Reconsidering rapid qubit purification by feedback

Abstract

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 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.

Optimality of Feedback Control Strategies for Qubit Purification

“…In the present case, it is less clear that this is a reasonable approximation, as singular controls are incompatible with the assumptions used to derive the quantum filtering equation [6,8]. Our goal here, however, is to demonstrate optimality of the problems ultimately considered in [1,2,3], so we will adopt their assumptions in the following.…”

“…The goal of the particular control problems studied in [1,2,3] is to maximize the rate of purification of the conditional state ρ t in a suitable sense. We consider precisely the same system as in [1,2,3]: monitoring one operator (σ z ) for a qubit with the ability to control the Hamiltonian.…”

“…The equations (2.5) and (2.6) are already too complicated to be amenable to optimal control. In order to proceed, the papers [1,2,3] assume that, rather than controlling Ω t , one can directly control u t at every time to take any value in the interval [−1, 1]. This could be seen as a singular limit of the above system, in the sense that an infinitely strong driving field Ω t would ostensibly allow one to set the angle θ t to any desired value.…”

“…It is clear from the discussion at the start of the preceding section that, under our assumptions, this problem is equivalent to minimizing the time taken to prepare a given pure state (in the y = 0 plane) with fidelity (1 + √ 1 − h)/2. From consideration of the stochastic evolution of log(1 − S t ), Wiseman and Ralph [3] claimed that the strategy u = 1 was optimal in the asymptotic (h ≪ 1) limit. This strategy requires instantaneously rotating (if necessary) the qubit onto the z-axis at t = 0, then letting Ω t = 0 for t > 0.…”

“…Wiseman and Ralph [3] show by explicit calculation that for u t = 1 the expected cost-to-go is 8) where for convenience we have defined, for x ∈ [0, 1],…”