Research opportunities arising from measurement and estimation of quantum systems

Wei CUI

School of Automation Science and Engineering,South China University of Technology,Guangzhou Guangdong 510640,China

With the advancement of quantum experimental technology,quantum computing has earned many attentions and made great development.Meanwhile,quantum measurement theory[1]has become an indispensable component of it,ultimately working for the quantum state estimation.Being tools for emerging quantum computing technology,those two areas provide many research opportunities.

Different from the classical world,quantum measurements cause dynamical changes that result in a limit to the obtained information about the quantum states.The relationships between information gained[2]and the dynamics induced(usually regarded as the damage to the systems caused by the measurement)thus become quite important since researchers always aim at finding the best measurements that gain as much information as possible with acceptable damage to the systems[3].

Another challenging direction in quantum measurement is the simultaneous measurement of non-commuting observables[4–6]which is impossible with projective measurement.Heisenberg’s uncertainty principle limits the intrinsic precision of a state as well.However,theoretical work has proved that it is possible to realize simultaneous non-commuting measurements with continuous weak measurement.Recently the dynamics[4]of such measurements and the temporal correlations[5,6]in the two measurement output signals have been analyzed.

Since most realistic quantum systems actually interact continuously with their environment via several noncom muting decoherence channels,the studies of simultaneous non-commuting measurements might help to explore the structure of quantum foundations.Also we wonder whether the non-commuting measurements would provide us with a better choice of the right kinds of measurements.

Quantum state estimation[7],which aims at using the measurement records to determine the states of quantum systems can be divided into two categories:quantum state tomography[8]and real-time quantum state estimation[9].

The task of quantum state tomography is to reconstruct the initial states of the quantum systems.The basic theories in this area have already been established in detail,however there are still several problems in its practical applications,which can be summarized as follows:The amount of data involved in the system would increase exponentially with the size of the quantum system;the development of quantum measurement requires quantum tomography methods that can fit the properties of measurement better.Quantum state tomography based on the maximum likelihood method[10]using continuous weak measurement has been proposed recently.

In order to deal with the exponential growth of the according qubitin the Hilbert space,linear regression method[8]and compressive sensing method[11–13]have been applied to quantum tomography.It is worth mentioning that the working principles of those two methods are fundamentally different.Linear regression method is applied to accomplish a full quantum state tomography and reduce the computational complexity,while quantum state tomography based on compressive sensing utilizes the fact that the density matrices keep a low rank,in the general case to accomplish an approximate quantum state tomography.What’s more,quantum state tomography based on compressive sensing provides a much shorter reconstruction time than that based on linear regressive method.

In quantum experiments,however,it is the current states rather than the initial states that play important roles,thus the real-time monitoring of quantum states is widely demanded as well.The filter equation[14]and the quantum Bayesian approach[15,16]are the two most widely used methods in real-time quantum state estimation.

To give a sense of how to do some work in this area,below we introduce our recent work[17]on the quantum Bayesian approach.Although the Bayesian approach has already been used as an effective and useful tool in quantum information technology,there is not a strict proof of its establishment.Therefore,we analyzed the suitable conditions under which the Bayesian approach can accurately update the quantum state,and considered the correlation between some basic and physically meaningful parameters as well as the performance of the method.

We are very glad to see more researchers start to devote themselves to those areas and contribute to the development of the quantum technology which is expected to change the lifestyles of human beings eventually.