Quantum Information and Computing

Preface

Contents

Pictures

We investigate, using the stochastic limit method, the coherent quantum control of a 3-level atom in Λ-configuration interacting with two laser fields. We prove that, in the generic situation, this interaction entangles the two lower energy levels of the atom into a single qubit, i.e. it drives at an exponentially fast rate the atom to a stationary state which is a coherent superposition of the two lower levels. By applying to the atom two laser fields with appropriately chosen intensities, one can create, in principle, any superposition of the two levels. Thus relaxation is not necessarily synonymous of decoherence.

No abstract received.

We propose a new technique to control quantum states by exploiting the decoherence due to the coupling with environment. With this technique we can get any target state, in a stable way and in an exponentially small time, as a stationary state of a semigroup (master equation) canonically derived from a microscopic Hamiltonian model. The stationary state of the system depends (i) on the interaction with the environment; (ii) on the initial state of the environment and it “inherits” some properties of it in a sense that will be explained in sec. 5 of the present paper. Moreover this is true not only for by thermal or vacuum environments but also for more general non-equilibrium environments. We prove that, by appropriately choosing this control parametes (interaction system–environment, initial environment state) one can drive the system to an arbitrary pre–assigned quantum state.

We propose multi-qubit logical gates, which implement logical operations on N qubits realized physically with the use of local manipulations of qubits before and after the one-time evolution of Ising chain (an array of N qubits connected with the Ising interactions), avoiding complicated tuning of the interactions between qubits. General rules of the action of our multi-qubit logical gate are derived by decomposing the whole process into a product of two-qubit logical operations. We construct the multi-qubit logical gate, which expressed as a group of controlled-NOT gates.

The problem of existence and uniqueness of a state of a joint system, when its restrictions to subsystems are specified, is studied for a Fermion system, where a novel feature as compared with Boson or spin systems is non-commutativity between algebras of subsystems located at disjoint regions. The case of two subsystems with one of the subsystem states pure is completely analyzed.

Methods from quantum filtering theory and classical control theory are combined to give an analytic solution to the problem of optimal control of the state of a Gaussian quantum free particle. The solutions to the filtering problem show an asymptotic localization of the continuously observed free particle and we also observe a duality with the solutions of optimal control.

We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P on a separable Hilbert space . The convergence is demonstrated in the space , which gives a partial answer to the question about existence of the limit describing quantum Zeno dynamics in the subspace Ran P, the range of P. A stronger result with the convergence in is proved in the case of a finite-dimensional P.

We discuss three different control strategies, all aimed at countering the effects of decoherence. The first strategy hinges upon the quantum Zeno effect, the second makes use of frequent unitary interruptions (“bang-bang” pulses), and the third of a strong, continuous coupling. Decoherence is suppressed if the frequency N of the measurements/pulses is large enough or if the coupling K is sufficiently strong. However, if N or K are large, but not extremely large, all these control procedures accelerate decoherence.

In this paper we derive the Clauser-Horne (CH) inequality for the full electron counting statistics in a mesoscopic multiterminal conductor and we discuss its properties. We first consider the idealized situation in which a flux of entangled electrons is generated by an entangler. Given a certain average number of incoming entangled electrons, the CH inequality can be evaluated for different numbers of transmitted particles. Strong violations occur when the number of transmitted charges on the two terminals is the same (Q₁ = Q₂), whereas no violation is found for Q₁ ≠ Q₂. We then consider two actual setups that can be realized experimentally. The first one consists of a three terminal normal beam splitter and the second one of a hybrid superconducting structure. Interestingly, we find that the CH inequality is violated for the three terminal normal device. The maximum violation scales as 1/Mand 1/M²for the entangler and normal beam splitter, respectively, 2M being the average number of injected electrons. As expected, we find full violation of the CH inequality in the case of the superconducting system.

A teleportation model using a beam splitter proposed by Fichtner and Ohya is investigated. Fidelity which represents a perfectness of the model is estimated for general entangled states.

In [15] Fredkin and Toffoli proposed a logical conservative gate on the base of which Milburn [24] constructed a quantum logical gate. Photon number states were used as the input state for the control gate. The control gate consisted of a Kerr medium. In the present paper we will realize the truth table that has to be satisified by the Fredkin–Toffoli gate solely by using beam splitting procedures. The Kerr medium is replaced by further beam splittings which are applied to both outputs of the first splitting procedure. The information 0 and 1 are encoded by coherent states on a general bosonic Fock space. The aim of the paper is to give examples for quantum channels transmitting some information such that the gate will fulfill the truth table. For the output we get simple explicit expressions.

Many definitions of relative information (often called relative entropy) have been proposed in order to characterise the ‘distance’ as information divergence between states. One such class of relative informations are the quasi-entropies proposed by Petz in ⁶⁶ (of which the standard Araki-Umegaki form, ⁵³, is a particular example). This paper axiomatises the idea of relative information and then proceeds to build a large class of examples that satisfy these axioms. By using the Radon-Nikodym derivative for completely positive maps as an ‘operational density’ for channels, along with a theorem by Belavkin and Ohya, ⁵⁷, for taking the supremum over local couplings these ideas are then lifted beyond the realm of states into that of quantum channels. This method for extending information quantities to channels was pioneered by V.P. Belavkin, with respect to fidelity, ⁶¹ (see ⁶⁰ for an in-depth presentation of the subject).

No abstract received.

We shall construct a noncanonical representation of a d-dimensional Brownian motion by using a method similar to that in ². This is also considered as a generalization of a result obtained in ⁶.

Having had a very short review of the white noise analysis, we consider some of its future directions with special emphasis on innovation theory.

Innovation of a stochastic process or a random field plays one of the most important roles in the stochastic analysis. This fact is illustrated from the view point of information theory.

Ohya and Volovich have proposed a new quantum algorithm with chaotic amplification to solve the SAT problem, which went beyond usual quantum algorithm. In this paper, we generalize quantum Turing machine in terms of general channel, not always unitary, transformation. Moreover, some computational classes in generalized quantum Turing machine are introduced, and it is shown that the Ohya-Volovich (OV) SAT algorithm can be described in this generalized Turing machine.

In this paper we discuss some problems of stroboscopic characterization of quantum states. In particular, the minimal number η of observables Q₁,…, Q_η, whose expectation values at some time instants t₁,…, t_r determine the trajectory of a d-level quantum system (“qudit”) governed by a semigroup of linear transformations are analyzed. We use the unitary and the Gaussian evolutions as examples. We assume that the macroscopic information about the system in question is given by the mean values _j(Q_i) = Tr(Q_iρ(t_j)) of n selfadjoint operators Q_1,…, Q_n at some time instants t₁ < t₂ < … < t_r, where n < d²− 1 and r ≤ deg μ(λ, 𝕃).Here μ(λ, 𝕃) stands for the minimal polynomial of the generator

A class of linear positive, trace preserving maps in M_n is given in terms of affine maps in ℝ^n2–1 which map the close unit ball into itself. The above class contains atomic maps considered by Tanahashi and Tomiyama.

This paper focus on the possibility of simulating the open system dynamics of a paradigmatic model, namely the damped harmonic oscillator, with single trapped ions. The key idea consists in using a controllable physical system, i.e. a single trapped ion interacting with an engineered reservoir, to simulate the dynamics of other open systems usually difficult to study. The exact dynamics of the damped harmonic oscillator under very general conditions is firstly derived. Some peculiar characteristic of the system's dynamics are then presented. Finally a way to implement with trapped ion the specific quantum simulator of interest is discussed.

A purification scheme which utilizes the action of repeated measurements on a (part of a total) quantum system is briefly reviewed and is applied to a few simple systems to show how it enables us to extract an entangled state as a target pure state. The scheme is rather simple (e.g., we need not prepare a specific initial state) and is shown to have wide applicability and flexibility, and is able to accomplish both the maximal fidelity and non-vanishing yield.

No abstract received.

It has been known for a while that the equality in several strong subadditivity inequalities for the von Neumann entropy of the local restriction of states of infinite product chains is equivalent to the Markov property initiated by Accardi. The goal of this paper is to analyse the situation further and to give the structure of states which satisfy strong subadditivity of quantum entropy with equality. This structure has implication for quantum Markov states.

The Lévy Laplacian is formulated as an operator acting on a class in the Lévy white noise L² space. This space includes regular functionals in terms of Gaussian white noise and it is large enough to discuss the stochastic process. This formulation is slightly outside the usual white noise distribution theory, while the Lévy Laplacian has been discussed within the framework of white noise analysis. From Cauchy processes an infinite dimensional stochastic process is constructed, of which the generator is the Lévy Laplacian.

Following P. Lévy, we consider a particular linear process given by

Quantum stochastic processes are characterized in terms of completely nonnegative quantum dynamical maps which form a convex set. The canonical form of such a map is in terms of R ≤ N, N × N matrices with their phase completely arbitrary. The maps constitute a convex set of which the extremal elements can be identified. Every such map may be viewed as the product of a unitary transformation in the adjoint representation, a classical stochastic semigroup in N variables followed by another unitary transformation.

The purpose of the present paper is to review the quantum analysis and exponential product formulas, together with the quantum transfer-matrix method and to propose the concept of effective information “entropy”.

Recently, nonequilibrium steady states (NESS) are intensively studied by several methods. In this article, NESS of a harmonic oscillator interacting with two free-boson reservoirs at different temperatures and/or chemical potentials are studied by the stochastic limit approach and C* algebraic approach. And their interrelation is investigated.

We present a strategy to control decoherence with multipulse application. We show that the degradation of a two-level system which linearly interacts with a reservoir characterized by a specific frequency is effectively suppressed by synchronizing the pulse-train application with the dynamical motion of the reservoir. We find that the non-Markovian nature of dynamical motion of the reservoir makes this strategy effective.

No abstract received.

The study of classical mutual entropy has been extensively done by several researchers like Kolmogorov and Gelfand. In quantum system, there are several definitions of the mutual entropy from classical inputs to quantum outputs, which are called semi-classical mutual entropy. In 1983, Ohya introduced a fully quantum mutual entropy by means of the relative entropy of Umegaki, and he extended it to general quantum systems by using the relative entropy of Araki and Uhlmann. It is showed that the Ohya mutual entropy contains the definitions of the semi-classical mutual entropy including the classical mutual entropy. Recently, Bennet, Nielsen, Shor et al. took the coherent information and so-called Lindblad - Nielsen's entropy for discuss a sort of coding theorem for communication processes. One can be concluded that Ohya mutual entropy is a most suitable one for studying the information transmission in quantum communication systems.