenient perturbative treatment of the nonlinearity.

We consider a one-dimensional matter-wave bright soliton, corresponding to the ground bound state of N bosonic particles of mass m having a binary attractive delta potential interaction on the open line. For a full N-body quantum treatment, we derive several results for the scattering of this quantum soliton on a short-range, bounded from below, external potential, restricting to the low energy, elastic regime where the centre-of-mass kinetic energy of the incoming soliton is lower than the internal energy gap of the soliton, that is the minimal energy required to extract particles from the soliton.

Boundary flow in the c = 1 2d CFT of a $\mathbb {Z}_2$ orbifold of a free boson on a circle is considered. Adding a bulk marginal operator to the c = 1 orbifold branch induces a boundary flow. We show that this flow is consistent for any bulk marginal operator and known initial given boundary condition. The supersymmetric $c=\frac{3}{2}$ case is also mentioned. For the circle branch of the moduli space, this has been shown by Fredenhagen et al (2007 J. Phys. A: Math. Theor.40 F17). The ground state multiplicity (g_b) is calculated and it is shown that it does indeed decrease.

Vacuum-energy calculations with ideal reflecting boundaries are plagued by boundary divergences, which presumably correspond to real (but finite) physical effects occurring near the boundary. Our working hypothesis is that the stress tensor for idealized boundary conditions with some finite cutoff should be a reasonable ad hoc model for the true situation. The theory will have a sensible renormalized limit when the cutoff is taken away; this requires making sense of the Einstein equation with a distributional source. Calculations with the standard ultraviolet cutoff reveal an inconsistency between energy and pressure similar to the one that arises in noncovariant regularizations of cosmological vacuum energy. The problem disappears, however, if the cutoff is a spatial point separation in a 'neutral' direction parallel to the boundary. Here we demonstrate these claims in detail, first for a single flat reflecting wall intersected by a test boundary, then more rigorously for a region of finite cross section surrounded by four reflecting walls. We also show how the moment-expansion theorem can be applied to the distributional limits of the source and the solution of the Einstein equation, resulting in a mathematically consistent differential equation where cutoff-dependent coefficients have been identified as renormalizations of properties of the boundary. A number of issues surrounding the interpretation of these results are aired.

The notion of operator resonances was previously introduced by Al Zamolodchikov within the framework of the conformal perturbation theory. The resonances are related to logarithmic divergences of integrals in the perturbation expansion, and manifest themselves in poles of the correlation functions and form factors of local operators considered as functions of conformal dimensions. The residues of the poles can be computed by means of some operator identities. Here, we study the resonances in the Liouville, sinh- and sine-Gordon models, considered as perturbations of a massless free boson. We show that the well-known higher equations of motion discovered by Al Zamolodchikov in the Liouville field theory are nothing but resonance identities for some descendant operators. The resonance expansion in the vicinity of a resonance point provides a regularized version of the corresponding operators. We try to construct the corresponding resonance identities and resonance expansions in the sinh- and sine-Gordon theories. In some cases it can be done explicitly, but in most cases we are only able to obtain a general form so far. We show nevertheless that the resonances are perturbatively exact, which means that each of them only appears in a single term of the perturbation theory.

The dynamics of viscous point vortices in two dimensions is studied both analytically and numerically. We consider a core-growth model based on the Lamb–Oseen vortices, the so-called multi-Gaussian model, to describe the evolution of viscous vortices. We focus mainly on the interaction of three viscous vortices. It is found that for three vortices, there are no self-similar motion except rigid rotation, and no collapse of the vortex centers, unlike the inviscid point vortices. We perform numerical computations for the Navier–Stokes equation and the multi-Gaussian model for the collapsing case of the inviscid three point vortices and examine in detail the viscous evolutions of the vortices from the models. The motions of the vortices are little influenced by viscosity, when their mutual distances are fairly large, but the dynamics is altered by viscosity as the vortices get close to each other and the cores of the vortices overlap. The multi-Gaussian model demonstrates the prevention of the total collapse of the vortices by the viscosity effect, and the merging process of two vortices, which are the main qualitative features of the Navier–Stokes solutions.

Numerical simulations show that microscopic rod-like bodies suspended in a turbulent flow tend to align with the vorticity vector, rather than with the dominant eigenvector of the strain-rate tensor. This paper investigates an analytically solvable limit of a model for alignment in a random velocity field with isotropic statistics. The vorticity varies very slowly and the isotropic random flow is equivalent to a pure strain with statistics which are axisymmetric about the direction of the vorticity. We analyse the alignment in a weakly fluctuating uniaxial strain field, as a function of the product of the strain relaxation time τ_s and the angular velocity ω about the vorticity axis. We find that when ωτ_s ≫ 1, the rods are predominantly either perpendicular or parallel to the vorticity.

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