Phase diagram for the Grover algorithm with static imperfections

We study effects of static inter-qubit interactions on the stability of the Grover quantum search algorithm. Our numerical and analytical results show existence of regular and chaotic phases depending on the imperfection strength e\varepsilon . The critical border e_c\varepsilon_c between two phases drops polynomially with the number of qubits n _q as e_c ~ n_q^-3/2\varepsilon_c \sim n_q^{-3/2} . In the regular phase (e < e_c)(\varepsilon < \varepsilon_c) the algorithm remains robust against imperfections showing the efficiency gain e_c / e\varepsilon_c / \varepsilon for e >~2^-n_q/2\varepsilon \gtrsim 2^{-n_q/2} . In the chaotic phase $(\varepsilon > \varepsilon_c)$(\varepsilon > \varepsilon_c) the algorithm is completely destroyed.     

Inequalities and separations among assisted capacities of quantum channels

We exhibit quantum channels whose classical and quantum capacities, when assisted by classical feedback, exceed their unassisted classical Holevo capacity. These channels are designed to be noisy in a way that can be corrected with the help of the output and a reference system entangled with part of the input. A similar construction yields quantum channels whose classical capacity, when assisted by two-way classical communication independent of the source, exceeds their classical capacity assisted by feedback alone. We give a hierarchy of capacity inequalities and open questions.     

$${\phi}$$ -Coordinated Quasi-Modules for Quantum Vertex Algebras

We develop a theory of f{\phi} -coordinated (quasi-) modules for a general nonlocal vertex algebra where f{\phi} is what we call an associate of the one-dimensional additive formal group. By specializing f{\phi} to a particular associate, we obtain a new construction of weak quantum vertex algebras in the sense of Li (Selecta Mathematica (New Series) 11:349–397, 2005). As an application, we associate weak quantum vertex algebras to quantum affine algebras, and we also associate quantum vertex algebras and f{\phi} -coordinated modules to a certain quantum βγ-system explicitly.     

Relevance of the Slip Condition for Fluid Flows Near an Irregular Boundary

We consider the Navier-Stokes equation in a domain with a rough boundary. The roughness is modeled by a small amplitude and small wavelength oscillation, with typical scale e << 1{\varepsilon \ll 1}. For periodic oscillation, it is well-known that the best homogenized (that is regular in e{\varepsilon}) boundary condition is of Navier type. Such result still holds for random stationary irregularities, as shown recently by the first author [5, 15]. We study here arbitrary irregularity patterns.     

Cosmological Post-Newtonian Expansions to Arbitrary Order

We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter e = v_T/c{\epsilon=v_T/c} (0 < e < e₀){(0< \epsilon < \epsilon_0)}, where c is the speed of light, and v _T is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab M @ [0,T)×\mathbb T³{M\cong [0,T)\times \mathbb {T}^3}, and converge as e\searrow 0{\epsilon \searrow 0} to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter e{\epsilon} to any specified order with expansion coefficients that satisfy e{\epsilon}-independent (nonlocal) symmetric hyperbolic equations.     

Protecting the Conformal Symmetry via Bulk Renormalization on Anti deSitter Space

The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field j{\varphi} on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field f{\phi} on d + 1-dimensional Anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the one-loop “fish diagram” on AdS₄ by differential renormalization, and calculate the anomalous dimension of the composite boundary field j²{\varphi^2} with bulk interaction kf⁴{\kappa \phi^4}.     

Adaptive Sub-sampling for Parametric Estimation of Gaussian Diffusions

We consider a Gaussian diffusion X _t (Ornstein-Uhlenbeck process) with drift coefficient γ and diffusion coefficient σ ², and an approximating process Y^e_tY^{\varepsilon}_{t} converging to X _t in L ₂ as ε→0. We study estimators [^(g)]_e\hat{\gamma}_{\varepsilon}, [^(s)]²_e\hat{\sigma}^{2}_{\varepsilon} which are asymptotically equivalent to the Maximum likelihood estimators of γ and σ ², respectively. We assume that the estimators are based on the available N=N(ε) observations extracted by sub-sampling only from the approximating process Y^e_tY^{\varepsilon}_{t} with time step Δ=Δ(ε). We characterize all such adaptive sub-sampling schemes for which [^(g)]_e\hat{\gamma}_{\varepsilon}, [^(s)]²_e\hat{\sigma}^{2}_{\varepsilon} are consistent and asymptotically efficient estimators of γ and σ ² as ε→0. The favorable adaptive sub-sampling schemes are identified by the conditions ε→0, Δ→0, (Δ/ε)→∞, and NΔ→∞, which implies that we sample from the process Y^e_tY^{\varepsilon}_{t} with a vanishing but coarse time step Δ(ε)≫ε. This study highlights the necessity to sub-sample at adequate rates when the observations are not generated by the underlying stochastic model whose parameters are being estimated. The adequate sub-sampling rates we identify seem to retain their validity in much wider contexts such as the additive triad application we briefly outline.     

The value of the cosmological constant

We make the cosmological constant, Λ, into a field and restrict the variations of the action with respect to it by causality. This creates an additional Einstein constraint equation. It restricts the solutions of the standard Einstein equations and is the requirement that the cosmological wave function possess a classical limit. When applied to the Friedmann metric it requires that the cosmological constant measured today, t _U , be L ~ t_U^-2 ~ 10^-122{\Lambda \sim t_{U}^{-2} \sim 10^{-122}} , as observed. This is the classical value of Λ that dominates the wave function of the universe. Our new field equation determines Λ in terms of other astronomically measurable quantities. Specifically, it predicts that the spatial curvature parameter of the universe is W_k0 o -k/a₀²H²=-0.0055{\Omega _{\mathrm{k0}} \equiv -k/a_{0}^{2}H^{2}=-0.0055} , which will be tested by Planck Satellite data. Our theory also creates a new picture of self-consistent quantum cosmological history.     

High energy radiation femtochemistry of water molecules: early electron-radical pairs processes

The damages triggered by ionizing radiation on chemical and biological targets depend on the survival probability of radicals produced in clusters of ionization-excitation events. In this paper, we report on femtolysis (FEMTOsecond radioLYSIS) of pure liquid water using an innovative laser produced high-energy, ultra-short electron bunches in the 2.5-15 MeV range and high energy radiation femtochemistry (HERF) measurements. The short-time monitoring of a primary reducing radical, hydrated electron e^-_aq^{-}_{aq}, has been performed in confined ionization spaces (nascent spurs). The calculated yield of hydrated electrons at early time, G(e^-_aq)_ETG({\rm e}^{-}_{aq})_{ET}, is estimated to be 6.5 ± 0.5 (number/100 eV) at t ~ 5 ps after the ultrafast energy deposition. This estimated value is high compare to (i) the available data of previous works that used scavenging techniques; (ii) the predictions of stochastic water radiolysis modelling for which the initial behaviour of hydrated electron is investigated in the framework of a classical diffusion regime of independent pairs. The HERF developments give new insights into the early ubiquitous radical escape probability in nascent aqueous spurs and emphasize the importance of short-lived solvent bridged electron-radical complexes [H₃O^+...{\rm H}_{3}{\rm O}^{+...}  e_aq^-{\rm e}_{aq}^{-} ..OH]_nH2O{\rm OH}]_{n{\rm H}_2{\rm O}} (non-independent pairs). A complete understanding of the G(e^-_aq)_ET{\rm e}^{-}_{aq})_{ET} value needs to account for quantum aspects of 1s-like trapped electron ground state and neoformed prototropic radicals that govern ultra-fast recombination processes within these non-independent pair configurations. Femtolysis data emphasize that within a time-dependent non-diffusion regime, spatio-temporal correlations between hydrated electron and nearest neighbours OH radical or hydrated proton (H₃O⁺{\rm H}_{3}{\rm O}^{+}) would assist ultrafast anisotropic 1D recombination within solvent bridged electron-radical complexes. The emerging HERF domain would provide guidance for understanding of ultrashort-lived sub-structure of tracks and stimulate future semi-quantum simulations on prethermal radical reactions.     

Solar System planetary tests of $${\dot c/c}$$

Analytical and numerical calculations show that a putative temporal variation of the speed of light c, with the meaning of space-time structure constant c _ST, assumed to be linear over timescales of about one century, would induce a secular precession of the longitude of the pericenter v{\varpi} of a test particle orbiting a spherically symmetric body. By comparing such a predicted effect to the corrections D[(v)\dot]{\Delta\dot\varpi} to the usual Newtonian/Einsteinian perihelion precessions of the inner planets of the Solar System, recently estimated by E.V. Pitjeva by fitting about one century of modern astronomical observations with the standard classical/relativistic dynamical force models of the EPM epehemerides, we obtained [(c)\dot]/c = (0.5±2)×10^-7 yr^-1{\dot c/c =(0.5\pm 2)\times 10^{-7} {\rm yr}^{-1}} . Moreover, the possibility that [(c)\dot]/c 1 0{\dot c/c\neq 0} over the last century is ruled out at 3−12σ level by taking the ratios of the perihelia for different pairs of planets. Our results are independent of any measurement of the variations of other fundamental constants which may be explained by a variation of c itself (with the meaning of electromagnetic constant c _EM). It will be important to repeat such tests if and when other teams of astronomers will estimate their own corrections to the standard Newtonian/Einsteinian planetary perihelion precessions with different ephemerides.     

On the maximum admissible error and key compression degree in quantum cryptography on two nonorthogonal states

The extreme admissible error probability at the receiving end to which the secure key distribution is possible has been found. This result takes into account all possible attacks on the distributed key, including those that involve large quantum memory and the capability of an eavesdropper to perform collective measurements simultaneously over the entire transmitted sequence of quantum states. The critical error is independent of the parameters of a particular attack and is determined only in terms of the overlapping degree ε = |〈u₁|u₀〉¦ of information states and the fundamental functions of classical and quantum information theories. The latter functions are the capacity H(Q) of a classical binary communication channel and classical capacity \(\bar C(\varepsilon )\) of a binary quantum communication channel. The key compression degree after error correction is also expressed in terms of only the classical capacity \(\bar C(\varepsilon )\) of the quantum communication channel.     

On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature

We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to +  (equal to ?). For β large enough we show that for any ${\varepsilon >0 }We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to +  (equal to −). For β large enough we show that for any ${\varepsilon >0 }${\varepsilon >0 } there exists c=c(b,e){c=c(\beta,\varepsilon)} such that the corresponding mixing time T _mix satisfies lim_L?￥ P(T_mix 3 exp(cL^e)) = 0{{\rm lim}_{L\to\infty}\,{\bf P}\left(T_{\rm mix}\ge {\rm exp}({cL^\varepsilon})\right) =0}. In the non-random case τ ≡ +  (or τ ≡ −), this implies that T_mix ￡ exp(cL^e){T_{\rm mix}\le {\rm exp}({cL^\varepsilon})}. The same bound holds when the boundary conditions are all +  on three sides and all − on the remaining one. The result, although still very far from the expected Lifshitz behavior T _mix = O(L ²), considerably improves upon the previous known estimates of the form T_mix ￡ exp(c L^{\frac 12 + e}){T_{\rm mix}\le {\rm exp}({c L^{\frac 12 + \varepsilon}})}. The techniques are based on induction over length scales, combined with a judicious use of the so-called “censoring inequality” of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to its equilibrium measure.     

Quantum Brownian Motion on Non-Commutative Manifolds: Construction,Deformation and Exit Times

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, [math.PR], 2009), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus A_q{\mathcal{A}_\theta} , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A_q{\mathcal{A}_\theta} is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.     

Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.     

Conformational dynamics and internal friction in homopolymer globules: equilibrium vs. non-equilibrium simulations

We study the conformational dynamics within homopolymer globules by solvent-implicit Brownian dynamics simulations. A strong dependence of the internal chain dynamics on the Lennard-Jones cohesion strength e \varepsilon and the globule size N _G is observed. We find two distinct dynamical regimes: a liquid-like regime (for e \varepsilon < e_s \varepsilon_{{\rm s}}^{} with fast internal dynamics and a solid-like regime (for e \varepsilon > e_s \varepsilon_{{\rm s}}^{} with slow internal dynamics. The cohesion strength e_s \varepsilon_{{\rm s}}^{} of this freezing transition depends on N _G . Equilibrium simulations, where we investigate the diffusional chain dynamics within the globule, are compared with non-equilibrium simulations, where we unfold the globule by pulling the chain ends with prescribed velocity (encompassing low enough velocities so that the linear-response, viscous regime is reached). From both simulation protocols we derive the internal viscosity within the globule. In the liquid-like regime the internal friction increases continuously with e \varepsilon and scales extensive in N _G . This suggests an internal friction scenario where the entire chain (or an extensive fraction thereof) takes part in conformational reorganization of the globular structure.     

The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information

An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the classical and quantum capacities of the channel. Although our formula involves regularization, i.e. taking a limit over many copies of the channel, it reduces to a single-letter expression in the case of generalized dephasing channels. Analogous formulas are conjectured for the simultaneous public-private capacity of a quantum channel and for the simultaneously 1-way distillable common randomness and entanglement of a bipartite quantum state.     

Natural Connections for Semi-Hamiltonian Systems: The Case of the $${\epsilon}$$-System

Given a semi-Hamiltonian system, we construct an F-manifold with a connection satisfying a suitable compatibility condition with the product. We exemplify this procedure in the case of the so-called e{\epsilon}-system. The corresponding connection turns out to be flat, and the flat coordinates give rise to additional chains of commuting flows.     

Big bounce from spin and torsion

The Einstein-Cartan-Sciama-Kibble theory of gravity naturally extends general relativity to account for the intrinsic spin of matter. Spacetime torsion, generated by spin of Dirac fields, induces gravitational repulsion in fermionic matter at extremely high densities and prevents the formation of singularities. Accordingly, the big bang is replaced by a bounce that occurred when the energy density e μ gT⁴{\epsilon \propto gT^4} was on the order of n²/m_Pl²{n^2/m_{\rm Pl}^2} (in natural units), where n μ gT³{n \propto gT^3} is the fermion number density and g is the number of thermal degrees of freedom. If the early Universe contained only the known standard-model particles (g ≈ 100), then the energy density at the big bounce was about 15 times larger than the Planck energy. The minimum scale factor of the Universe (at the bounce) was about 10³² times smaller than its present value, giving ≈ 50 μm. If more fermions existed in the early Universe, then the spin-torsion coupling causes a bounce at a lower energy and larger scale factor. Recent observations of high-energy photons from gamma-ray bursts indicate that spacetime may behave classically even at scales below the Planck length, supporting the classical spin-torsion mechanism of the big bounce. Such a classical bounce prevents the matter in the contracting Universe from reaching the conditions at which a quantum bounce could possibly occur.     

Quantum Transport in Crystals: Effective Mass Theorem and K·P Hamiltonians

In this paper the effective mass approximation and the k·p multi-band models, describing quantum evolution of electrons in a crystal lattice, are discussed. Electrons are assumed to move in both a periodic potential and a macroscopic one. The typical period e{\epsilon} of the periodic potential is assumed to be very small, while the macroscopic potential acts on a much bigger length scale. Such homogenization asymptotic is investigated by using the envelope-function decomposition of the electron wave function. If the external potential is smooth enough, the k·p and effective mass models, well known in solid-state physics, are proved to be close (in the strong sense) to the exact dynamics. Moreover, the position density of the electrons is proved to converge weakly to its effective mass approximation.     

Non-Abelian Vortices on Compact Riemann Surfaces

We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field f{\phi} with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface Σ, their moduli space of solutions is closely related to a moduli space of τ-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix f{\phi} , we show that the vortex solutions are entirely characterized by the location in Σ of the zeros of det f{\phi} and by the choice of a vortex internal structure at each of these zeros. We describe explicitly the vortex internal spaces and show that they are compact and connected spaces.