We consider the problem of maximizing the transition probability in an n-level quantum system from a given initial state to a given final state using nonselective quantum measurements. We find a sequence of measurements that is a critical point of the transition probability and, moreover, a local maximum in each variable on the set of one-dimensional projectors. We consider the class of one-dimensional projectors because these projectors describe the measurements of populations of pure states of the system.
We consider the problem of determining the state of a quantum system via a series of controlled evolutions and nonselective measurements of a given observable. Generalizing previous results to the case of possibly degenerate observables, we give a bound on the maximum number of parameters of the state that can be determined. We give a general algorithm for the determination of the maximum number of parameters. We also discuss how indirect measurement, when the system and probe are jointly controllable, allows one to determine all the parameters of the quantum state
Nonselective quantum measurements, i.e., measurements without reading the results, are often considered as a resource for manipulating quantum systems. In this work, we investigate optimal acceleration of the Landau-Zener (LZ) transitions by nonselective quantum measurements. We use the measurements of a population of a diabatic state of the LZ system at certain time instants as control and find the optimal time instants which maximize the LZ transition. We find surprising nonmonotonic behavior of the maximal transition probability with increase of the coupling parameter when the number of measurements is large. This transition probability gives an optimal approximation to the fundamental quantum Zeno effect (which corresponds to continuous measurements) by a fixed number of discrete measurements. The difficulty for the analysis is that the transition probability as a function of time instants has a huge number of local maxima. We resolve this problem both analytically by asymptotic analysis and numerically by the development of efficient algorithms mainly based on the dynamic programming. The proposed numerical methods can be applied, besides this problem, to a wide class of measurement-based optimal control problems.
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.
Optimal and finite positive operator valued measurements on a finite number
N of identically prepared systems have recently been presented. With physical realization in mind, we propose here optimal and minimal generalized quantum measurements for two-level systems. We explicitly construct them up to
N=7 and verify that they are minimal up to
N=5.
A tilted nonselective measurement can close the upper transition of a three-level quantum refrigerator, but it also generates coherence that feeds back on the populations and suppresses the cooling current. We formulate randomized open-loop control of the measurement azimuthal phase
α, sampled independently at each event from a probability distribution
P(α), while keeping the event rate, polar angle, and single-event population-transfer probability fixed. Direct population transfer is phase independent, whereas the coherence source carries the factor
e−iα. Exact elimination of the stationary coherence gives
0≤keff[P]≤Am, where
keff[P] is the effective population-transfer rate between the upper levels and
Am is the bare direct mixing rate. When the fixed measurement couples populations and coherence, the upper bound is attained if and only if
⟨eiα⟩P=0, where
⟨⋅⟩P denotes an average over the phase distribution. The minimal realization uses two equally weighted phases separated by
π: the population-to-coherence source and its return to the populations then vanish at the generator level, while the full direct mixing remains. Because the cold current increases strictly with
keff in the refrigeration window, the same condition maximizes the stationary cooling power over the independent random-phase distributions considered here. For one representative parameter set, phase cycling increases the fixed-gap cooling power by about
32% relative to a fixed-phase measurement, while leaving the ratio of extracted cold heat to supplied measurement energy unchanged.