Quantum mechanical hamiltonian models of turing machines

Quantum mechanical Hamiltonian models, which represent an aribtrary but finite number of steps of any Turing machine computation, are constructed here on a finite lattice of spin-1/2 systems. Different regions of the lattice correspond to different components of the Turing machine (plus recording system). Successive states of any machine computation are represented in the model by spin configuration states. Both time-independent and time-dependent Hamiltonian models are constructed here. The time-independent models do not dissipate energy or degrade the system state as they evolve. They operate close to the quantum limit in that the total system energy uncertainty/computation speed is close to the limit given by the time-energy uncertainty relation. However, the model evolution is time global and the Hamiltonian is more complex. The time-dependent models do not degrade the system state. Also they are time local and the Hamiltonian is less complex.     

Chaotic polaronic and bipolaronic states in the adiabatic Holstein model

A rigorous proof for the existence of bipolaronic states is given for the adiabatic Holstein model for any lattice at any dimension, periodic or not, and for an arbitrary band filling, provided that the electron-phonon coupling (in dimensionless units) is large enough. The existence of mixed polaronic-bipolaronic states is also proven, but for larger electron-phonon coupling. These states consist of arbitrary distributions of bipolarons (or of bipolarons and polarons) localized in real space which can be simply labeled by pseudospin configurations as for a lattice gas model. The theory not only applies to periodic crystals, but also to quasicrystals, amorphous structures, polymer network, etc.When these bipolaronic and mixed polaronic-bipolaronic states exist, it is proven that: (1) These bipolaronic (and mixed polaronic-bipolaronic) states exhibit a nonzero phonon gap with a nonvanishing lower bound and an electronic gap at the Fermi energy. (2) These structures are insulating. The perturbation generated by any local change in the bipolaronic or polaronic distribution or by any charged impurity or defect decays exponentially at long distance. (3) These bipolaronic (and mixed polaronic-bipolaronic) states persist for any uniform magnetic field. (4) For large enough electron-phonon coupling, the ground state of the extended adiabatic Holstein model is a bipolaronic state when there is no uniform magnetic field or when it is small enough. It becomes a mixed polaronic-bipolaronic state for large enough magnetic field (note that the mixed polaronic-bipolaronic states are magnetic).In one-dimensional models, the ground state is an incommensurate (or commensurate) charge density wave (CDW) as predicted by Peierls (this result is not rigorous, but has been confirmed numerically). It is proven that the ground state becomes a bipolaronic charge density wave (BCDW) at large enough electron-phonon coupling. The existence of a transition by breaking of analyticity (TBA), which was numerically observed as a function of the electron-phonon coupling, is then confirmed. In that case, the shape of the effective bipolaron can be numerically calculated. It is observed that its size diverges at the TBA. The physical properties of BCDWs are rather different from those predicted by standard charge density wave theory. Bipolaronic charge density waves can also exist in models which are not only low-dimensional, but purely two- or three-dimensional.The technique for proving these theorems is an application of the concept of anti-integrability initially developed for Hamiltonian dynamical systems. It consists in proving that the eigenstates of the (trivial) Hamiltonian (called antiintegrable) obtained by canceling all electronic and lattice kinetic terms survive as a uniformly continuous function of the electronic kinetic energy terms in the Hamiltonian up to a certain threshold.     

Super-Derivations and Associated Standard Super-Potentials

Abstractly defined super-derivations on Fermionic systems on a lattice are studied. The existence and uniqueness of the associated standard super-potential are shown for every super-derivation with the subalgebra of all local operators as its domain. The relation between the standard super-potential of a super-derivation and the standard potential for the square of the super-potential (which is shown to be a derivation in the case of finite range super-potentials) is obtained (by use of local super-Hamiltonian for the super-derivation and local Hamiltonian for the square). As a consequence, a necessary and sufficient condition for a super-derivation to be nilpotent is obtained in terms of the corresponding standard super potential. Examples of translation invariant nilpotent super-derivations are given in the case of super-potentials of finite ranges on a one-dimensional lattice. A merit of considering the super-potential associated with a super-derivation is that the former can be used as free parameters for the latter.     

Translation invariant tensor product states in a finite lattice system

We show that the matrix (or more generally tensor) product states in a finite translation invariant system can be accurately constructed from a same set of local matrices (or tensors) that are determined from an infinite lattice system in one or higher dimensions. This provides an efficient approach for studying translation invariant tensor product states in finite lattice systems. Two methods are introduced to determine the size-independent local tensors.     

Joint Extension of States of Subsystems for a CAR System

 The problem of existence and uniqueness of a state of a joint system with given restrictions to subsystems is studied for a Fermion system, where a novel feature is non-commutativity between algebras of subsystems. For an arbitrary (finite or infinite) number of given subsystems, a product state extension is shown to exist if and only if all states of subsystems except at most one are even (with respect to the Fermion number). If the states of all subsystems are pure, then the same condition is shown to be necessary and sufficient for the existence of any joint extension. If the condition holds, the unique product state extension is the only joint extension. For a pair of subsystems, with one of the given subsystem states pure, a necessary and sufficient condition for the existence of a joint extension and the form of all joint extensions (unique for almost all cases) are given. For a pair of subsystems with non-pure subsystem states, some classes of examples of joint extensions are given where non-uniqueness of joint extensions prevails. Received: 17 May 2002 / Accepted: 16 January 2003 Published online: 17 April 2003 Communicated by D. Buchholz and K.Fredenhagen     

Generalized squeezed states

Squeezed states are one of the most useful quantum optical models having various applications in different areas, especially in quantum information processing. Generalized squeezed states are even more interesting since, sometimes, they provide additional degrees of freedom in the system. However, they are very difficult to construct and, therefore, people explore such states for individual setting and, thus, a generic analytical expression for generalized squeezed states is yet inadequate in the literature. In this article, we propose a method for the generalization of such states, which can be utilized to construct the squeezed states for any kind of quantum models. Our protocol works accurately for the case of the trigonometric Rosen–Morse potential, which we have considered as an example. Presumably, the scheme should also work for any other quantum mechanical model. In order to verify our results, we have studied the nonclassicality of the given system using several standard mechanisms. Among them, the Wigner function turns out to be the most challenging from the computational point of view. We, thus, also explore a generalization of the Wigner function and indicate how to compute it for a general system like the trigonometric Rosen–Morse potential with a reduced computation time.     

On Fermion Grading Symmetry for Quasi-Local Systems

We discuss fermion grading symmetry for quasi-local systems with graded commutation relations. We introduce a criterion of spontaneously symmetry breaking (SSB) for general quasi-local systems. It is formulated based on the idea that each pair of distinct phases (appeared in spontaneous symmetry breaking) should be disjoint not only for the total system but also for every complementary outside system of a local region specified by the given quasi-local structure. Under a completely model independent setting, we show the absence of SSB for fermion grading symmetry in the above sense. We obtain some structural results for equilibrium states of lattice systems. If there would exist an even KMS state for some even dynamics that is decomposed into noneven KMS states, then those noneven states inevitably violate our local thermal stability condition.     

Standard Potentials for Non-Even Dynamics

The notion of standard potentials is introduced for a general dynamics. This is a generalization of earlier works of Araki and Moriya which is restricted to even dynamics. Most formulae in the present analysis are the same as the case of even dynamics: The time derivative of a local observable is times the sum of commutators with all potentials, and application of the conditional expectation to the local algebra for a region I to a potential for a region J leave the potential unchanged if and annihilate it otherwise (the standardness of the potential). However, the convergence condition for the potential takes a different form for the odd part of the potential. The equivalence of various characterizations of equilibrium states remain valid, except that the variational principle is out of the game for non-even dynamics because the translation invariance and non-evenness of dynamics are incompatible as is already known.     

Laziness of Symmetric and Separable States

An N-partite state is considered lazy, if the entropy rate of one subsystem with respect to time is zero under any coupling to the other subsystems. In this paper, we show that all biaxial or purely second rank tensor polarized systems are lazy. Such a system can be produced in the laboratory by the interaction of a spin-1 nuclei with non-zero quadrupole moment like H ², N ¹⁴ with an external quadrupole field found in suitable crystal lattice. We then investigate the ’laziness’(property of the system to be lazy) of N-qubit mixed symmetric separable states and enumerate the conditions for them to be lazy. Further, we study the laziness of direct product states on application of a global and local noisy channels.     

The Penna model for biological ageing on a lattice: spatial consequences of child-care

We introduce a square lattice into the Penna bit-string model for biological ageing and study the evolution of the spatial distribution of the population considering different strategies of child-care. Two of the strategies are related to the movements of a whole family on the lattice: in one case the mother cannot move if she has any child younger than a given age, and in the other case if she moves, she brings these young children with her. A stronger condition has also been added to the second case, considering that young children die with a higher probability if their mothers die, this probability decreasing with age. We show that a highly non uniform occupation can be obtained when child-care is considered, even for an uniform initial occupation per site. We also compare the standard survival rate of the model with that obtained when the spacial lattice is considered (without any kind of child-care). Received 30 October 1998 and Received in final form 27 November 1998     

First-principle calculations of structural, electronic and optical properties of BaTiO3 and BaZrO3 under hydrostatic pressure

A theoretical study of structural, electronic and optical properties of cubic BaTiO₃ and BaZrO₃ perovskites is presented, using the full-potential linear augmented plane wave (FP-LAPW) method as implemented in the WIEN2K code. In this approach the local density approximation (LDA) is used for the exchange-correlation (XC) potential. Results are given for lattice constant, bulk modulus, its pressure derivative, band structure, density of states, pressure coefficients of energy gaps and refractive indices. The results are compared with previous calculations and experimental data.     

球形Dirac方程的空间格点求解及假态问题

本文在空间格点上利用虚时间步长方法求解了球形Dirac方程, 着重研究了出现的假态问题. 利用三点数值导数公式离散方程中一阶导数项, 可以证明对于量子数为 κ 和 -κ的单粒子能级能量是完全相同的, 其中一个为物理解, 另一个为假态. 通过在径向Dirac方程中引入Wilson 项, 可以解决假态问题, 得到全部物理解. 文章以 Woods-Saxon 势为例, 考虑 Wilson 项后, 得到与打靶法一致的结果.     

Entanglement in the symmetric sector of n qubits

We discuss the entanglement properties of symmetric states of n qubits. The Majorana representation maps a generic such state into a system of n points on a sphere. Entanglement invariants, either under local unitaries (LU) or stochastic local operations and classical communication (SLOCC), can then be addressed in terms of the relative positions of the Majorana points. In the LU case, an overcomplete set of invariants can be built from the inner product of the radial vectors pointing to these points; this is detailed for the well-documented three-qubits case. In the SLOCC case, a cross ratio of related M?bius transformations are shown to play a central role, exemplified here for four qubits. Finally, as a side result, we also analyze the manifold of maximally entangled 3 qubit state, both in the symmetric and generic case.     

Twist-teleportation-based local discrimination of maximally entangled states

In this work, we study the local distinguishability of maximally entangled states(MESs). In particular, we are concerned with whether any fixed number of MESs can be locally distinguishable for sufficiently large dimensions. Fan and Tian et al. have already obtained two satisfactory results for the generalized Bell states(GBSs) and the qudit lattice states when applied to prime or prime power dimensions. We construct a general twist-teleportation scheme for any orthonormal basis with MESs that is inspired by the method used in [Phys. Rev. A 70, 022304(2004)]. Using this teleportation scheme, we obtain a sufficient and necessary condition for one-way distinguishable sets of MESs, which include the GBSs and the qudit lattice states as special cases.Moreover, we present a generalized version of the results in [Phys. Rev. A 92, 042320(2015)] for the arbitrary dimensional case.     

Diffusion in a heterogeneous medium for low concentrations

The equations describing diffusion on a heterogeneous lattice for low concentrations are considered taking into account lattice site blocking. It is shown that lattice site blocking cannot be disregarded in the case of a strongly heterogeneous lattice even for low concentrations. It is established that the equation with a fractional time derivative holds only in a bounded time interval. Anomalous diffusion, which is described by the equation with a fractional time derivative at the initial stage, must be described over long time periods by an ordinary diffusion equation with a concentration-dependent diffusion coefficient.     

Generation of displaced squeezed superpositions of coherent states

We study the method of generation of states that approximate superpositions of large-amplitude coherent states (SCSs) with high fidelity in free-traveling fields. Our approach is based on the representation of an arbitrary single-mode pure state, and SCSs in particular, in terms of displaced number states with an arbitrary displacement amplitude. The proposed optical scheme is based on alternation of photon additions and displacement operators (in the general case, N photon additions and N − 1 displacements are required) with a seed coherent state to generate both even and odd displaced squeezed SCSs regardless of the parity of the used photon additions. It is shown that the optical scheme studied is sensitive to the seed coherent state if the other parameters are unchanged. Output states can approximate either even squeezed SCS or odd SCS shifted relative to each other by some value. This allows constructing a local rotation operator, in particular, the Hadamard gate, which is a mainframe element for quantum computation with coherent states. We also show that three-photon additions with two intermediate displacement operators are sufficient to generate even displaced squeezed SCS with the amplitude 1.7 and fidelity more than 0.99. The effects deteriorating the quality of output states are considered.     

Dynamics and potentials

A dynamics (i.e. a one-parameter group of automorphisms) of a system described by a C*-algebra with a local structure in terms of C*-subalgebras A(I) for local domains I of the physical space (a discrete lattice) is normally constructed out of potentials P(I), each of which is a self-adjoint element of the subalgebra A(I), such that the the first time derivative of the dynamical change of any local observable A is i times the convergent sum of the commutator [P(I), A] over all finite regions I. We will invert this relation under the assumption (obviously assumed in the usual approach) that local observables all have the first time derivative, i.e. we prove the existence of potentials for any given dynamics satisfying the above-stated condition. Furthermore, by imposing a further condition for the potential P(I) to be chosen for each I that it does not have a portion which can be shifted to potentials for any proper subset of I, we also show (1) the existence, (2) uniqueness, (3) an automatic convergence property for the sum over I, and (4) a quite convenient property for the chosen potential. The so-obtained properties (3) and (4) are not assumed and are very useful, though they were never noticed nor used before.     

Topological order at nonzero temperature

We propose a definition for topological order at nonzero temperature in analogy to the usual zero temperature definition that a state is topologically ordered, or "nontrivial", if it cannot be transformed into a product state (or a state close to a product state) using a local (or approximately local) quantum circuit. We prove that any two-dimensional Hamiltonian which is a sum of commuting local terms is not topologically ordered at T > 0. We show that such trivial states cannot be used to store quantum information using certain stringlike operators. This definition is not too restrictive, however, as the four dimensional toric code does have a nontrivial phase at nonzero temperature.     

Topological and Nontopological Edge States Induced by Qubit-Assisted Coupling Potentials

In the usual Su–Schrieffer–Heeger (SSH) chain, the topology of the energy spectrum is divided into two categories in different parameter regions. Here, the topological and nontopological edge states induced by qubit-assisted coupling potentials in circuit quantum electrodynamics (QED) lattice modeled as a SSH chain are studied. It is found that, when the coupling potential added on only one end of the system raises to a certain extent, the strong coupling potential will induce a new topologically nontrivial phase accompanied by the appearance of a nontopological edge state, and the novel phase transition leads to the inversion of odd–even effect directly. Furthermore, the topological phase transitions when two unbalanced coupling potentials are injected into both ends of the circuit QED lattice are studied, and it is found that the system exhibits three distinguishing phases with multiple flips of energy bands. These phases are significantly different from the previous phase induced via unilateral coupling potential due to the existence of a pair of nontopological edge states. The scheme provides a feasible and visible method to induce different topological and nontopological edge states through controlling the qubit-assisted coupling potentials in circuit QED lattice both in experiment and theory.