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The Proceedings of Conference of Kansai Branch

716 Quantum Optimal Control of Purification via Measurement Feedback

Joe Imae, Yui TAKENAKA, Tomoaki Kobayashi

Published Jan 1, 2013• 0 citations
Publisher
DOI
10.1299/jsmekansai.2013.88._7-16_

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This thesis addresses the of developing a counter-part of the well established classical theory of control. We dwell on the fundamental fact that states are generally not perfectly distinguishable, and measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the setting. We center our investigations around the of transforming the state of a system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the quantum tracking problem and show how it can be formulated as an optimization that can be approached both numerically and analytically. This provides a simple route to the characterization of the trade-off between information gain and disturbance, and is seen to have several applications in information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of measurements. We also demonstrate that our optimal controllers are immediately applicable in several information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.

•Aug 1, 2008•0 citations
A quantum system may be purified, i.e., projected into a pure state, faster if one applies feedback operations during the measurement process. However, the existing results suggest that such an enhancement is only possible when the measurement efficiency exceeds 0.5, which is difficult to achieve experimentally. We address the task of finding the global optimal feedback control for purifying a single qubit in the presence of measurement inefficiency. We use the Bloch vector length, a more physical and practical quantity than purity, to assess the quality of the state, and employ a backward-iteration algorithm to find the globally optimal strategy. Our results show that a speedup is available for quantum efficiencies well below 0.5, which opens the possibility of experimental implementation in existing systems.
It was first shown by Jacobs, in 2003, that the process of qubit state purification by continuous measurement of one observable can be enhanced, on average, by unitary feedback control. Here, we quantify this by the reduction in any one of the family of Rényi entropies , with , at some terminal time, revealing the rich structure of stochastic quantum control even for this simple problem. We generalize Jacobs' original argument, which was for the (unique) impurity measure with a linear evolution map under his protocol, by replacing linearity with convexity, thereby making it applicable to Rényi entropies for α in a finite interval about one. Even with this generalization, Jacobs' argument fails to identify the surprising fact, which we prove by Bellmanʼs principle of dynamic programming, that his protocol is globally optimal for all Rényi entropies whose decrease is locally maximized by that protocol. Also surprisingly, even though there is a range of Rényi entropies whose decrease is always locally maximized by the null-control protocol, that null-control protocol cannot be shown to be globally optimal in any instance. These results highlight the non-intuitive relation between local and global optimality in stochastic quantum control.
This thesis addresses the of developing a counter-part of the well established classical theory of control. We dwell on the fundamental fact that states are generally not perfectly distinguishable, and measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the setting. We center our investigations around the of transforming the state of a system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the quantum tracking problem and show how it can be formulated as an optimization that can be approached both numerically and analytically. This provides a simple route to the characterization of the trade-off between information gain and disturbance, and is seen to have several applications in information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of measurements. We also demonstrate that our optimal controllers are immediately applicable in several information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.