Identification and estimation of quantum linear input-output systems

Abstract

The system identification problem is to estimate dynamical parameters from the output data, obtained by performing measurements on the output fields. We investigate system identification for quantum linear systems. Our main objectives are to address the following general problems: (1) Which parameters can be identified by measuring the output? (2) How can we construct a system realisation from sufficient input-output data? (3) How well can we estimate the parameters governing the dynamics? We investigate these problems in two contrasting approaches; using time-dependent inputs (Sec. 3.7.1 or time-stationary (quantum noise) inputs (Sec. 3.7.2). In the time-dependent approach, the output fields are characterised by the transfer function. We show that indistinguishable minimal systems in the transfer function are related by symplectic transformations acting on the space of system modes (Ch. 6). We also present techniques enabling one to find a physical realisation of the system from the input-output data. We present realistic schemes for estimating passive quantum linear systems at the Heisenberg limit (Ch. 7) under energy resource constraint. ‘Realistic’ is our primary concern here, in the sense that there exists both experimentally feasible states and practical measurement choices that enable this heightened performance for all passive quantum linear systems. We consider both single parameter and multiple parameter estimation. In the stationary approach, the characteristic quantity is the power spectrum. We define the notion of global minimality for a given power spectrum, and characterise globally minimal systems as those with fully mixed stationary state (Sec. 6.1). The power spectrum depends on the system parameters via the transfer function. Our main result here is that under global minimality the power spectrum uniquely determines the transfer function, so the system can be identified up to a symplectic transformation (see Secs. 6.5, 6.4 6.11). We also give methods for constructing a globally minimal subsystem directly from the power spectrum (see Sec. 6.3). These results hold for pure inputs, we discuss extensions to mixed inputs and the use of additional input channels; using an appropriately chosen input in the latter case ensures that the system is always globally minimal (hence identifiable). Finally, we discuss a particular feedback control estimation problem in Chs. 8 and 9. In general, information about a parameter within a quantum linear system may be obtained at a linear rate with respect to time (in both approaches above); the so-called standard scaling. However, we see that when the system destabilises, so that its system matrix has eigenvalues very close to the imaginary axis, the quantum Fisher information is enhanced, to quadratic (Heisenberg) level. We give feedback methods enabling one to destabilise the system and give adaptive procedures for realising the Heisenberg bounds