1 Emergence of classicality under decoherence in near steady-state robust quantum transport S. Schirmer, Member, IEEE, E. Jonckheere, Life Fellow, IEEE, S. O’Neil, and F. Langbein, Member, IEEE Abstract—In the quantum analogue of the classical feedback problem of driving the state to a desired terminal target, several highly nonclassical features emerge—in particular, the lack of closed-loop stability in case of purely coherent dynamics where Schr¨ odinger equation prevails and a challenge to the classical limitation of conflict between precision of targeting and sensitivity of targeting error to parameter uncertainty. Here we show that given a quantum system evolving according to a fixed Hamiltonian and pure dephasing we recover classicality, in the sense that the closed-loop system is stable and that the classical limitations re-emerge. I. I NTRODUCTION Recently, a new paradigm for quantum control based on energy landscape shaping has been proposed and applied to derive feedback control laws for selective transfer of excita- tions between nodes of a network consisting of N coupled spins [1]. The controllers D(|IN〉 , |OUT〉) are designed to maximize the fidelity F := |〈OUT| U (t f ) |IN〉| of transfer from the input quantum state |IN〉 to the output state |OUT〉 at a specified readout time t f , where U (t) is the unitary time- evolution operator of the system, relying only on static bias fields {D n } N n=1 applied to their respective spins to shift the energy levels of the system [2], obviating the need for, and side-stepping the back-action of, measurements. The presumption that the bias fields are spin-addressable calls into question the issue of robustness of the design. Even though current technological advances are targeting such ob- jective [3], field focusing errors are always going to be present to some extent. Next to this bias spillage issue, uncertainties in the (possibly engineered) spin couplings are also present. Differential sensitivity analyses against such uncertainties have led to the surprising result that controllers with the best fidelity yields a design with performance least sensitive to coupling errors—a feature that goes against conventional control wis- dom [4]. This classical-quantum discrepancy is not surprising given that the optimal controllers described above are selective, that is, no input other than |IN〉 can drive the system to |OUT〉. Furthermore, unlike classical robust control, the target states are not asymptotically stable but rather the system is oscillating around the target state, similar to the case of S. Schirmer is with Swansea University, Swansea, UK, lw1660@gmail.com. E. Jonckheere is with the University of Southern California, Los Angeles, CA, jonckhee@usc.edu. S. O’Neil is with US Military Academy, West Point, NY, sean.o’neil@westpoimnt.edu. F. Langbein is with Cardiff University, Cardiff, UK, LangbeinFC@cardiff.ac.uk. Anderson localization [5], although localization is achieved through control biases and no randomness is required. The aforementioned uncertainties, while they apply to non- classical quantum systems, are still classical in the sense that they model parameter uncertainties in the plant and in the controller, to borrow classical phraseology. In addition to clas- sical uncertainties, quantum systems exhibit uncertainties that do not fit within the classical robustness paradigm: namely, preparation errors in the initial state |IN〉 and, last but not least, uncertainties in the decoherence process. Ref. [3] specifically addresses the large, structured properties of the uncertainties using the µ-function [6] and only tacitly introduces initial state preparation error as extraneous perturbation. Ref [7] focuses on the effect of uncertainties lumped in the decoherence process, disregarding all other uncertainties. In the present paper, which follows in the footsteps of [8], we consider another mix. We consider coupling errors, bias spillage, and initial preparation error, under a fixed Lindblad decoherence process, where the rates have been randomly drawn to avoid biases. The primary objective of this somewhat preliminary study is to expose a shift from anticlassical to classical robustness properties as a result of the decoher- ence. To be somewhat more specific, we look at how the transmission T from the initial state preparation error to the error ε := 1 − F 2 is affected by large perturbation in the couplings and bias fields. Those uncertainties are embedded in T as an upper linear fractional transformation F u (G, Δ) of the diagonal model Δ of the uncertainty under a 2 × 2 coupling matrix G, as shown by Fig. 1. The sensitivity of T to large variation is quantified using µ(G). We consider a variety of controllers achieving a variety of F ’s and µ(G)’s and examine discordance or concordance between F and µ(G) using statistical techniques. Discordance, that is, F increases (decreases) while µ(G) decreases (increases) across some ordering of the controllers is anticlassical and manifests itself for coherent systems, whereas under decoherence the trend shifts to classical concordance, that is, F and µ(G) either simultaneously increases or decrease across a family of ordered controllers. This anticlassical to classical transition under decoherence is consistent with some arguments that decoherence may provide the quantum-to-classical bridge that makes the macro world classical (e.g., [9], [10] and references therein), although some of these arguments are controversial. II. SPIN NETWORKS UNDER DECOHERENCE In this section we briefly review the theory of quantum spin networks subject decoherence and derive the relevant single