Coherently tracking the covariance matrix of an open quantum system

Abstract

Coherent feedback control of quantum systems has demonstrable advantages over measurementbased control, but so far there has been little work done on coherent estimators and more specifically coherent observers. Coherent observers are input the coherent output of a specified quantum plant, and are designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. We previously developed a class of mean tracking (MT) observers for open harmonic oscillators that only converged in mean position and momentum; Here we develop a class of covariance matrix tracking (CMT) coherent observers that track both the mean and covariance matrix of a quantum plant. We derive necessary and sufficient conditions for the existence of a CMT observer, and find there are more restrictions on a CMT observer than there are on a MT observer. We give examples where we demonstrate how to design a CMT observer and show it can be used to track properties like the entanglement of a plant. As the CMT observer provides more quantum information than a MT observer, we expect it will have greater application in future coherent feedback schemes mediated by coherent observers. Investigation of coherent quantum estimators and observers is important in the ongoing discussion of quantum measurement; As they provide estimation of a system's quantum state without explicit use of the measurement postulate in their derivation.

Quantum state transfer through time reversal of an optical channel

“…As we have performed a time reversal on the optical channel, the input field to the receiver is both non-Markov and non-causal [36]. There has been extensive work on solving causal filtering problems with coherent quantum components [41][42][43][44][45], but very little on coherent non-causal filters. This is primarily because it has been unclear how to coherently implement a non-causal filter.…”

Direct and indirect couplings in the interconnection of open two level quantum systems

“…Finally, by substituting the terms above to (21), we can conclude the necessary and sufficient conditions (22).…”

“…On the other hand, a coherent feedback controller is directly fed by the output of the quantum plant (non-commutative quantum signal), without the need for a measurement device, and therefore this scheme retains the coherence in the whole process [12], This [15], [30]. Moreover, coherent filtering and estimation has been studied [1], [21], with the aim of ruling out drawbacks caused by measurements.…”

Pole placement design for quantum systems via coherent observers

“…A coherent observer is another system of quantum harmonic oscillators which we engineer such that at least the system variables track those of the quantum plant asymptotically in the sense of mean values [27], [28], [35]. In classical control theory, it is well known that if not all state variables of a linear plant are available for feedback, an observer may be needed for feedback design [8], [40].…”

“…In particular, if we desire an observable (a self-adjoint operator defined on a Hilbert space to represent physical quantities in quantum mechanics) to be asymptotically stable in the mean with specific transient response, a mean tracking coherent observer can be employed to provide a reliable estimate. In addition, a coherent observer and the plant are correlated in the sense that some quantum features can be observed in the joint system [27], [28]. As the first step in studying observer-based coherent feedback control design, in this paper we are concerned with the pole-placement technique using coherent observers.…”