On Models of Nonlinear Evolution Paths in Adiabatic Quantum Algorithms

In this paper,we study two different nonlinear interpolating paths in adiabatic evolution algorithms for solving a particular class of quantum search problems where both the initial and final Hamiltonian are one-dimensional projector Hamiltonians on the corresponding ground state.If the overlap between the initial state and final state of the quantum system is not equal to zero,both of these models can provide a constant time speedup over the usual adiabatic algorithms by increasing some another corresponding "complexity".But when the initial state has a zero overlap with the solution state in the problem,the second model leads to an infinite time complexity of the algorithm for whatever interpolating functions being applied while the first one can still provide a constant running time.However,inspired by a related reference,a variant of the first model can be constructed which also fails for the problem when the overlap is exactly equal to zero if we want to make up the "intrinsic" fault of the second model - an increase in energy.Two concrete theorems are given to serve as explanations why neither of these two models can improve the usual adiabatic evolution algorithms for the phenomenon above.These just tell us what should be noted when using certain nonlinear evolution paths in adiabatic quantum algorithms for some special kind of problems.     

On the Quantum Probability Flux Through Surfaces

We remark that the often ignored quantum probability current is fundamental for a genuine understanding of scattering phenomena and, in particular, for the statistics of the time and position of the first exit of a quantum particle from a given region, which may be simply expressed in terms of the current. This simple formula for these statistics does not appear as such in the literature. It is proposed that the formula, which is very different from the usual quantum mechanical measurement formulas, be verified experimentally. A full understanding of the quantum current and the associated formula is provided by Bohmian mechanics.     

On the Lattice Structure of Probability Spaces in Quantum Mechanics

Let $\mathcal{C}$ be the set of all possible quantum states. We study the convex subsets of $\mathcal{C}$ with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes’ Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets concerning the geometrical structure of the convex subsets. It is seen that the above mentioned framework is also capable of generalization to any statistical theory via the so-called convex operational models’ approach. In particular, we show how to extend the geometrical structure underlying entanglement to any statistical model, an extension which may be useful for studying correlations in different generalizations of quantum mechanics.     

Adiabatic Elimination in Quantum Stochastic Models

We consider a physical system with a coupling to bosonic reservoirs via a quantum stochastic differential equation. We study the limit of this model as the coupling strength tends to infinity. We show that in this limit the solution to the quantum stochastic differential equation converges strongly to the solution of a limit quantum stochastic differential equation. In the limiting dynamics the excited states are removed and the ground states couple directly to the reservoirs.     

On the Role of Spin in Quantum Mechanics

From the invariance properties of the Schrödinger equation and the isotropy of space we show that a generic (non-relativistic) quantum system is endowed with an external motion, which can be interpreted as the motion of the centre of mass, and an internal one, whose presence disappears in the classical limit. The latter is caused by the spin of the particle, whatever is its actual value (different from zero). The quantum potential in the Schrödinger equation, which is responsible of the quantum effects of the system, is then completely determined from the properties of the internal motion, and its unusual properties have a simple and physical explanation in the present context. From the impossibility to fix the initial conditions relevant for the internal motion follows, finally, the need of a probabilistic interpretation of quantum mechanics.     

A topological investigation of the Quantum Adiabatic Phase

Using algebraic topology, the appearance of the Quantum Adiabatic Phase over various parameter manifolds is investigated. The relation with nontrivial gauge bundles (both abelian and non-abelian) is studied and it is shown that the phase appears as a result of homotopically non-trivial mappings, induced by the Hamiltonian in the space of wave-functions. The cohomological picture is developed and some topological considerations concerning field theory anomalies in the Hamiltonian picture are presented. A proof of the Nielsen-Ninomiya theorem is given inspired from the notion of the adiabatic phase.Work supported in part by the U.S. Department of Energy under contract DEAC 03-81-ER 40050     

On the Role of the Controller in Controlled Quantum Teleportation

We consider a set of quantum channels which are though partially entangled but can perform perfectly quantum teleportation of two-qubit states with the assistance of a controller. The quantum channels are designed so that only the controller is able to correctly control the task and without his/her cooperation the receiver cannot obtain with certainty a state with quality better than that obtained classically. The key point enhancing the role of the controller is that he/she is the only one who is allowed to know the quantum channel parameters.     

利用绝热过程实现量子信息转移

基于绝热过程,仅需一个真空腔即可分别实现未知单原子态、双原子纠缠态及GHZ态的转移.在这些方案中,量子信息都存储在原子的基态,且系统仅在暗态空间中演化,原子激发态上无布居,这使得原子的自发辐射效应大大受到抑制.相比之前的方案,这些方案所需资源更少,操作更简单.另外,还讨论了其在实验上实现的可行性.     

Symmetry-Protected Quantum Adiabatic Evolution in Spontaneous Symmetry-Breaking Transitions

Quantum adiabatic evolution describes the dynamical evolution of a slowly driven Hamiltonian. In most systems undergoing spontaneous symmetry-breaking transitions, the symmetry-protected quantum adiabatic evolution can still appear, even when the two lowest eigenstates become degenerate. Here, a general derivation to revisit the symmetry-dependent transition and the symmetry-dependent adiabatic condition (SDAC) is given. Further, based on the SDAC, an adiabatic-parameter-fixed sweeping scheme is used for achieving fast adiabatic evolution, which is more efficient than the linear sweeping scheme. In the limit of small adiabatic parameter, an analytic inequality is obtained for the ground state fidelity only dependent on the adiabatic parameter. The general statements are then demonstrated via two typical systems. Besides, the robustness of the symmetry-dependent adiabatic evolution against weak symmetry-breaking sources is studied. The findings can be tested via the techniques in quantum annealing and may provide promising applications in practical quantum technologies.     

Quantum State Discrimination with Prior Knowledge in Noisy Quantum Channels

Discrimination between two states of a qubit is investigated, which is performed under the influence of noisy quantum channels. When prior knowledge about on the quantum states is available, the detection probability of quantum measurement is compared with that of pure guessing during the irreversible time evolution. In the case of a Markovian channel, the superiority of quantum measurement to pure guessing is lost at finite time which is determined by the prior probability and the fidelity of the quantum states. For a non-Markovian channel, however, it is possible to recover the superiority of quantum measurement even if it is lost. The effect of a system-environment initial correlation on the quantum state discrimination is also investigated.     

Probability Representation of Quantum Observables and Quantum States

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.     

Quantum Algorithms in Hilbert Database

A fast quantum algorithm for a search and pattern recognition in a Hilbert space memory structure is proposed. All the memory information is mapped onto a unitary operator acting upon a quantum state which represents a piece of information to be retrieved. As a result of only one quantum measurement, the address of the required information encoded in a number of the corresponding row of the unitary matrix is determined. By combining direct and dot products, the dimensionality of the memory space can be made exponentially large, using only linear resources. However, since the preprocessing, i.e., mapping the memory information into a Hilbert space can appear to be exponentially expensive, the proposed algorithm will be effective for NASA applications when the preprocessing is implemented on the ground, while the memory search is performed on remote objects.     

Hilbert Modules in Quantum Electro Dynamics and Quantum Probability

A physical system of the form with a distinguished state on may be described in a natural way on a Hilbert -module. Following the ideas of Accardi and Lu [1], we apply this possibility to a concrete system consisting of a boson field in the vacuum state coupled to a free electron. We show that the physical system is described adequately on a new type of Fock module: the symmetric Fock module. It turns out that a module has to fulfill an algebraic condition in order to allow for the construction of a symmetric Fock module. We prove in a central limit theorem that in the stochastic limit the moments of the collective operators (i.e. more or less the time-integrated interaction Hamiltonian) converge to the moments of free creators and annihilators on a full Fock module. In the sense of Voiculescu [22] and Speicher [20] these operators form a free white noise over the algebra . Received: 28 October 1996 / Accepted: 21 July 1997     

Probability Representation of Quantum States

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born’s rule and recently suggested method of dequantizer–quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical–like equations for the probability distributions determining the quantum system states. Relations to phase–space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.     

Sufficient Condition for Validity of Quantum Adiabatic Theorem

In this paper,we attempt to give a sufficient condition of guaranteeing the validity of the proof of the quantum adiabatic theorem.The new sufficient condition can clearly remove the inconsistency and the counterexample of the quantum adiabatic theorem pointed out by Marzlin and Sanders.     

Return Probability of the Fibonacci Quantum Walk

In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.     

Return Probability of the Fibonacci Quantum Walk

In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.     

Quantum Damped Oscillator in Probability Representation

The probability distribution describing quantum states of the damped oscillator in the framework of the Caldirola-Kanai model is introduced. The probability distributions for coherent states and Fock states of the damped oscillator are found explicitly. The transition probability for the damped oscillator is expressed in terms of distributions describing initial and final states.     

Classical and Quantum Probability in the ∈-Model

We describe the probabilistic study of a hiddenvariable model in which the origin of the quantumprobability is due to fluctuations of the internal stateof the measuring apparatus. By varying the intensity of these fluctuations from zero to a maximalvalue, we describe in a heuristic manner the transitionfrom classical behavior to quantum behavior. Wecharacterize this transition in terms of theAccardi–Fedullo inequalities. This is a review article in whichwe gather our recent contributions to the subject, mostof which have not been published in articleform.     

量子环中量子比特内的电子概率密度分布

通过较精确地求解能量本征方程获得量子环中量子比特内的电子概率密度分布。对InAs量子环的数值计算表明:电子概率密度分布与电子的坐标(半径、高度,角度)及时间有关。当其中三个变量给定时,电子概率密度随另一个变量的变化规律分别为:随半径的增大做非周期性振荡;随高度的变化而变化,在半高处出现的概率最大;随角度作周期变化,在角度等于π处出现的概率最大;随时间作周期性振荡。