Quantum field theory of binary alternatives

We describe a framework for quantum field theory that is based on quantized binary alternatives. We discuss the relation of the dynamics of quantum fields and the time evolution of the Heisenberg operators defined by the creation and annihilation operators of the alternatives. An explicit expression for the vacuum vector of the quantum fields is derived. Finally we discuss eigenstates of the momentum operators.     

Absence of wave operators for one-dimensional quantum walks

Letters in Mathematical Physics - We show that there exist pairs of two time evolution operators which do not have wave operators in a context of one-dimensional discrete time quantum walks. As a...     

哈密顿算符是SU(1,1)和SU(2)算子含时线性组合量子系统的时间演变及厄密不变量

研究了哈密顿算符是SU(1,1)和SU(2)算子线性组合且系数显含时间量子系统的时间演化,我们发现适当选取厄密不变量,不仅可得到统一的量子态时间演化封闭解,而且可得到时间演化幺正算符。用时间演化算符我们讨论了含时双光子压缩态和SU(1,1)相干态以及SU(2)压缩态的压缩性质。     

Exact phase space localized projectors from energy eigenstates

In investigations of the emergence of classicality from quantum theory, a useful step is the construction of quantum operators corresponding to the classical notion that the system resides in a region of phase space. The simplest such constructions are approximate projection operators. Here, we show how to construct exact projection operators which are localized on regions of phase. We elucidate the properties of such operators and explore their time evolution. For the harmonic oscillator we find sets of phase space localized histories which are exactly decoherent for any initial state and have probability 1 for classical evolution.     

基于相位匹配的量子行走搜索算法及电路实现

量子行走是经典随机行走在量子力学框架下的对应, 理论上可以用来解决一类无序数据库的搜索问题. 因为携带信息的量子态的扩散速度与经典相比有二次方式的增长, 所以量子行走优于经典随机行走, 量子行走的特性值得加以利用. 量子行走作为一种新发现的物理现象的数学描述, 引发了一种新的思维方式, 孕育了一种新的理论计算模型. 最新研究表明, 量子行走本身也是一种通用计算模型, 可被视为设计量子算法的高级工具, 因此受到部分计算机理论科学领域学者的关注和研究. 对于多数问题求解方案的量子算法的设计, 理论上可以只在量子行走模型下进行考虑. 基于Grover算法的相位匹配条件, 本文提出了一个新的基于量子行走的搜索算法. 理论演算表明: 一般情况下本算法的时间复杂度与Grover算法相同, 但是当搜索的目标数目多于总数的1/3时, 本算法搜索成功的概率要大于Grover算法. 本文不但利用Grover算法中相位匹配条件构造了一个新的量子行走搜索算法, 而且在本研究室原有的量子电路设计研究成果的基础上给出了该算法的量子电路表述.     

Dissipative quantum Church-Turing theorem

We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.     

Holevo-Ordering and the Continuous-Time Limit for Open Floquet Dynamics

We consider an atomic beam reservoir as a source of quantum noise. The atoms are modelled as two-state systems and interact one-at-a-time with the system. The Floquet operators are described in terms of the Fermionic creation, annihilation and number operators associated with the two-state atom. In the limit where the time between interactions goes to zero and the interaction is suitably scaled, we show that we may obtain a causal (that is, adapted) quantum stochastic differential equation of Hudson—Parthasarathy type, driven by creation, annihilation and conservation processes. The effect of the Floquet operators in the continuous limit is exactly captured by the Holevo ordered form for the stochastic evolution     

Tense Operators on Basic Algebras

The concept of tense operators on a basic algebra is introduced. Since basic algebras can serve as an axiomatization of a many-valued quantum logic (see e.g. Chajda et al. in Algebra Univer. 60(1):63–90, 2009), these tense operators are considered to quantify time dimension, i.e. one expresses the quantification “it is always going to be the case that” and the other expresses “it has always been the case that”. We set up the axiomatization and basic properties of tense operators on basic algebras and involve a certain construction of these operators for left-monotonous basic algebras. Finally, we relate basic algebras with tense operators with another quantum structures which are the so-called dynamic effect algebras.     

Baxter operators for arbitrary spin

We construct Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite- or finite-dimensional s?₂ representations. All algebraic relations of Baxter operators and transfer matrices are deduced uniformly from Yang-Baxter relations of the local building blocks of these operators. This results in a systematic and very transparent approach where the cases of finite- and infinite-dimensional representations are treated in analogy. Simple relations between the Baxter operators of both cases are obtained. We represent the quantum spaces by polynomials and build the operators from elementary differentiation and multiplication operators. We present compact explicit formulae for the action of Baxter operators on polynomials.     

Revisiting non-degenerate parametric down-conversion

The quantum dynamics of a two-mode non-resonant parametric down-conversion process is studied by recasting the time evolution equations for the basic operators in an equivalent spin equation form with simpler exact solutions for a pump field with harmonic time dependence. Expectation values of suitable operators for studying important features such as squeezing and quantum revivals are presented in simple forms.      

Random fields and quantum dynamics

We shall discuss quantum mechanical operators depending on the time or on the manifold in space. There is a similarity to the case of stochastic processes or random fields, where the innovation approach is one of the powerful tools to investigate their probabilistic structure. Having had some review of the innovation, similar attempt is made for some cases in quantum dynamics.     

Quantum Theory of Solid State Plasma Dielectric Response

We review the quantum mechanical derivation of the random phase approximation (RPA) for solid state plasmas, starting from the Hamilton equations for canonically paired “second quantized” creation and annhilation field operators of interacting quantum many‐body systems. Discussing variational differentiation, the coupled equations of motion for the quantum field operators are derived. The concept of Green's functions is reviewed and interpreted, first for retarded Green's functions, and their equations of motion are developed from the equations of motion for the field operators. Thermodynamic Green's functions are discussed, and their periodicity/antiperiodicity properties in imaginary time are carefully examined with discussion of Matsubara Fourier series and representation in terms of a spectral weight function. The analytic continuation from imaginary time to real time is treated. Finally, we define nonequilibrium Green's functions and discuss the linearized timedependent Hartree approximation leading to the random phase approximation. An interesting application to the case of Graphene in a perpendicular magnetic field is discussed in detail, along with applications to normal systems, in terms of attendant phenomenology involving electron‐hole pair excitations and plasmons (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The quantum groupSUq(2) andq-analog of angular momentum operators in quantum mechanics

We present three operators in quantum mechanics that obey the commutation relations of quantum groupSUq(2). These operators are nonlinear combinations of the conventional angular momentum operators and are called the quantumq-analog angular momentum operators. When the quantum deformation parameterr = Inq vanishes, these quantumq-analog angular momentum operators reduce to the usual angular momentum operators.     

High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically. The first author was supported by NSERC, FQRNT and Dawson fellowship.     

Normal Ordering and Abnormal Nonsense

The technique of the normal ordering of non-commuting operators is an important tool in the solution of problems involving creation and annihilation operators in quantum physics, such as in many-body theory or quantum optics. We point out the inconsistencies in previous definitions of the two standard normal ordering procedures for such operators, and show how consistent definitions may be made.     

The q-Deformation of the Superoscillators and Their q-Supercoherent States

We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q-deformed operators, we also give the q-deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q-supercoherent states associated with the q-deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q-supercoherent states are discussed as well.     

On Grover’s search algorithm from a quantum information geometry viewpoint

We present an information geometric characterization of Grover’s quantum search algorithm. First, we quantify the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric. We then show that the quantum searching problem can be recast in an information geometric framework where Grover’s dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover’s algorithm. We also discuss possible deviations from Grover’s algorithm within this quantum information geometric setting.     

Quantum Zeno and anti-Zeno paradoxes

Continuous observation of a time independent projection operator is known to prevent change of state (the quantum Zeno paradox). We discuss the recent result that generic continuous measurement of time dependent projection operators will in fact ensure change of state: an anti-Zeno paradox.     

Black holes and the nature of quantum gravity

Hawking and Wald have recently argued that the process of quantum black hole evaporation requires the violation of the fundamental physical law which asserts that the time evolution of quantum states is governed by unitary operators. I show this violation can be avoided by a change in the global boundary conditions. It is remotely possible that astronomical observation could establish whether or not the universe has these boundary conditions in which quantum mechanical time evolution is governed by unitary operators.     

Background independent duals of the harmonic oscillator

We show that a class of topological field theories are quantum duals of the harmonic oscillator. This is demonstrated by establishing a correspondence between the creation and annihilation operators and nonlocal gauge invariant observables of the topological field theory. The example is used to discuss some issues concerning background independence and the relation of vacuum energy to the problem of time in quantum gravity.