Volume 57, Issue 6, June 2016

We present a new decoding protocol to realize transmission of classical information through a quantum channel at asymptotically maximum capacity, achieving the Holevo bound and thus the optimal communication rate. At variance with previous proposals, our scheme recovers the message bit by bit, making use of a series of “yesno” measurements, organized in bisection fashion, thus determining which codeword was sent in log2 N steps, N being the number of codewords.
 ARTICLES

 Partial Differential Equations

Global solutions to the twodimensional Riemann problem for a system of conservation laws
View Description Hide DescriptionWe study the global solutions to the twodimensional Riemann problem for a system of conservation laws. The initial data are three constant states separated by three rays emanating from the origin. Under the assumption that each ray in the initial data outside of the origin projects exactly one planar contact discontinuity, this problem is classified into five cases. By the selfsimilar transformation, the reduced system changes type from being elliptic near the origin to being hyperbolic far away in selfsimilar plane. Then in hyperbolic region, applying the generalized characteristic analysis method, a Goursat problem is solved to describe the interactions of planar contact discontinuities. While, in elliptic region, a boundary value problem arises. It is proved that this boundary value problem admits a unique solution. Based on these preparations, five explicit solutions and their corresponding criteria can be obtained in selfsimilar plane.

Stability estimate for the aligned magnetic field in a periodic quantum waveguide from DirichlettoNeumann map
View Description Hide DescriptionIn this article, we study the boundary inverse problem of determining the aligned magnetic field appearing in the magnetic Schrödinger equation in a periodic quantum cylindrical waveguide, by knowledge of the DirichlettoNeumann map. We prove a Hölder stability estimate with respect to the DirichlettoNeumann map, by means of the geometrical optics solutions of the magnetic Schrödinger equation.
 Representation Theory and Algebraic Methods

Twisted logarithmic modules of free field algebras
View Description Hide DescriptionGiven a nonsemisimple automorphism φ of a vertex algebra V, the fields in a φtwisted Vmodule involve the logarithm of the formal variable, and the action of the Virasoro operator L 0 on such a module is not semisimple. We construct examples of such modules and realize them explicitly as Fock spaces when V is generated by free fields. Specifically, we consider the cases of symplectic fermions (odd superbosons), free fermions, and βγsystem (even superfermions). In each case, we determine the action of the Virasoro algebra.
 Quantum Mechanics

Effective nonMarkovian description of a system interacting with a bath
View Description Hide DescriptionWe study a harmonic system coupled to a chain of first neighbor interacting oscillators. After deriving the exact dynamics of the system, we prove that one can effectively describe the exact dynamics by considering a suitable shorter chain. We provide the explicit expression for such an effective dynamics and we provide an upper bound on the error one makes considering it instead of the dynamics of the full chain. We eventually prove how error, time scale and number of modes in the truncated chain are related.

Kato expansion in quantum canonical perturbation theory
View Description Hide DescriptionThis work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator. Our approach leads to an explicit expression for a generator of a blockdiagonalizing Dyson’s ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and blockdiagonalized Hamiltonian. We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnus methods for high perturbative orders.

The Wentzel–Kramers–Brillouin approximation method applied to the Wigner function
View Description Hide DescriptionAn adaptation of the Wentzel–Kramers–Brilluoin method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase of a wave function and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi–classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl–Teller potential have been found.

A generalized JaynesCummings model: The relativistic parametric amplifier and a single trapped ion
View Description Hide DescriptionWe introduce a generalization of the JaynesCummings model and study some of its properties. We obtain the energy spectrum and eigenfunctions of this model by using the tilting transformation and the squeezed number states of the onedimensional harmonic oscillator. As physical applications, we connect this new model to two important and novelty problems: the relativistic parametric amplifier and the quantum simulation of a single trapped ion.

Algebraic solutions of shapeinvariant positiondependent effective mass systems
View Description Hide DescriptionKeeping in view the ordering ambiguity that arises due to the presence of positiondependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with positiondependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and LévyLeblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class of nonlinear oscillators has been considered. This class includes the particular example of a onedimensional oscillator with different positiondependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.
 Quantum Information and Computation

Fault tolerant filtering and fault detection for quantum systems driven by fields in single photon states
View Description Hide DescriptionThe purpose of this paper is to solve the fault tolerant filtering and fault detection problem for a class of open quantum systems driven by a continuousmode bosonic input field in single photon states when the systems are subject to stochastic faults. Optimal estimates of both the system observables and the fault process are simultaneously calculated and characterized by a set of coupled recursive quantum stochastic differential equations.

A TrotterSuzuki approximation for Lie groups with applications to Hamiltonian simulation
View Description Hide DescriptionWe present a product formula to approximate the exponential of a skewHermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from wellknown results. We apply our results to construct product formulas useful for the quantum simulation of some continuousvariable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.

On a quantum entropy power inequality of Audenaert, Datta, and Ozols
View Description Hide DescriptionWe give a short proof of a recent inequality of Audenaert, Datta, and Ozols, and determine cases of equality.

Achieving the Holevo bound via a bisection decoding protocol
View Description Hide DescriptionWe present a new decoding protocol to realize transmission of classical information through a quantum channel at asymptotically maximum capacity, achieving the Holevo bound and thus the optimal communication rate. At variance with previous proposals, our scheme recovers the message bit by bit, making use of a series of “yesno” measurements, organized in bisection fashion, thus determining which codeword was sent in log2 N steps, N being the number of codewords.

Local unitary equivalence of quantum states and simultaneous orthogonal equivalence
View Description Hide DescriptionThe correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂ^{d1} ⊗ ℂ^{d2} are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.

Quantum Maxflow/Mincut
View Description Hide DescriptionThe classical maxflow mincut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network and, more specifically, as a linear map from the input space to the output space. The quantum maxflow is defined to be the maximal rank of this linear map over all choices of tensors. The quantum mincut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network. We show that unlike the classical case, the quantum maxflow=mincut conjecture is not true in general. Under certain conditions, e.g., when the capacity on each edge is some power of a fixed integer, the quantum maxflow is proved to equal the quantum mincut. However, concrete examples are also provided where the equality does not hold. We also found connections of quantum maxflow/mincut with entropy of entanglement and the quantum satisfiability problem. We speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity.

Secondorder asymptotics for quantum hypothesis testing in settings beyond i.i.d.—quantum lattice systems and more
View Description Hide DescriptionQuantum Stein’s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states (ρ or σ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n) identical copies of the state (ρ ^{⊗n} or σ ^{⊗n}) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability αn of erroneously inferring the state to be σ, the probability βn of erroneously inferring the state to be ρ decays exponentially in n, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d. Examples of these include Gibbs states of quantum spin systems (with finiterange, translationinvariant interactions) at high temperatures, and quasifree states of fermionic lattice gases.

Dissipative entanglement of quantum spin fluctuations
View Description Hide DescriptionWe consider two noninteracting infinite quantum spin chains immersed in a common thermal environment and undergoing a local dissipative dynamics of Lindblad type. We study the time evolution of collective mesoscopic quantum spin fluctuations that, unlike macroscopic meanfield observables, retain a quantum character in the thermodynamical limit. We show that the microscopic dissipative dynamics is able to entangle these mesoscopic degrees of freedom, through a purely mixing mechanism. Further, the behaviour of the dissipatively generated quantum correlations between the two chains is studied as a function of temperature and dissipation strength.

On the geometry of mixed states and the Fisher information tensor
View Description Hide DescriptionIn this paper, we will review the coadjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1form, settling the issues about its very definition and explicit computation. Moreover, the fibration of coadjoint orbits, seen as spaces of mixed states, is also discussed.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Canonical quantization of lattice HiggsYangMills fields: Krein essential selfadjointness of the Hamiltonian
View Description Hide DescriptionUsing a Krein indefinite metric in Fock space, the Hamiltonian for cutoff models of canonically quantized HiggsYangMills fields interpolating between the GuptaBleulerFeynman and Landau gauges is shown to be essentially maximal accretive and essentially Krein selfadjoint.
 General Relativity and Gravitation

A spacetime with pseudoprojective curvature tensor
View Description Hide DescriptionThe object of the present paper is to study spacetimes admitting pseudoprojective curvature tensor. At first we prove that a pseudoprojectively flat spacetime is Einstein and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying Einstein’s field equation with cosmological constant is covariant constant. Next, we prove that if the perfect fluid spacetime with vanishing pseudoprojective curvature tensor obeys Einstein’s field equation without cosmological constant, then the spacetime has constant energy density and isotropic pressure, and the perfect fluid always behaves as a cosmological constant and also such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field U. Moreover, it is shown that a pseudoprojectively flat spacetime satisfying Einstein’s equation without cosmological constant for a purely electromagnetic distribution is an Euclidean space. We also prove that under certain conditions a perfect fluid spacetime with divergencefree pseudoprojective curvature is a RobertsonWalker spacetime and the possible local cosmological structure of such a spacetime is of type I, D or O. We also study dustlike fluid spacetime with vanishing pseudoprojective curvature tensor.
 Dynamical Systems

The continuous transition of Hamiltonian vector fields through manifolds of constant curvature
View Description Hide DescriptionWe ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature κ (spheres, for κ > 0, and hyperbolic spheres, for κ < 0) pass continuously through the value κ = 0 if the potential functions U κ, κ ∈ ℝ, which define them satisfy the property , where U 0 corresponds to the Euclidean case. We prove that the answer to this question is positive, both in the 2 and 3dimensional cases, which are of physical interest, and then apply our conclusions to the gravitational Nbody problem.