Showing papers in "Journal of Physics A in 2014"

Quantum metrology from a quantum information science perspective

Abstract: We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramer–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Biorthogonal quantum mechanics

Abstract: The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

New Editor-in-Chief for Journal of Physics A: Mathematical and Theoretical

Abstract: We are delighted to announce that Professor Martin Evans of University of Edinburgh has been appointed as the new Editor-in-Chief of Journal of Physics A: Mathematical and Theoretical. Martin Evans has been Editor of the Statistical Physics section of the journal since 2009. Prior to this, he served as a Board Member for the journal. His areas of research include statistical mechanics of nonequilibrium systems, phase transitions and scaling regimes in nonequilibrium statistical physics, glassy dynamics, phase transitions and ordering in driven diffusive systems, mass transport models, condensation models, zero range processes and exclusion processes. We very much look forward to working with Martin to continue to improve the journal's quality and interest to the readership. We would like to thank our outgoing Editor-in-Chief, Professor Murray Batchelor. Murray has worked hard and provided excellent guidance in improving the quality of the journal and the service that the journal provides to authors, referees and readers. During the last five years, we have raised the quality threshold for acceptance in the journal and currently reject over 70% of submissions. As a result, papers published in Journal of Physics A: Mathematical and Theoretical are amongst the best in the field. We have also maintained and improved on our excellent receipt-to-first-decision times, which now average under 40 days for papers. With the help of Martin Evans and our distinguished Editorial Board, we will be working to further improve the quality of the journal whilst continuing to offer excellent services to our readers, authors and referees. We hope that you benefit from reading the journal. If you have any comments or questions, please do not hesitate to contact us at jphysa@iop.org. Rebecca Gillan Publisher

An integrable deformation of the AdS5 × S5 superstring

Abstract: The S-matrix on the world-sheet theory of the string in AdS has previously been shown to admit a deformation where the symmetry algebra is replaced by the associated quantum group. The case where q is real has been identified as a particular deformation of the Green-Schwarz sigma model. An interpretation of the case with q a root of unity has, until now, been lacking. We show that the Green-Schwarz sigma model admits a discrete deformation which can be viewed as a rather simple deformation of the gauged WZW model, where . The deformation parameter q is then a kth root of unity where k is the level. The deformed theory has the same equations-of-motion as the Green-Schwarz sigma model but has a different symplectic structure. We show that the resulting theory is integrable and has just the right amount of kappa-symmetries that appear as a remnant of the fermionic part of the original gauge symmetry. This points to the existence of a fully consistent deformed string background.

Conformal Carroll groups

Abstract: Conformal extensions of Levy–Leblondʼs Carroll group, based on geometric properties analogous to those of Newton–Cartan space-time are proposed. The extensions are labeled by an integer k. This framework includes and extends our recent study of the Bondi–Metzner–Sachs (BMS) and Newman–Unti (NU) groups. The relation to conformal Galilei groups is clarified. Conformal Carroll symmetry is illustrated by ‘Carrollian photons’. Motion both in the Newton–Cartan and Carroll spaces may be related to that of strings in the Bargmann space.

Diffusion with resetting in arbitrary spatial dimension

Abstract: We consider diffusion in arbitrary spatial dimension d with the addition of a resetting process wherein the diffusive particle stochastically resets to a fixed position at a constant rate r. We compute the nonequilibrium stationary state which exhibits non-Gaussian behaviour. We then consider the presence of an absorbing target centred at the origin and compute the survival probability and mean time to absorption of the diffusive particle by the target. The mean absorption time is finite and has a minimum value at an optimal resetting rate r which depends on dimension. Finally we consider the problem of a finite density of diffusive particles, each resetting to its own initial position. While the typical survival probability of the target at the origin decays exponentially with time regardless of spatial dimension, the average survival probability decays asymptotically as exp ( − A(ln t)d) where A is a constant. We explain these findings using an interpretation as a renewal process and arguments invoking extreme value statistics.

Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

Abstract: We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2l + 3 recurrence relation where l is the length of the partition λ. Explicit expressions for such recurrence relations are given.

Loopholes in Bell inequality tests of local realism

Abstract: Bell inequalities are intended to show that local realist theories cannot describe the world. A local realist theory is one where physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Quantum-mechanical predictions for certain experiments violate the Bell inequality while a local realist theory cannot, and this shows that a local realist theory cannot give those quantum-mechanical predictions. However, because of unexpected circumstances or ?loopholes? in available experiment tests, local realist theories can reproduce the data from these experiments. This paper reviews such loopholes, what effect they have on Bell inequality tests, and how to avoid them in experiment. Avoiding all these simultaneously in one experiment, usually called a ?loophole-free? or ?definitive? Bell test, remains an open task, but is very important for technological tasks such as device-independent security of quantum cryptography, and ultimately for our understanding of the world.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?50 years of Bell?s theorem?.

Revealing networks from dynamics: an introduction

Abstract: What can we learn from the collective dynamics of a complex network about its interaction topology? Taking the perspective from nonlinear dynamics, we briefly review recent progress on how to infer structural connectivity (direct interactions) from accessing the dynamics of the units. Potential applications range from interaction networks in physics, to chemical and metabolic reactions, protein and gene regulatory networks as well as neural circuits in biology and electric power grids or wireless sensor networks in engineering. Moreover, we briefly mention some standard ways of inferring effective or functional connectivity.

Quantifying entanglement resources

Abstract: We present an overview of the quantitative theory of single-copy entanglement in finite-dimensional quantum systems. In particular we emphasize the point of view that different entanglement measures quantify different types of resources, which leads to a natural interdependence of entanglement classification and quantification. Apart from the theoretical basis, we outline various methods for obtaining quantitative results on arbitrary mixed states.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Generalized probability theories: what determines the structure of quantum theory?

Abstract: The framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics. It provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical structure of quantum theory. This review paper tries to present the framework and recent results to a broader readership in an accessible manner. To achieve this, we follow a constructive approach. Starting from a few basic physically motivated assumptions we show how a given set of observations can be manifested in an operational theory. Furthermore, we characterize consistency conditions limiting the range of possible extensions. In this framework classical and quantum theory appear as special cases, and the aim is to understand what distinguishes quantum mechanics as the fundamental theory realized in nature. It turns out that non-classical features of single systems can equivalently result from higher-dimensional classical theories that have been restricted. Entanglement and non-locality, however, are shown to be genuine non-classical features.

Mathematical and physical aspects of complex symmetric operators

Abstract: Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-Hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics and complex variables.

Exact WKB analysis and cluster algebras

Abstract: We develop the mutation theory in the exact WKB analysis using the framework of cluster algebras. Under a continuous deformation of the potential of the Schrodinger equation on a compact Riemann surface, the Stokes graph may change the topology. We call this phenomenon the mutation of Stokes graphs. Along the mutation of Stokes graphs, the Voros symbols, which are monodromy data of the equation, also mutate due to the Stokes phenomenon. We show that the Voros symbols mutate as variables of a cluster algebra with surface realization. As an application, we obtain the identities of Stokes automorphisms associated with periods of cluster algebras. The paper also includes an extensive introduction of the exact WKB analysis and the surface realization of cluster algebras for nonexperts.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Cluster algebras in mathematical physics'.

Entanglement witnesses: construction, analysis and classification

Abstract: From the physical point of view entanglement witnesses (EWs) define a universal tool for analysis and classification of quantum entangled states. From the mathematical perspective bipartite EWs provide highly non-trivial generalization of positive operators and establish elegant correspondence with the theory of positive maps in matrix algebras. We concentrate on theoretical analysis of various important notions like decomposability, atomicity, optimality, extremality and exposedness. Several methods of construction are provided as well. Our discussion is illustrated by many examples enabling the reader to see the intricate structure of these objects. It is shown that the theory of EWs finds elegant geometric formulation in terms of convex cones and related geometric structures.

Time scale of diffusion in molecular and cellular biology

Abstract: Diffusion is the driver of critical biological processes in cellular and molecular biology. The diverse temporal scales of cellular function are determined by vastly diverse spatial scales in most biophysical processes. The latter are due, among others, to small binding sites inside or on the cell membrane or to narrow passages between large cellular compartments. The great disparity in scales is at the root of the difficulty in quantifying cell function from molecular dynamics and from simulations. The coarse-grained time scale of cellular function is determined from molecular diffusion by the mean first passage time of molecular Brownian motion to a small targets or through narrow passages. The narrow escape theory (NET) concerns this issue. The NET is ubiquitous in molecular and cellular biology and is manifested, among others, in chemical reactions, in the calculation of the effective diffusion coefficient of receptors diffusing on a neuronal cell membrane strewn with obstacles, in the quantification of the early steps of viral trafficking, in the regulation of diffusion between the mother and daughter cells during cell division, and many other cases. Brownian trajectories can represent the motion of a molecule, a protein, an ion in solution, a receptor in a cell or on its membrane, and many other biochemical processes. The small target can represent a binding site or an ionic channel, a hidden active site embedded in a complex protein structure, a receptor for a neurotransmitter on the membrane of a neuron, and so on. The mean time to attach to a receptor or activator determines diffusion fluxes that are key regulators of cell function. This review describes physical models of various subcellular microdomains, in which the NET coarse-grains the molecular scale to a higher cellular-level, thus clarifying the role of cell geometry in determining subcellular function.

High-order quantum algorithm for solving linear differential equations

Abstract: Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems. We examine the use of high-order methods (where the error over a time step is a high power of the size of the time step) to improve the efficiency. These provide scaling close to Δt2 in the evolution time Δt. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possible to extract global features of the solution.

On nonlocality as a resource theory and nonlocality measures

Abstract: With the advent of device-independent quantum information processing, nonlocality is nowadays regarded as a resource that can be used for various tasks. Using the analogy of entanglement theory, we approach nonlocality from this perspective. In order to do so, we analyze in full detail the operations that can be implemented in this scenario and under which nonlocality cannot increase. This provides a theoretical basis from which to study how to order and quantify nonlocal behavior. Finally, we review several nonlocality measures and discuss their validity from this point of view.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell's theorem’.

General N-soliton solution to a vector nonlinear Schrödinger equation

Abstract: We consider a general N-soliton solution to a vector nonlinear Schrodinger (NLS) equation of all possible combinations of nonlinearities including all-focusing, all-defocusing and mixed types. Based on the KP hierarchy reduction method, we firstly construct general two-bright-one-dark and one-bright-two-dark soliton solutions in a three-coupled NLS equation, then we extend our analysis to a vector NLS equation to obtain a general N-soliton solution in Gram determinant form. This formula unifies the bright, dark and bright-dark soliton solutions, which have been widely studied in the literature. The conditions for the existence of all types of soliton solutions with all possible combinations of nonlinearities are elucidated.

Cluster polylogarithms for scattering amplitudes

Abstract: Motivated by the cluster structure of two-loop scattering amplitudes in $\mathcal{N}=4$ Yang-Mills theory we define cluster polylogarithm functions. We find that all such functions of weight four are made up of a single simple building block associated with the A(2) cluster algebra. Adding the requirement of locality on generalized Stasheff polytopes, we find that these A(2) building blocks arrange themselves to form a unique function associated with the A(3) cluster algebra. This A(3) function manifests all of the cluster algebraic structure of the two-loop n-particle MHV amplitudes for all n, and we use it to provide an explicit representation for the most complicated part of the n = 7 amplitude as an example.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’.

The statistical mechanics of the coagulation–diffusion process with a stochastic reset

Abstract: The effects of a stochastic reset, to its initial configuration, is studied in the exactly solvable one-dimensional coagulation–diffusion process. A finite resetting rate leads to a modified non-equilibrium stationary state. If, in addition, the input of particles at a fixed given rate is admitted, then a competition between the resetting and the input rates leads to a non-trivial behaviour of the particle-density in the stationary state. From the exact inter-particle probability distribution, a simple physical picture emerges: the reset mainly changes the behaviour at larger distance scales, while at smaller length scales, the non-trivial correlation of the model without a reset dominate.

General rogue waves in the focusing and defocusing Ablowitz–Ladik equations

Abstract: General rogue waves in the focusing and defocusing Ablowitz–Ladik equations are derived by the bilinear method. In the focusing case, it is shown that rogue waves are always bounded. In addition, fundamental rogue waves reach peak amplitudes which are at least three times that of the constant background, and higher-order rogue waves can exhibit patterns such as triads and circular arrays with different individual peaks. In the defocusing case, it is shown that rogue waves also exist. In addition, these waves can blow up to infinity in finite time.

Lie point and variational symmetries in minisuperspace Einstein gravity

Abstract: We consider the application of the theory of symmetries of coupled ordinary differential equations to the case of reparametrization invariant Lagrangians quadratic in the velocities; such Lagrangians encompass all minisuperspace models. We find that, in order to acquire the maximum number of possibly existing symmetry generators, one must (a) consider the lapse N(t) among the degrees of freedom and (b) allow the action of the generator on the Lagrangian and/or the equations of motion to produce a multiple of the constraint, rather than strictly zero. The result of this necessary modification of the standard theory (concerning regular systems) is that the Lie point symmetries of the equations of motion are exactly the variational symmetries (containing the time reparametrization symmetry) plus the well known scaling symmetry. These variational symmetries are seen to be the simultaneous conformal Killing fields of both the metric and the potential, thus coinciding with the conditional symmetries defined in phase space. In a parametrization of the lapse for which the potential becomes constant, the generators of the aforementioned symmetries become the Killing fields of the scaled supermetric and its homothetic field, respectively.

A gaudin-like determinant for overlaps of Néel and XXZ Bethe states

Abstract: We derive a determinant expression for overlaps of Bethe states of the XXZ spin chain with the Neel state, the ground state of the system in the antiferromagnetic Ising limit. Our formula, of determinant form, is valid for generic system size. Interestingly, it is remarkably similar to the well-known Gaudin formula for the norm of Bethe states, and to another recently-derived overlap formula appearing in the Lieb-Liniger model.

Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator

Abstract: We revisit the tree-level closed superstring amplitude and identify its α'-expansion as series with single-valued multiple zeta values as coefficients. The latter represent a subclass of multiple zeta values originating from single-valued multiple polylogarithms at unity. Moreover, the α'-expansion of the closed superstring amplitude can be cast into the same algebraic form as the open superstring amplitude: the closed superstring amplitude is essentially the single-valued version of the open superstring amplitude. This fact points to a deeper connection between gauge and gravity amplitudes than is implied by Kawai–Lewellen–Tye relations. Furthermore, we argue that the Deligne associator carries the relevant information on the closed superstring amplitude. In particular, we give an explicit representation of the Deligne associator in terms of Gamma functions modulo squares of commutators of the underlying Lie algebra. This form of the associator can be interpreted as the four-point closed superstring amplitude.

Quantum quenches from excited states in the Ising chain

Abstract: We consider the non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that stationary values of local correlation functions can be described by the generalized Gibbs ensemble. Then we study the full time evolution of the transverse magnetization by means of stationary phase methods. The equal-time two-point longitudinal correlation function is analytically derived for a particular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches.

Quantum randomness extraction for various levels of characterization of the devices

Abstract: The amount of intrinsic randomness that can be extracted from measurement on quantum systems depends on several factors: notably, the power given to the adversary and the level of characterization of the devices of the authorized partners. After presenting a systematic introduction to these notions, in this paper we work in the class of least adversarial power, which is relevant for assessing setups operated by trusted experimentalists, and compare three levels of characterization of the devices. Many recent studies have focused on the so-called ?device-independent? level, in which a lower bound on the amount of intrinsic randomness can be certified without any characterization. The other extreme is the case when all the devices are fully characterized: this ?tomographic? level has been known for a long time. We present for this case a systematic and efficient approach to quantifying the amount of intrinsic randomness, and show that setups involving ancillas (positive-operator valued measures, pointer measurements) may not be interesting here, insofar as one may extract randomness from the ancilla rather than from the system under study. Finally, we study how much randomness can be obtained in presence of an intermediate level of characterization related to the task of ?steering?, in which Bob?s device is fully characterized while Alice?s is a black box. We obtain our results here by adapting the NPA hierarchy of semidefinite programs to the steering scenario.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?50 years of Bell?s theorem?.

What Bell Did

Abstract: On the 50th anniversary of Bell?s monumental 1964 paper, there is still widespread misunderstanding about exactly what Bell proved. This misunderstanding derives in turn from a failure to appreciate the earlier argument of Einstein, Podolsky and Rosen. I retrace the history and logical structure of these arguments in order to clarify the proper conclusion, namely that any world that displays violations of Bell?s inequality for experiments done far from one another must be non-local. Since the world we happen to live in displays such violations, actual physics is non-local.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?50 years of Bell?s theorem?.

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials

Abstract: The critical curves of the q-state Potts model can be determined exactly for regular two-dimensional lattices G that are of the three-terminal type. This comprises the square, triangular, hexagonal and bow–tie lattices. Jacobsen and Scullard have defined a graph polynomial PB(q, v) that gives access to the critical manifold for general lattices. It depends on a finite repeating part of the lattice, called the basis B, and its real roots in the temperature variable v = eK − 1 provide increasingly accurate approximations to the critical manifolds upon increasing the size of B. Using transfer matrix techniques, these authors computed PB(q, v) for large bases (up to 243 edges), obtaining determinations of the ferromagnetic critical point vc > 0 for the (4, 82), kagome, and (3, 122) lattices to a precision (of the order 10−8) slightly superior to that of the best available Monte Carlo simulations. In this paper we describe a more efficient transfer matrix approach to the computation of PB(q, v) that relies on a formulation within the periodic Temperley–Lieb algebra. This makes possible computations for substantially larger bases (up to 882 edges), and the precision on vc is hence taken to the range 10−13. We further show that a large variety of regular lattices can be cast in a form suitable for this approach. This includes all Archimedean lattices, their duals and their medials. For all these lattices we tabulate high-precision estimates of the bond percolation thresholds pc and Potts critical points vc. We also trace and discuss the full Potts critical manifold in the (q, v) plane, paying special attention to the antiferromagnetic region v < 0. Finally, we adapt the technique to site percolation as well, and compute the polynomials PB(p) for certain Archimedean and dual lattices (those having only cubic and quartic vertices), using very large bases (up to 243 vertices). This produces the site percolation thresholds pc to a precision of the order of 10−9.

Néel-XXZ state overlaps: odd particle numbers and Lieb–Liniger scaling limit

Abstract: We specialize a recently-proposed determinant formula (Brockmann, De Nardis, Wouters and Caux 2014 J. Phys. A: Math. Theor. 47 145003) for the overlap of the zero-momentum N?el state with Bethe states of the spin-1/2 XXZ chain to the case of an odd number of downturned spins, showing that it is still of ?Gaudin-like? form, similar to the case of an even number of down spins. We generalize this result to the overlap of q-raised N?el states with parity-invariant Bethe states lying in a nonzero magnetization sector. The generalized determinant expression can then be used to derive the corresponding determinants and their prefactors in the scaling limit to the Lieb?Liniger (LL) Bose gas. The odd number of down spins directly translates to an odd number of bosons. We furthermore give a proof that the N?el state has no overlap with non-parity-invariant Bethe states. This is based on a determinant expression for overlaps with general Bethe states that was obtained in the context of the XXZ chain with open boundary conditions (Pozsgay 2013 arXiv:1309.4593, Kozlowski and Pozsgay 2012 J. Stat. Mech. P05021, Tsuchiya 1998 J. Math. Phys. 39 5946). The statement that overlaps with non-parity-invariant Bethe states vanish is still valid in the scaling limit to LL which means that the Bose?Einstein condensate state (De Nardis, Wouters, Brockmann and Caux 2014 Phys. Rev. A 89 033601) has zero overlap with non-parity-invariant LL Bethe states.

Construction of all general symmetric informationally complete measurements

Abstract: We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.