Identifiability of Autonomous and Controlled Open Quantum Systems

TL;DR

This work presents a unified framework for identifying open quantum systems by connecting GKSL master equations with linear and bilinear dynamical system representations. It delivers concrete identifiability criteria under discrete or irregular sampling, and provides constructive procedures to recover Hamiltonian and decoherence parameters from measurement data, up to transformation equivalence. The approach covers both autonomous (LDS) and driven (BDS) open quantum dynamics, and includes a symmetric-Kossakowski-matrix special case with simpler reconstruction. Through a two-qubit example and algorithmic prescriptions, the paper demonstrates how classical system-identification concepts can be applied to quantum devices, with implications for calibration, tomography, and controlled quantum technologies.

Abstract

Open quantum systems are a rich area of research in the intersection of quantum mechanics and stochastic analysis. By considering a variety of master equations, we unify multiple views of autonomous and controlled open quantum systems and, through considering their measurement dynamics, connect them to classical linear and bilinear system identification theory. This allows us to formulate corresponding notions of quantum state identifiability for these systems which, in particular, applies to quantum state tomography, providing conditions under which the probed quantum system is reconstructible. Interestingly, the dynamical representation of the system lends itself to considering two types of identifiability: the full master equation recovery and the recovery of the corresponding system matrices of the linear and bilinear systems. Both of these concepts are discussed in detail, and conditions under which reconstruction is possible are given. We set the groundwork for a number of constructive approaches to the identification of open quantum systems.

Figures (2)

• Figure 1: A broad overview of the key stages in learning and recovering the parameters of open quantum systems. (a) Design choices include: the times $T$ at which one samples the quantum state (e.g. uniform or random), the control pulses, $u_{j}(t)$, of length $T$ used for the introduction of states at times $T$ (e.g. different frequencies), measurement basis (e.g. computational basis or some other arbitrary positive operator-valued measurement (POVM) basis). (b) Estimates: In order to identify the system parameter one must perform quantum state tomography (QST) at times $T$, which involves estimating the state from many copies of the state obtained using one and the same pulse of length $T$. From this, one can run an identification algorithm to recover the parameters of the bilinear dynamical system (BDS) representation and to to estimate the parameters of the master equation. (See \ref{['sec:parameter_reconstruction']}). (c) Identifiability: in a crucial test, we validate that we have estimated the correct system. The same theorems can also be used to guide the design choices above. See Sections \ref{['sec:level3']} and \ref{['sec:sum_identification']}.
• Figure 2: A dendrogram that displays the links between various common master equations of Open Quantum Systems and how they can be approximated into the Lindblad form and from there to a linear or bilinear dynamical system. The equations outlined by a red (blue) box are non-Markovian (Markovian) and the uncontained text are the assumptions under which the equations are valid. The representations shown are: (a) Nakajima-Zwanzig \ref{['sec:pertubative']} (b) Dynamical maps \ref{['sec:Dynamical_maps']} (c) Microscopic descriptions \ref{['sec:pertubative']}, (d) \ref{['sec:Lindbladian']} and (e) Dynamical systems \ref{['sec:level3']}.

Theorems & Definitions (36)

• Proposition 1: consequence of CP-divisibility for master equation formulations wolf06Rivas2011OpenQSKossakowski10
• Remark 1
• Remark 2
• Definition 2
• Definition 3
• Definition 4: non-uniformly sampled linear dynamical system
• Definition 5: non-uniformly sampled LDS
• Lemma 6: retention of observability and controllability
• proof
• Definition 7: single-rate models associated with a multi-rate model of the form \ref{['eq_non_unif_sampled_syst']}