Abstract Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in operation quantum device best way possible, has evolved into one cornerstones enabling technologies. The last few years have seen rapid evolution expansion field. We review here recent progress our understanding controllability open systems development application control techniques to also address key challenges sketch roadmap future developments.

For monotone systems evolving on the positive orthant of $\mathbb{R}^n_+$ two types Lyapunov functions are considered: Sum- and max-separable functions. One can be written as a sum, other maximum scalar arguments. Several constructive existence results for both given. Notably, one construction provides function that is defined at least an arbitrarily large compact set, based little more than knowledge about trajectory. Another class planar yields global sum-separable function, provided right...

Abstract One of the main theoretical challenges in quantum computing is design explicit schemes that enable one to effectively factorize a given final unitary operator into product basic operators. As this equivalent constructive controllability task on Lie group special operators, faces interesting classes bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In paper we give review recent Lie‐theoretical developments finite‐dimensional play key...

An efficient and robust computational framework for solving closed quantum spin optimal-control exact-controllability problems with control constraints is presented. Closed systems are of fundamental importance in modern technologies such as nuclear magnetic resonance (NMR) spectroscopy, imaging, computing. These modeled by the Liouville--von Neumann master (LvNM) equation describing time evolution density operator representing state system. A unifying setting provided to discuss results....

This paper shows how C-numerical-range related new strucures may arise from practical problems in quantum control--and vice versa, an understanding of these structures helps to tackle hot topics information. We start out with overview on the role C-numerical ranges current research theory: mechanical task maximising projection a point unitary orbit initial state onto target C relates radius A via trace function |\tr \{C^\dagger UAU^\dagger\}|. In control n qubits one be interested (i) having...

Knowledge about to what extent quantum dynamical systems can be steered by coherent controls is indispensable for future developments in technology. The purpose of this paper analyze such controllability aspects finite dimensional bilinear control systems. We use a unified approach based on Lie-algebraic methods from nonlinear theory revisit known and establish new results closed open In particular, we provide simplified characterization different notions described the Liouville-von Neumann...

Separable Lyapunov functions play vital roles, for example, in stability analysis of large-scale systems. A function is called max-separable if it can be decomposed into a maximum with one-dimensional arguments. Similarly, sum-separable sum such functions. In this paper shown that monotone system on compact state space, asymptotic implies existence function. We also construct two systems non-compact which does not exist. One them has The other not.

Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups thermodynamic processes. We characterise a wide class gksl-generators for quantum maps within thermal operations and argue that every infinitesimal generator (a one-parameter semigroup of) Markovian belongs to this class. completely classify visualise them their non-Markovian counterparts case single qubit. Second, use description in framework bilinear control systems reachable sets coherently...

We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian controls and elucidate their geometry in terms of Lie semigroups. For dissipative interactions with the environment different coherent controls, we particularly specify tangent cones (Lie wedges) respective semigroups channels. These are counterpart infinitesimal generator a single one-parameter semigroup. They comprise all directions underlying system can be steered to thus give insight into...

Abstract Motivated by applications in quantum information and control, a new type of C-numerical range, the relative range denoted as WK (C, A), is introduced. It arises upon replacing unitary group U(N) definition classical any its compact connected subgroups K ⊂ U(N). The geometric properties are analyzed detail. Counterexamples prove that geometry more intricate than case: e.g., W A) neither star-shaped nor simply connected. Yet, well-known result on rotational symmetry extends to shown...

We investigate the possibilities of steering probability density functions state variables in linear control systems, using a combination open loop and time-varying output feedback strategies. This is an intrinsically nonlinear problem which makes contact with earlier work by Brockett (2012) on controlling mean variance systems via transformations. extend Brockett's Liouville equation to more difficult case, as well parallel connected systems. Our methods depend certain controllability...

We show that the discrete-time evolution of an open quantum system generated by a single channel T can be embedded in enlarged closed system, i.e., we construct unitary dilation quantum-dynamical semigroup (Tn)n∈N0. In case cyclic T, auxiliary space may chosen (partially) finite-dimensional. further investigate control systems finitely many commuting channels and prove similar result as channel.

In infinite dimensions and on the level of trace-class operators C rather than matrices, we show that closure C-numerical range WC(T) is always star-shaped with respect to set tr⁡(C)We(T), where We(T) denotes essential numerical bounded operator T. Moreover, convex if either normal collinear eigenvalues or T essentially self-adjoint. case compact operators, C-spectrum a subset range, which itself hull C-spectrum. This coincides if, in addition, are collinear.

The solutions to the celebrated Kossakowski-Lindblad equation extended by coherent controls yield Markovian quantum maps. More precisely, set of all its forms a semigroup completely positive trace-preserving maps taking specific form Lie semigroup. Non-trivial symmetries these semigroups are shown preclude accessibility in dissipative systems. This is open-system analogue closed systems, where triviality (quadratic) Hamiltonian part suffices decide that system fully controllable. findings...

We introduce a generalized Rayleigh-quotient $\rho_A$ on the direct product of Grassmannians $\mathrm{Gr}({\bf m},{\bf n})$ enabling unified approach to well-known optimization tasks from different areas numerical linear algebra, such as best low-rank approximations tensors (data compression), geometric measures entanglement (quantum computing), and subspace clustering (image processing). compute Riemannian gradient $\rho_A$, characterize its critical points, prove that they are generically...

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by dynamic system in question. Here we consider infinite-dimensional open dynamical following a unital Kossakowski–Lindblad master equation extended controls. More precisely, their time evolution shall governed inevitable potentially unbounded Hamiltonian drift term H 0 , finitely many bounded control Hamiltonians j allowing for (at least) piecewise constant...

The contribution of this paper is twofold. First a simplified derivation the maximum principle for optimal control problems with delay presented based on first principles. Then result applied to an investigation into dynamic resource allocation problem in population dynamics involving delayed action.

We generalize the C-numerical range WC(T ) from trace-class to Schatten-class operators, i.e. C ∈ B p (H) and T q with 1/p + 1/q = 1, show that its closure is always star-shaped respect origin.For (1, ∞], this equivalent saying of image unitary orbit under any continous linear functional L (B (H)) ′ one has star-shapedness tr(T )We(L), where We(L) denotes essential L.Moreover, WC (T convex if or normal collinear eigenvalues.If are both normal, then C-spectrum a subset range, which itself...