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arXiv:1511.05206 · Physical Review A

Peculiarities of qubit initial-state preparation by nonselective measurements on an overcomplete basis

V. V. Ignatyuk

Published Dec 9, 2015• 4 citations
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Abstract

We consider the qubit initial-state preparation due to the nonselective measurements on an overcomplete basis, when the number of outcomes $N=3$. To be specific, we have chosen the dephasing model and applied the conditions for a post-measurement state to ensure a gradual coherence enhancement at the initial stage of the system evolution. It is shown that contrary to the von Neumann--L\"uders projection scheme with $N=2$, the most mixed post-measurement state of the qubit in the general case can depend on the bath temperature. This remarkable feature allows one to consider a ``temperature-controlled'' initial purification of the open quantum system. We also consider some kinds of the repeated nonselective measurements from the viewpoint of an eventual system purification and analyze the important peculiarities of such schemes.

DOI
10.1103/physreva.92.062115

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We consider the qubit initial-state preparation due to the nonselective measurements on an overcomplete basis, when the number of outcomes N=3N=3N=3. To be specific, we have chosen the dephasing model and applied the conditions for a post-measurement state to ensure a gradual coherence enhancement at the initial stage of the system evolution. It is shown that contrary to the von Neumann--L"uders projection scheme with N=2N=2N=2, the most mixed post-measurement state of the qubit in the general case can depend on the bath temperature. This remarkable feature allows one to consider a ``temperature-controlled'' initial purification of the open quantum system. We also consider some kinds of the repeated nonselective measurements from the viewpoint of an eventual system purification and analyze the important peculiarities of such schemes.
The influence of the initial preparation on dephasing in open quantum dynamics is studied using an exactly solvable model of a two-level system (qubit) interacting with a bosonic bath. It is found that, for some classes of nonselective preparation measurements, qubit-bath correlations lead to a significant enhancement of coherence in the qubit at the initial stage of evolution. The time behavior of the qubit purity and entropy in the regime of enhancement of coherence is considered for different temperatures and coupling strengths.
The effects of the measurement apparatus on quantum coherence are studied by considering a purely dephasing model of a qubit. The initial state is prepared from a thermal state of the whole system by performing a nonselective measurement on the qubit. The magnitude of the initial postmeasurement coherence is bounded by the value [Formula: see text], which is realized with special measurement schemes and in the low-temperature limit. The coherence magnitude identically vanishes, increases or decreases with an approximately constant rate over a determined short-time scale, according to the choice of the preparation measurement. The maximization of the short-time increasing or decreasing rate is favored by the choice of other special measurement schemes and the high-temperature limit. The measurement apparatus allows to manipulate the quantum coherence of the qubit over short times via nonselective preparation measurements.
The evolution of quantum coherence in a qubit is analyzed, over short and long times, in case the qubit interacts with a bosonic environment. The distribution of the frequency modes exhibits a low-frequency band gap. The initial conditions under study of the qubit and the bosonic environment are factorized or prepared with selective or nonselective measurements. The modulus of the coherence term of the reduced density matrix of the qubit evolves algebraically over short times and, due to the band gap, tends to the nonvanishing asymptotic value with damped oscillations. Depending on the measurement scheme, the nonselective preparation measurement can enhance quantum coherence or accelerate the decoherence process, over short times, according to dominant linear laws. Over long times, the nonselective preparation measurement produces the slowest-decaying envelope of the damped oscillations among the decoherence processes under study. Similar short- and long-time evolution is exhibited by the trace distance measure of two general states of the qubit.
The appearance of decoherence-recoherence events is analyzed in a dephasing model of a qubit which interacts with a bosonic environment. The initial condition is prepared from the thermal state of the whole system by performing a nonselective measurement on the qubit. Decoherence-recoherence events appear in the reduced dynamics of the qubit uniquely if special nonselective preparation measurements are adopted and the spectral density of the system fulfills determined conditions. General lower bounds are found for the number of decoherence-recoherence events in terms of the properties of the measurement scheme and of the spectral density. The number is finite for canonical spectral densities which are super-ohmic, perturbed super-ohmic, or ohmic at low frequencies and are arbitrarily tailored at high frequencies. The number is infinite in the sub-ohmic regime and for perturbations of the sub-ohmic or ohmic regime. Decoherence-recoherence times are evaluated, analytically and numerically, for ohmic-like spectral densities with exponential high-frequency cutoffs.
The recoherence times t * and the maximum values of the recoherence increments γ extr are studied as functions of the bath parameters for a single qubit dephasing model, prepared initially by a special kind of the nonselective measurements. The recoherence-decoherence events (RDE), occurring at the initial stage of the system evolution, are found to be both similar and different from the system dynamics at large times. For instance, in contrast to the RDE observed on large timescales, the sub-Ohmic and Ohmic coupling regimes are more favorable for the short-time recoherence than the super-Ohmic one. On the other hand, the short-time behavior of the recoherence and the long-time dynamics of the decoherence are closely related: The domain of the ohmicity indexes, where the decoherence changes its type (from the complete to incomplete one), is, simultaneously, that of the weakest recoherence. The obtained results give us some hints about the basic characteristics of the environment, which might provide the most optimal values of t * and γ extr in some sense.
A general scheme is presented for controlling quantum systems using evolution driven by nonselective von Neumann measurements, with or without an additional tailored electromagnetic field. As an example, a two-level quantum system controlled by nonselective quantum measurements is considered. The control goal is to find optimal system observables such that consecutive nonselective measurement of these observables transforms the system from a given initial state into a state which maximizes the expected value of a target operator (the objective). A complete analytical solution is found including explicit expressions for the optimal measured observables and for the maximal objective value given any target operator, any initial system density matrix, and any number of measurements. As an illustration, upper bounds on measurement-induced population transfer between the ground and the excited states for any number of measurements are found. The anti-Zeno effect is recovered in the limit of an infinite number of measurements. In this limit the system becomes completely controllable. The results establish the degree of control attainable by a finite number of measurements.