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

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 α S , with α < < ∞ 0 , 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 α S 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-Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.intuitive relation between local and global optimality in stochastic quantum control.

Rapid readout of a register of qubits using open-loop quantum control

“…Consider an arbitrary state ρ with an infidelity ∆. The upper bound on Eq (20). is obtained by considering the minimally mixed state with the same infidelity: ρ 2 = diag(1 − ∆, ∆, 0, · · · , 0)[5].…”

“…Conversely the state ρ F is invariant under permutations of the remaining 2 n − 1 eigenvalues [with respect to Eq. (20)].The using the states ρ 2 and ρ F the it is simple to calculate the bounds on Eq (20)…”

Focus on coherent control of complex quantum systems

“…In particular, the development of fast high fidelity quantum limited measurements for scalable systems such as superconducting qubits has generated renewed interest in systematic development of feedback control for quantum systems. In 'Global versus local optimality in feedback-controlled qubit purification: new insights from minimizing Renyi entropies' [4], Teo et al use the relation of minimal Renyi entropies to optimal feedback-controlled qubit purification to elucidate the relationship between local and global optimality of measurement-based feedback protocols. In 'Rapid steady-state convergence for quantum systems using time-delayed feedback control' [5], Grimsmo et al explore the use of time-delayed feedback in coherent feedback control without measurement and show that this may be used to speed-up the establishment of steady states.…”

What is the optimal way to prepare a Bell state using measurement and feedback?

“…Substituting Eqs. (18) and (19) into Eqs. (15) and (16), we find the condition for P H to be globally optimal is that it satisfies the maximization condition…”

“…This proves that the half-parity protocol P H is globally optimal for maximizing the concurrence at fixed time T . Because c(P H , x, t) is linear in C, one can also show that global optimality follows directly from local optimality in this case [19], but this proof method is not applicable in general.…”