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arXiv:1312.1015 · New Journal of Physics

Global versus local optimality in feedback-controlled qubit purification: new insights from minimizing Rényi entropies

Colin Teo, Joshua Combes, Howard M. Wiseman

Published Oct 16, 2014• 5 citations

Abstract

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.

DOI: 10.1088/1367-2630/16/10/105010

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This paper reconsiders the claimed rapidity of a scheme for the purification of the quantum state of a qubit, proposed recently in Jacobs (2003 Phys. Rev. A 67 030301(R)). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs' rapid purification protocol uses Hamiltonian feedback control to maximize the average purity of the qubit for a given time , with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it minimizes the average time taken for a qubit to reach a given purity . In fact, the average time taken for the no-feedback protocol beats that for Jacobs' protocol by a factor of two. We discuss how this is compatible with Jacobs' result and the usefulness of the different approaches.

Abstract. This paper reconsiders the properties of a scheme for the rapid purification of the quantum state of a qubit, proposed recently in Jacobs 2003 Phys. Rev. A 67 030301(R). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs ’ rapid purification protocol uses Hamiltonian feedback control to maximise the average purity of the qubit for a given time, with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it mininises the average time taken for a qubit to reach a given purity. In fact, the average time taken for the no-feedback protocol beats that for Jacobs ’ protocol by a factor of two. We discuss how this is compatible with Jacobs ’ result, and the usefulness of the different approaches. Reconsidering Rapid Qubit Purification by Feedback 2 1.

Entanglement of purification is a fundamental measure of total correlations whose additivity remains unresolved. We study its additivity for classical states on two qubits at different Rényi orders. For every

α\in[0,1)

, we prove nonadditivity within this family, witnessed by two copies of a single state. We first solve the one-copy optimization exactly for the entire family at every Rényi order. We then restrict the two-copy optimization to a natural finite set of purifications and exhibit one whose entropy is strictly below the product value. In contrast, for

α\in[2,\infty]

we prove additivity under tensor products within this family. The interval

α\in[1,2)

, including the von Neumann case

α=1

, remains open, and we conjecture additivity there throughout the same family.

For quantum systems with high purity, we find all observables that, when continuously monitored, maximize the instantaneous reduction in the average linear entropy. This allows us to obtain all locally optimal feedback protocols with strong feedback, and explicit expressions for the best such protocols for systems of size N < or =4. We also show that for a qutrit the locally optimal protocol is the optimal protocol for observables with equispaced eigenvalues, providing the first fully optimal feedback protocol for a 3-state system.

Rapid purification by feedback---specifically, reducing the mean impurity faster than by measurement alone---can be achieved by choosing the eigenbasis of the density matrix to be unbiased relative to the measurement basis. Here we further examine the protocol introduced by Combes and Jacobs [Phys. Rev. Lett. 96, 010504 (2006)] involving continuous measurement of the observable

{J}_{z}

for a

D

-dimensional system. We rigorously rederive the lower bound

(2/3)(D+1)

on the achievable speedup factor and also an upper bound, namely

{D}^{2}/2

, for all feedback protocols that use measurements in unbiased bases. Finally, we extend our results to

n

independent measurements on a register of

n

qubits and derive an upper bound on the achievable speedup factor that scales linearly with

n

.

Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.