Albertini and Ticozzi (Automatica 47, 2451, 2011) proved that discrete-time quantum dynamics under a fixed generalized measurement and outcome-conditioned unitary feedback is generically asymptotically controllable at the state level, and finite-time state controllable (in at most 2N steps) for two-outcome measurements with singular Kraus operators, but left the channel-level theory open. This problem asks for a complete characterization of feedback controllability in this model: necessary and sufficient conditions on a fixed measurement operator-sum representation for full Kraus-map controllability — equivalently, an exact description of the reachable semigroup of completely positive trace-preserving (CPTP) maps — together with the extension of the finite-time constructive state-transfer protocol to measurements with more than two outcomes.

Background

In finite-dimensional quantum mechanics, the state of an NN-level system is a density operator ρ\rho (a positive semidefinite, trace-one matrix on a Hilbert space H≅CN\mathcal{H}\cong\mathbb{C}^N), and the most general state transformation compatible with quantum theory is a completely positive trace-preserving (CPTP) map, also called a quantum channel. By the Kraus–Stinespring theorem every such map admits an operator-sum representation (OSR) E(ρ)=∑kMkρMk†\mathcal{E}(\rho)=\sum_k M_k\rho M_k^\dagger with Kraus operators satisfying ∑kMk†Mk=I\sum_k M_k^\dagger M_k=I; the OSR is unique only up to unitary mixing of the Kraus operators. A generalized measurement with outcomes kk is described by precisely such a family {Mk}\{M_k\}: outcome kk occurs with probability pk=Tr(MkρMk†)p_k=\mathrm{Tr}(M_k\rho M_k^\dagger) and the conditional post-measurement state is MkρMk†/pkM_k\rho M_k^\dagger/p_k.

Albertini and Ticozzi (arXiv:1012.1397; Automatica 47, 2451–2456, 2011) studied controllability of discrete-time Markovian quantum dynamics when the controller can, at each time step, enact a fixed generalized measurement and then apply a unitary control action UkU_k conditioned on the observed outcome kk. One feedback step enacts the CPTP map

ρ(t+1)=∑kUk Mk ρ(t) Mk† Uk†.\rho(t+1)=\sum_k U_k\,M_k\,\rho(t)\,M_k^\dagger\,U_k^\dagger.

Several graduated controllability notions apply. Density-operator to density-operator controllability (DDC) means arbitrary state-to-state transfers between density operators, in finite time or asymptotically; Kraus-map controllability (KC) is the stronger requirement that any CPTP map can be enacted as a finite composition of controlled steps. The results of Albertini and Ticozzi delimit what feedback achieves: (i) if the measurement OSR is not a probabilistic average of unitaries — that is, not every MkM_k is of the form Mk=qkVkM_k=q_k V_k with VkV_k unitary (the paper's "form (12)", i.e. a random-unitary measurement) — feedback allows asymptotic purification and approximate DDC, whereas for form-(12) measurements the reachable set is trapped in the convex hull of the states isospectral to the initial state, so a full-rank initial state cannot be purified at all; (ii) for two-outcome measurements whose Kraus operators can be simultaneously diagonalized by feedback unitaries and are both singular, an explicit constructive protocol achieves DDC in finite time, in at most 2N2N steps; (iii) for exact KC, matching of the canonical QR decompositions of the OSRs gives only a sufficient condition. Here the canonical QR decomposition is the paper's standardized QR-type factorization of each Kraus operator into a unitary part and an upper-triangular factor RR (their Proposition 4.1): a target measurement can be simulated in one feedback step when its canonical RR factors match those of the feedback-loop measurement up to reordering, but because a target CPTP map admits many equivalent OSRs, canonical-form matching is only sufficient for realizability by a feedback protocol. The paper also gives an explicit two-level example showing that a form-(12) measurement cannot enact an amplitude-damping target channel (the standard qubit relaxation channel): every one-step feedback map it generates is unital — it maps the identity operator to itself — and so is every finite composition of unital maps, while amplitude damping is not unital. Section 4.2 of the paper states the open problem: "What is the class of CPTP maps one can realize via feedback? ... Providing conditions for exact KC, or characterizing the reachable set of propagators are, to the best of our knowledge, open problems." The Conclusions pose two sharp questions: if the measurement OSR is not of the form (12), is the system feedback KC? And can the two-outcome finite-time construction be extended to OSRs with more than two Kraus operators? Adjacent open-loop facts frame the question: sampled closed-system control inherits continuous-time controllability (by Sontag's sampled-controllability theorems, which establish that a discrete-time system obtained by sampling a controllable continuous-time system remains controllable), and Wu–Pechen–Brif–Rabitz (J. Phys. A 40, 5681, 2007) had characterized KC for open-loop Kraus-map dynamics, leaving the feedback case open.

Problem Statement

Determine the scope and limits of measurement-based feedback control for discrete-time quantum dynamics with a fixed generalized measurement, at both the state level and the channel level.

(1) Channel-level characterization. For a fixed measurement OSR {Mk}\{M_k\} on an NN-level system, with arbitrary outcome-conditioned unitaries {Uk}\{U_k\} available at each step, characterize the semigroup of CPTP maps reachable as finite compositions of feedback steps ρ↦∑kUkMkρMk†Uk†\rho\mapsto\sum_k U_k M_k\rho M_k^\dagger U_k^\dagger. In particular, find necessary and sufficient conditions on {Mk}\{M_k\} under which the system is feedback Kraus-map controllable, i.e. every CPTP map is reachable. This includes the source paper's explicit question — is every measurement whose OSR is not of the probabilistic-average-of-unitaries form (12), Mk=qkVkM_k=q_k V_k with VkV_k unitary, feedback KC? — and its concrete instances, such as whether the two-outcome singular measurements that already yield finite-time state controllability yield KC (for example, whether the identity channel itself is exactly realizable by a finite feedback protocol). The general characterization is the required deliverable of this item; the source paper's explicit question and the listed instances are corollaries that a complete answer must settle.

(2) State-level constructive extension. Extend the finite-time, two-outcome constructive protocol to measurements with more than two Kraus operators: determine for which multi-outcome OSRs arbitrary density-operator-to-density-operator transfers can be achieved in finitely many feedback steps.

Both parts serve one common objective: a complete feedback-controllability theory for the discrete-time measurement-feedback loop, replacing the currently sufficient-only channel conditions (canonical-form matching) and state-level-only constructions. Continuous-time quantum-dynamical-semigroup analogues, sampling-cost estimates for open-loop control, and controllability under locality or quantized-control constraints are outside this problem's completion boundary.

References

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