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We propose a novel optimization scheme designed to find optimally correctable subspace codes for a known quantum noise channel. To each candidate subspace code we first associate a universal recovery map, as if the code was perfectly correctable, and aim to maximize a performance functional that combines a modified channel fidelity with a tuneable regularization term that promotes simpler codes. With this choice optimization is performed only over the set of codes, and not over the set of recovery operators. The set of codes of fixed dimensions is parametrized as the complex-valued Stiefel manifold of the same dimension: the resulting non-convex optimization problem is then solved by gradient-based local algorithms. When perfectly correctable codes cannot be found, a second optimization routine is run on the recovery Kraus map, also parametrized in a suitable Stiefel manifold via Stinespring representation. To test the approach, correctable codes are sought in different scenarios and compared to existing ones: three qubits subjected to bit-flip errors (single and correlated), four qubits undergoing local amplitude damping and five qubits subjected to local depolarizing channels. Approximate codes are found and tested for the previous examples as well pure non-Markovian dephasing noise acting on a $7/2$ spin, induced by a $1/2$ spin bath, and the noise of the first three qubits of IBM's \texttt{ibm\_kyoto} quantum computer. The fidelity results are competitive with existing iterative optimization algorithms, with respect to which we maintain a strong computational advantage, while obtaining simpler codes.

This work focuses on the problem of exact model reduction of positive linear systems, by leveraging minimal realization theory. While even determining the existence of a positive reachable realization remains an open problem, we are able to fully characterize the cases in which the new model is obtained with non-negative reduction matrices, and hence positivity of the reduced model is robust with respect to small perturbations of original system. The characterization is obtained by specializing monotone matrix theory to positive matrices. In addition, we provide positive reduction also when minimal ones are not available, by exploiting algebraic techniques.

Leveraging an algebraic approach built on minimal realizations and conditional expectations in quantum probability, we propose a method to reduce the dimension of quantum filters in discrete-time, while maintaining the correct distributions on the measurement outcomes and the expectations of some relevant observable. The method is presented for general quantum systems whose dynamics depend on measurement outcomes, hinges on a system-theoretic observability analysis, and is tested on prototypical examples.

Obtaining reliable state preparation protocols is a key step towards practical implementation of many quantum technologies, and one of the main tasks in quantum control. In this work, different reinforcement learning approaches are used to derive a feedback law for state preparation of a desired state in a target system. In particular, we focus on the robustness of the obtained strategies with respect to different types and amount of noise. Comparing the results indicates that the learned controls are more robust to unmodeled perturbations with respect to simple feedback strategy based on optimized population transfer, and that training on simulated nominal model retain the same advantages displayed by controllers trained on real data. The possibility of effective off-line training of robust controllers promises significant advantages towards practical implementation.

In this work we analyze and bound the effect of modeling errors on the stabilization of pure states or subspaces for quantum stochastic evolutions. Different approaches are used for open-loop and feedback control protocols. For both, we highlight the key role of dynamical invariance of the target: if the perturbation preserves invariance, it is possible to prove that it also preserves its attractivity, under some additional assumptions. In addition, we prove boundedness in mean of the solutions of perturbed systems under open-loop protocols. For the feedback strategies, in the general case without assumptions on invariance, we provide bounds on the perturbation effect in expectation and in probability, as well as specific bounds for non-demolition nominal systems.

A general approach to obtain reduced models for a wide class of discrete-time quantum systems is proposed. The obtained models not only reproduce exactly the output of a given quantum model, but are also guaranteed to satisfy physical constraints, namely complete positivity and preservation of total probability. A fundamental framework for exact model reduction of quantum systems is constructed leveraging on algebraic methods, as well as novel results on quantum conditional expectations in finite-dimensions. The proposed reduction algorithm is illustrated and tested on prototypical examples, including the quantum walk realizing Grover's algorithm., Comment: 16 pages

We consider the problem of deciding whether a given state preparation, i.e., a source of quantum states, is accurate, namely produces states close to a target one within a prescribed threshold. We show that, when multiple measurements need to be used, the order of measurements is critical for quickly assessing accuracy. We propose and compare different strategies to compute optimal or suboptimal measurement sequences either relying solely on a priori information, i.e., the target state for state preparation, or actively adapting the sequence to the previously obtained measurements. Numerical simulations show that the proposed algorithms reduce significantly the number of measurements needed for verification, and indicate an advantage for the adaptive protocol especially assessing faulty preparations.

Switching controlled dynamics allows for fast, flexible control design methods for quantum stabilization of pure states and subspaces, which naturally include both Hamiltonian and dissipative control actions. A novel approach to measurement-based, dissipative feedback design is introduced, and extends the applicability of switching techniques with respect to previously proposed ones, as it does not need stringent invariance assumptions, while it still avoids undesired chattering or Zeno effects by modulating the control intensity. When the switching dynamics do leave the target invariant, on the other hand, we show that exponential convergence to the target can be enforced without modulation, and switching times that can be either fixed or stochastic with hysteresis to avoid chattering. The effectiveness of the proposed methods is illustrated via numerical simulations of simple yet paradigmatic examples, demonstrating how switching strategies converge faster than open-loop engineered dissipation., Comment: 27 pages, 3 figures

The problem of reducing a Hidden Markov Model (HMM) to one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach. Realization theory tools are extended to HMMs by leveraging suitable algebraic representations of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that exactly reproduce the single-time distribution of a given output process, while in the second the full (multi-time) distribution is preserved. The reduction method exploits not only the structure of the observed output, but also its initial condition, whenever the latter is known or belongs to a given subclass. Optimal algorithms are derived for a class of HMM, namely observable ones., Comment: 16 pages, no figures

We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction. As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev's toric code., Comment: 29 pages, 4 figures

We propose a new method for pure-state and subspace preparation in quantum systems, which employs the output of a continuous measurement process and switching dissipative control to improve convergence speed, as well as robustness with respect to the initial conditions. In particular, we prove that the proposed closed-loop strategy makes the desired target globally asymptotically stable both in mean and almost surely, and we show it compares favorably against a time-based and a state-based switching control law, with significant improvements in the case of faulty initialization., Comment: 6 pages, 4 figures, updated main results and proofs

We consider the problem of characterizing states of a multipartite quantum system from restricted, quasi-local information, with emphasis on uniquely determined pure states. By leveraging tools from dissipative quantum control theory, we show how the search for states consistent with an assigned list of reduced density matrices may be restricted to a proper subspace, which is determined solely by their supports. The existence of a quasi-local observable which attains its unique minimum over such a subspace further provides a sufficient criterion for a pure state to be uniquely determined by its reduced states. While the condition that a pure state is uniquely determined is necessary for it to arise as a non-degenerate ground state of a quasi-local Hamiltonian, we prove the opposite implication to be false in general, by exhibiting an explicit analytic counterexample. We show how the problem of determining whether a quasi-local parent Hamiltonian admitting a given pure state as its unique ground state is dual, in the sense of semidefinite programming, to the one of determining whether such a state is uniquely determined by the quasi-local information. Failure of this dual program to attain its optimal value is what prevents these two classes of states to coincide., Comment: 17 pages, 1 figure

We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer an open question on how the graph topology ultimately bounds their mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks., Comment: main paper: 5 pages, 2 figures. supplemental material: 9 pages, 1 figure. comments welcome

We investigate whether a generic multipartite pure state can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the simplest tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems' dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N-partite case, we provide conditions on the subsystems' size and the nature of the locality constraint, under which random pure states cannot be quasi-locally stabilized generically. Beside allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic, our results offer insights into the extent to which random pure states may arise as unique ground states of frustration free parent Hamiltonians. We further argue that, to high probability, pure quantum states sampled from a t-design enjoy the same stabilizability properties of Haar-random ones as long as suitable dimension constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a connection between the tasks of quasi-local state stabilization and unique state reconstruction from local tomographic information, and provide a constructive procedure for determining a generic N-partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties, improving over existing results., Comment: 36 pages (including appendix), 2 figures

Isoperimetric inequalities form a very intuitive yet powerful characterization of the connectedness of a state space, that has proven successful in obtaining convergence bounds. Since the seventies they form an essential tool in differential geometry, graph theory and Markov chain analysis. In this paper we use isoperimetric inequalities to construct a bound on the convergence time of any local probabilistic evolution that leaves its limit distribution invariant. We illustrate how this general result leads to new bounds on convergence times beyond the explicit Markovian setting, among others on quantum dynamics., Comment: 8 pages, 2 figures

Lifted Markov chains are Markov chains on graphs with added local "memory" and can be used to mix towards a target distribution faster than their memoryless counterparts. Upper and lower bounds on the achievable performance have been provided under specific assumptions. In this paper, we analyze which assumptions and constraints are relevant for mixing, and how changing these assumptions affects those bounds. Explicitly, we find that requesting mixing on either the original or the full lifted graph, and allowing for reducible lifted chains or not, have no essential influence on mixing time bounds. On the other hand, allowing for suitable initialization of the lifted dynamics and/or imposing invariance of the target distribution for any initialization do significantly affect the convergence performance. The achievable convergence speed for a lifted chain goes from diameter-time to no acceleration over a standard Markov chain, with conductance bounds limiting the effectiveness of the intermediate cases. In addition, we show that the relevance of imposing ergodic flows depends on the other criteria. The presented analysis allows us to clarify in which scenarios designing lifted dynamics can lead to better mixing, and provide a flexible framework to compare lifted walks with other acceleration methods., Comment: 15 pages, 5 figures. Comments welcome

A variety of tasks in quantum control, ranging from purification and cooling, to quantum stabilization and open-system simulation, rely on the ability to implement a target quantum channel over a specified time interval within prescribed accuracy. This can be achieved by engineering a suitable unitary dynamics of the system of interest along with its environment -- which, depending on the available level of control, is fully or partly exploited as a coherent quantum controller. After formalizing a controllability framework for completely positive trace-preserving quantum dynamics, we provide sufficient conditions on the environment state and dimension that allow for the realization of relevant classes of quantum channels -- including extreme channels, stochastic unitaries, or simply any channel. The results hinge on generalizations of Stinespring's dilation via a subsystem principle. In the process, we show that a conjecture by Lloyd on the minimal dimension of the environment required for arbitrary open-system simulation, albeit formally disproved, can in fact be salvaged -- provided that classical randomization is included among the available resources. Existing measurement-based feedback protocols for universal simulation, dynamical decoupling, and dissipative state preparation are recast within the proposed coherent framework as concrete applications, and the resources they employ discussed in the light of the general results., Comment: 27 pages, 5 figures

Preparation of pure states on networks of quantum systems by controlled dissipative dynamics offers important advantages with respect to circuit-based schemes. Unlike in continuous-time scenarios, when discrete-time dynamics are considered, dead-beat stabilization becomes possible in principle. Here, we focus on pure states that can be stabilized by distributed, unsupervised dynamics in finite time on a network of quantum systems subject to realistic quasi-locality constraints. In particular, we define a class of quasi-locality notions, that we name "tree-like hypergraphs," and show that the states that are robustly stabilizable in finite time are then unique ground states of a frustration-free, commuting quasi-local Hamiltonian. A structural characterization of such states is also provided, building on a simple yet relevant example., Comment: 6 pages, 3 figures

Open quantum systems evolving according to discrete-time dynamics are capable, unlike continuous-time counterparts, to converge to a stable equilibrium in finite time with zero error. We consider dissipative quantum circuits consisting of sequences of quantum channels subject to specified quasi-locality constraints, and determine conditions under which stabilization of a pure multipartite entangled state of interest may be exactly achieved in finite time. Special emphasis is devoted to characterizing scenarios where finite-time stabilization may be achieved robustly with respect to the order of the applied quantum maps, as suitable for unsupervised control architectures. We show that if a decomposition of the physical Hilbert space into virtual subsystems is found, which is compatible with the locality constraint and relative to which the target state factorizes, then robust stabilization may be achieved by independently cooling each component. We further show that if the same condition holds for a scalable class of pure states, a continuous-time quasi-local Markov semigroup ensuring rapid mixing can be obtained. Somewhat surprisingly, we find that the commutativity of the canonical parent Hamiltonian one may associate to the target state does not directly relate to its finite-time stabilizability properties, although in all cases where we can guarantee robust stabilization, a (possibly non-canonical) commuting parent Hamiltonian may be found. Beside graph states, quantum states amenable to finite-time robust stabilization include a class of universal resource states displaying two-dimensional symmetry-protected topological order, along with tensor network states obtained by generalizing a construction due to Bravyi and Vyalyi. Extensions to representative classes of mixed graph-product and thermal states are also discussed., Comment: 20 + 9 pages, 9 figures

We study sequences (both cyclic and randomized) of idempotent completely-positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods. We characterize the robustness features of the protocol against randomization and provide basic bounds on its convergence speed. The general results are then specialized to stabilizing en- tangled states in finite-dimensional multipartite quantum systems subject to a resource constraint, a problem of key interest for quantum information applications. We conclude by suggesting further developments, including techniques to enlarge the set of stabilizable states and ensure efficient, finite-time preparation., Comment: 12 pages, no figures

We study the stability of quantum pure states and, more generally, subspaces for stochastic dynamics that describe continuously--monitored systems. We show that the target subspace is almost surely invariant if and only if it is invariant for the average evolution, and that the same equivalence holds for the global asymptotic stability. Moreover, we prove that a strict linear Lyapunov function for the average evolution always exists, and latter can be used to derive sharp bounds on the Lyapunov exponents of the associated semigroup. Nonetheless, we also show that taking into account the measurements can lead to an improved bound on stability rate for the stochastic, non-averaged dynamics. We discuss explicit examples where the almost sure stability rate can be made arbitrary large while the average one stays constant., Comment: Structure and style update after review

Recently, consensus-type problems have been formulated in the quantum domain. Obtaining average quantum consensus consists in the dynamical symmetrization of a multipartite quantum system while preserving the expectation of a given global observable. In this paper, two improved ways of obtaining consensus via dissipative engineering are introduced, which employ on quasi local preparation of mixtures of symmetric pure states, and show better performance in terms of purity dynamics with respect to existing algorithms. In addition, the first method can be used in combination with simple control resources in order to engineer pure Dicke states, while the second method guarantees a stronger type of consensus, namely single-measurement consensus. This implies that outcomes of local measurements on different subsystems are perfectly correlated when consensus is achieved. Both dynamics can be randomized and are suitable for feedback implementation., Comment: 11 pages, 3 figures

We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, which is naturally motivated by consistency with pure-state stabilization results yet provably not necessary in general. Several applications are discussed, of relevance to both dissipative quantum engineering and information processing, and non-equilibrium quantum statistical mechanics. In particular, we show that a large class of graph product states (including arbitrary thermal graph states) as well as Gibbs states of commuting Hamiltonians are frustration-free stabilizable relative to natural quasi-locality constraints. Likewise, we provide explicit examples of non-commuting Gibbs states and non-trivially entangled mixed states that are stabilizable despite the lack of an underlying commuting structure, albeit scalability to arbitrary system size remains in this case an open question., Comment: 44 pages, main results are improved, several proofs are more streamlined, application section is refined

Control strategies for dissipative preparation of target quantum states, both pure and mixed, and subspaces are obtained by switching between a set of available semigroup generators. We show that the class of problems of interest can be recast, from a control--theoretic perspective, into a switched-stabilization problem for linear dynamics. This is attained by a suitable affine transformation of the coherence-vector representation. In particular, we propose and compare stabilizing time-based and state-based switching rules for entangled state preparation, showing that the latter not only ensure faster convergence with respect to non-switching methods, but can designed so that they retain robustness with respect to initialization, as long as the target is a pure state or a subspace., Comment: 15 pages, 4 figures

We investigate convergence properties of discrete-time semigroup quantum dynamics, including asymptotic stability, probability and speed of convergence to pure states and subspaces. These properties are of interest in both the analysis of uncontrolled evolutions and the engineering of controlled dynamics for quantum information processing. Our results include two Hilbert space decompositions that allow for deciding the stability of the subspace of interest and for estimating of the speed of convergence, as well as a formula to obtain the limit probability distribution for a set of orthogonal invariant subspaces., Comment: 14 pages, no figures, to appear in Journal of Physics A, 2015

Preparing a quantum system in a pure state is ultimately limited by the nature of the system's evolution in the presence of its environment and by the initial state of the environment itself. We show that, when the system and environment are initially uncorrelated and arbitrary joint unitary dynamics is allowed, the system may be purified up to a certain (possibly arbitrarily small) threshold if and only if its environment, either natural or engineered, contains a "virtual subsystem" which has the same dimension and is in a state with the desired purity. Beside providing a unified understanding of quantum purification dynamics in terms of a "generalized swap process," our results shed light on the significance of a no-go theorem for exact ground-state cooling, as well as on the quantum resources needed for achieving an intended purification task., Comment: 10 pages, 2 figures

The minimum Kullback entropy principle (mKE) is a useful tool to estimate quantum states and operations from incomplete data and prior information. In general, the solution of a mKE problem is analytically challenging and an approximate solution has been proposed and employed in different context. Recently, the form and a way to compute the exact solution for finite dimensional systems has been found, and a question naturally arises on whether the approximate solution could be an effective substitute for the exact solution, and in which regimes this substitution can be performed. Here, we provide a systematic comparison between the exact and the approximate mKE solutions for a qubit system when average data from a single observable are available. We address both mKE estimation of states and weak Hamiltonians, and compare the two solutions in terms of state fidelity and operator distance. We find that the approximate solution is generally close to the exact one unless the initial state is near an eigenstate of the measured observable. Our results provide a rigorous justification for the use of the approximate solution whenever the above condition does not occur, and extend its range of application beyond those situations satisfying the assumptions used for its derivation., Comment: 9 pages, 6 figures

This work interprets and generalizes consensus-type algorithms as switching dynamics leading to symmetrization of some vector variables with respect to the actions of a finite group. We show how the symmetrization framework we develop covers applications as diverse as consensus on probability distributions (either classical or quantum), uniform random state generation, and open-loop disturbance rejection by quantum dynamical decoupling. Robust convergence results are explicitly provided in a group-theoretic formulation, both for deterministic and for randomized dynamics. This indicates a way to directly extend the robustness and randomization properties of consensus-type algorithms to more fields of application.

We propose constructive approaches for the optimization of binary classical communication over a general noisy qubit quantum channel, for both the error probability and the classical capacity functionals. After showing that the optimal measurements are always associated to orthogonal projections, we construct a parametrization of the achievable transition probabilities via the coherence vector representation. We are then able to rewrite the problem in a form that can be solved by standard, efficient numerical algorithms and provides insights on the form of the solutions., Comment: 13 pages, 7 figures, figures rearrangement, clearer formalization of the procedure for the evaluation of the region of transition probabilities, minor changes

We characterize and construct time-independent Markovian dynamics that drive a finite-dimensional multipartite quantum system into a target (pure) entangled steady state, subject to physical locality constraints. In situations where the desired stabilization task can not be attained solely based on local dissipative means, we allow for local Hamiltonian control or, if the latter is not an option, we suitably restrict the set of admissible initial states. In both cases, we provide algorithms for constructing a master equation that achieves the intended objective and show how this can genuinely extend the manifold of stabilizable states. In particular, we present quasi-local control protocols for dissipatively engineering multipartite GHZ "cat" states and W states on $n$ qubits. For GHZ states, we find that no scalable procedure exists for achieving stabilization from arbitrary initial states, whereas this is possible for a target W state by a suitable combination of a two-body Hamiltonian and dissipators. Interestingly, for both entanglement classes, we show that quasi-local stabilization may be scalably achieved conditional to initialization of the system in a large, appropriately chosen subspace., Comment: 18 pages, no figures

This paper extends the consensus framework, widely studied in the literature on distributed computing and control algorithms, to networks of quantum systems. We define consensus situations on the basis of invariance and symmetry properties, finding four different probabilistic generalizations of classical consensus states. We then extend the gossip consensus algorithm to the quantum setting and prove its convergence properties, showing how it converges to symmetric states while preserving the expectation of permutation-invariant global observables., Comment: 14 pages, 2 figures

We propose a general framework for solving quantum state estimation problems using the minimum relative entropy criterion. A convex optimization approach allows us to decide the feasibility of the problem given the data and, whenever necessary, to relax the constraints in order to allow for a physically admissible solution. Building on these results, the variational analysis can be completed ensuring existence and uniqueness of the optimum. The latter can then be computed by standard, efficient standard algorithms for convex optimization, without resorting to approximate methods or restrictive assumptions on its rank., Comment: 9 pages, no figures

The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies., Comment: tutorial/review paper, 20 pages, 1 figure. Included in the IEEE Trans. Aut. Contr. special issue on quantum control

Single photon source represent a fundamental building block for optical implementations of quantum information tasks ranging from basic tests of quantum physics to quantum communication and high-resolution quantum measurement. In this paper we investigate the performance of a multiplexed system based on asymmetric configuration of multiple heralded single photon sources. {To compare the effectiveness of different designs we introduce a single-photon source performance index that is based on the value of single photon probability required to achieve a guaranteed signal to noise ratio.} The performance and scalability comparison with both currently existing multiple-source architectures and faint laser configurations reveals an advantage the proposed scheme offers in realistic scenarios. This analysis also provides insights on the potential of using such architectures for integrated implementation., Comment: 11 pages, 13 figures

We propose a general scheme for dissipatively preparing arbitrary pure quantum states on a multipartite qubit register in a finite number of basic control blocks. Our "splitting-subspace" approach relies on control resources that are available in a number of scalable quantum technologies (complete unitary control on the target system, an ancillary resettable qubit and controlled-not gates between the target and the ancilla), and can be seen as a "quantum-controller" implementation of a sequence of classical feedback loops. We show how a large degree of flexibility exists in engineering the required conditional operations, and make explicit contact with a stabilization protocol used for dissipative quantum state preparation and entanglement generation in recent experiments with trapped ions., Comment: 10 pages, 2 figures. Submitted to CDC 2012

We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under fixed locality constraints. Beside recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law., Comment: 11 pages, no figures

In this paper we investigate parametrization-free solutions of the problem of quantum pure state preparation and subspace stabilization by means of Hamiltonian control, continuous measurement and quantum feedback, in the presence of a Markovian environment. In particular, we show that whenever suitable dissipative effects are induced either by the unmonitored environment or by non Hermitian measurements, there is no need for feedback control to accomplish the task. Constructive necessary and sufficient conditions on the form of the open-loop controller can be provided in this case. When open-loop control is not sufficient, filtering-based feedback control laws steering the evolution towards a target pure state are provided, which generalize those available in the literature.

We consider finite-dimensional Markovian open quantum systems, and characterize the extent to which time-independent Hamiltonian control may allow to stabilize a target quantum state or subspace and optimize the resulting convergence speed. For a generic Lindblad master equation, we introduce a dissipation-induced decomposition of the associated Hilbert space, and show how it serves both as a tool to analyze global stability properties for given control resources and as the starting point to synthesize controls that ensure rapid convergence. The resulting design principles are illustrated in realistic Markovian control settings motivated by quantum information processing, including quantum-optical systems and nitrogen-vacancy centers in diamond., Comment: 13 pages, 4 figures

Controllability properties for discrete-time, Markovian quantum dynamics are investigated. We find that, while in general the controlled system is not finite-time controllable, feedback control allows for arbitrary asymptotic state-to-state transitions. Under further assumption on the form of the measurement, we show that finite-time controllability can be achieved in a time that scales linearly with the dimension of the system, and we provide an iterative procedure to design the unitary control actions.

We introduce the notion of trace-norm isometric encoding and explore its implications for passive and active methods to protect quantum information against errors. Beside providing an operational foundations to the "subsystems principle" [E. Knill, Phys. Rev. A 74, 042301 (2006)] for faithfully realizing quantum information in physical systems, our approach allows additional explicit connections between noiseless, protectable, and correctable quantum codes to be identified. Robustness properties of isometric encodings against imperfect initialization and/or deviations from the intended error models are also analyzed., Comment: 10 pages, 1 figure

Based on recent work on the asymptotic behavior of controlled quantum Markovian dynamics, we show that any generic quantum state can be stabilized by devising constructively a simple Lindblad-GKS generator that can achieve global asymptotic stability at the desired state. The applications of such result is demonstrated by designing a direct feedback strategy that achieves global stabilization of a qubit state encoded in a noise-protected subspace., Comment: Revised version with stronger proofs showing uniqueness can be achieved in all cases by using the freedom to the choose diagonal elements of both the Hamiltonian and Lindblad operator, and exploiting the fact that the non-existence of two orthogonal eigenvectors of the Lindblad operator is sufficient but not necessary for global asymptotic stability of the target state

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a` la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler pseudo-distance, which gives rise to a convex optimization problem. After developing the variational analysis, we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem, based on the iteration of a nonlinear map in a bounded subset of the dual space. Our main result is the proof of local convergence of the latter, established as a consequence of the Central Manifold Theorem. Supported by numerical evidence, we conjecture that, in the mentioned bounded set, the convergence is actually global.

We analyze the asymptotic behavior of discrete-time, Markovian quantum systems with respect to a subspace of interest. Global asymptotic stability of subspaces is relevant to quantum information processing, in particular for initializing the system in pure states or subspace codes. We provide a linear-algebraic characterization of the dynamical properties leading to invariance and attractivity of a given quantum subspace. We then construct a design algorithm for discrete-time feedback control that allows to stabilize a target subspace, proving that if the control problem is feasible, then the algorithm returns an effective control choice. In order to prove this result, a canonical QR matrix decomposition is derived, and also used to establish the control scheme potential for the simulation of open-system dynamics., Comment: 12 pages, 1 figure

The time reversal of a completely-positive, nonequilibrium discrete-time quantum Markov evolution is derived via a suitable adjointness relation. Space-time harmonic processes are introduced for the forward and reverse-time transition mechanisms, and their role for relative entropy dynamics is discussed., Comment: 19 pages