Reginaldo de Jesus Napolitano

Alternative fidelity measure between quantum states

We propose an alternative fidelity measure (namely, a measure of the degree of similarity) between quantum states and benchmark it against a number of properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple function of the linear entropy and the Hilbert-Schmidt inner product between the given states and is thus, in comparison, not as computationally demanding. It also features several remarkable properties such as being jointly concave and satisfying all of Jozsas axioms. The trade-off, however, is that it is supermultiplicative and does not behave monotonically under quantum operations. In addition, metrics for the space of density matrices are identified and the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is established.

Continuously decoupling single-qubit operations from a perturbing thermal bath of scalar bosons

We investigate the use of continuously applied external fields to maximize the fidelity of quantum logic operations performed on a decohering qubit. Assuming a known error operator and an environment represented by a scalar boson field at a finite temperature, we show how decoherence during logical operations can be efficiently reduced by applying a superposition of two external vector fields: one rotating orthogonally to the direction of the other, which remains static. The required field directions, frequency of rotation, and amplitudes to decouple noise dynamically are determined by the coupling constants and the desired logical operation. We illustrate these findings numerically for a Hadamard quantum gate and an environment with ohmic spectral density.

Protecting a quantum state from environmental noise by an incompatible finite-time measurement

We show that measurements of finite duration performed on an open two-state system can protect the initial state from a phase-noisy environment, provided the measured observable does not commute with the perturbing interaction. When the measured observable commutes with the environmental interaction, the finite-duration measurement accelerates the rate of decoherence induced by the phase noise. For the description of the measurement of an observable that is incompatible with the interaction between system and environment, we have found an approximate analytical expression, valid at zero temperature and weak coupling with the measuring device. We have tested the validity of the analytical predictions against an exact numerical approach, based on the superoperator-splitting method, that confirms the protection of the initial state of the system. When the coupling between the system and the measuring apparatus increases beyond the range of validity of the analytical approximation, the initial state is still protected by the finite-time measurement, according with the exact numerical calculations.

A simple derivation of the Lindblad equation

We present a derivation of the Lindblad equation -an important tool for the treatment of nonunitary evolutions -that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings formato We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad formo The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.Apresentamos uma derivacao da equacao de Lindblad -uma ferramenta importante no tratamento de evolucoes nao-unitarias -acessivel a estudantes de graduacao em fisica ou matematica com nocoes basicas de mecânica quântica. Consideramos aqui um caso especifico, correspondente a uma situacao bem simples, onde o sistema principal interage com um banho de osciladores harmonicos a temperatura nula, com hamiltoniano de interacao que se assemelha ao modelo de Jaynes-Cummings. Iniciamos com a equacao de Born-Markov e, atraves do traco parcial dos graus de liberdade do banho, obtemos uma equacao na forma de Lindblad. Essa situacao especifica e bem instrutiva, pois permite verificar que os lindblads representam a contribuicao do sistema principal ao hamiltoniano de interacao com o banho, e que a aproximacao markoviana e vital para o surgimento do lindbladiano. A deducao formal da equacao de Lindblad para situacoes gerais requer o uso do formalismo de semigrupos dinâmicos quânticos e consideracoes mais abrangentes sobre o ambiente e a temperatura do que as utilizadas aqui.

How Much Time Does a Measurement Take

We consider the problem of measurement using the Lindblad equation, which allows the introduction of time in the interaction between the measured system and the measurement apparatus. We use analytic results, valid for weak system-environment coupling, obtained for a two-level system in contact with a measurer (Markovian interaction) and a thermal bath (non-Markovian interaction), where the measured observable may or may not commute with the system-environment interaction. Analysing the behavior of the coherence, which tends to a value asymptotically close to zero, we obtain an expression for the time of measurement which depends only on the system-measurer coupling, and which does not depend on whether the observable commutes with the system-bath interaction. The behavior of the coherences in the case of strong system-environment coupling, found numerically, indicates that an increase in this coupling decreases the measurement time, thus allowing our expression to be considered the upper limit for the duration of the process.

Elliptical orbits in the phase-space quantization

The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfelds original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.

The master equation for the reduced open-system dynamics, including a Lindbladian description of finite-duration measurement

Abstract.We consider the problem of the measurement of a system occurring during a finite-time interval, while environmentally induced noise decreases the system-state coherence. We assume a Markovian measuring device and, therefore, use a Lindbladian description for the measurement dynamics. For studying the case of noise produced by a non-Markovian environment, whose definition does not include the measuring apparatus, we use the Redfield approach to the interaction between system and environment. In the present hybrid theory, to trace out the environmental degrees of freedom, we introduce an analytic method based on superoperator algebra and Nakajima-Zwanzig projectors. The resulting master equation, describing the reduced system dynamics, is illustrated in the case of a qubit under phase noise during a finite-time measurement.

Using continuous measurement to protect a universal set of quantum gates within a perturbed decoherence-free subspace

We consider a universal set of quantum gates encoded within a perturbed decoherence-free subspace of four physical qubits. Using second-order perturbation theory and a measuring device modelled by an infinite set of harmonic oscillators, simply coupled to the system, we show that continuous observation of the coupling agent induces inhibition of the decoherence due to spurious perturbations. We thus advance the idea of protecting or even creating a decoherence-free subspace for processing quantum information.

Efficient finite-time measurements under thermal regimes

Contrary to conventional quantum mechanics, which treats measurement as instantaneous, here we explore a model for finite-time measurement. The main two-level system interacts with the measurement apparatus in a Markovian way described by the Lindblad equation, and with an environment, which does not include the measuring apparatus. To analyse the environmental effects on the final density operator, we use the Redfield approach, allowing us to consider a non-Markovian noise. In the present hybrid theory, to trace out the environmental degrees of freedom, we use a previously developed analytic method based on superoperator algebra and Nakajima-Zwanzig superoperators. Here, we analyse two types of system-environment interaction, phase and amplitude damping, which allows us to conclude that, in general, a finite-time quantum measurement performed during a certain period is more efficient than an instantaneous measurement performed at the end of it, because the rate of change of the populations is attenuated by the system-measurement apparatus interaction.

Exact dynamics of single-qubit-gate fidelities under the measurement-based quantum computation scheme

Measurement-based quantum computation is an efficient model to perform universal computation. Nevertheless, theoretical questions have been raised, mainly with respect to realistic noise conditions. In order to shed some light on this issue, we evaluate the exact dynamics of some single qubit gate fidelities using the measurement-based quantum computation scheme when the qubits which are used as resource interact with a common dephasing environment. We report a necessary condition for the fidelity dynamics of a general pure N-qubit state, interacting with this type of error channel, to present an oscillatory behavior and we show that for the initial canonical cluster state the fidelity oscillates as a function of time. This state fidelity oscillatory behavior brings significant variations to the values of the computational results of a generic gate acting on that state depending on the instants we choose to apply our set of projective measurements. As we shall see, considering some specific gates that are frequently found in the literature, neither fast application of the set of projective measurements necessarily implies high gate fidelity, nor slow application thereof necessarily implies low gate fidelity. Our condition for the occurrence of the fidelity oscillatory behavior shows that the oscillation presented by the cluster state is due exclusively to its initial geometry. Other states that can be used as resources for measurement-based quantum computation can present the same initial geometrical condition. Therefore, it is very important for the present scheme to know when the fidelity of a particular resource state will oscillate in time and, if this is the case, what are the best times to perform the measurements.