Conventional approaches for controlling open quantum systems use coherent control which affects the system's evolution through the Hamiltonian part of the dynamics. Such control, although being extremely efficient for a large variety of problems, has limited capabilities, e.g., if the initial and desired target states have density matrices with different spectra or if a control field needs to be designed to optimally transfer different initial states to the same target state. Recent research works suggest extending coherent control by including active manipulation of the non-unitary (i.e., incoherent) part of the evolution. This paper summarizes recent results specifically for incoherent control by the environment (e.g., incoherent radiation or a gaseous medium) with a kinematic description of controllability and landscape analysis.
Abstract In this work, we consider a pair of qubits controlled by coherent and incoherent controls. The dynamics of the two-qubit system is driven by a Gorini–Kossakowsky–Sudarchhan–Lindblad master equation where coherent control enters into the Hamiltonian and incoherent control inters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. Two classes of interaction between the system and the coherent field are considered. For this system, we analyze the control problem of generating a given target density matrix which is formulated as minimizing the Hilbert–Schmidt distance between the final density matrix and the target density matrix. Incoherent control is modeled as a sum of constant in time Gaussians with centers related with the transitions frequencies between the energy levels of the qubits. Coherent control in general formulation is considered as measurable function and in numerical experiments as piecewise constant function with constraints on magnitudes and variations. Finite-dimensional numerical optimization is performed using the dual annealing method; the corresponding results are described for some initial and target density matrices and for some set of the parameters of the control problem.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at
2n×2n unitary matrices, which describe operations on a closed
n-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general
2n×2n unitary matrix to the product of no more than
2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element
Φ among a given set of quantum channels
S that will produce the optimal value of a scalar function
D(ρ1,Φ(ρ2)), on two fixed quantum states
ρ1 and
ρ2. Some of the functions we considered for
D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when
S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers
a1≥…≥a2n≥0, we want to determine if there exist a
2n×2n density matrix
ρ having eigenvalues
a1,…,a2n and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on
a1,…,a2n for
n≤6 and state some observations on general values of
n. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that
Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given
ρ(1),…,ρ(k)∈Dn and
σ(1),…,σ(k)∈Dm, (b) constructing a multipartite state
ρ having a prescribed set of reduced states
ρ1,…,ρr on
r of its subsystems, (c) constructing a multipartite state
ρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix
A can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets
K1 and
K2 of $\IC$ given by
K1K2={ab:a∈K1,b∈K2}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that
K1K2 is star-shaped when
K1 and
K2 are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set
K1K2 is star-shaped if
K1 is a line segment or a circular disk.
This paper considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem of generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [48] for generic
N-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage duration can be significantly less than the unmodified first stage duration, but at the cost of performing optimization in the class of piecewise constant controls.
This work considers two-qubit open quantum systems driven by coherent and incoherent controls. Incoherent control induces time-dependent decoherence rates via time-dependent spectral density of the environment which is used as a resource for controlling the system. The system evolves according to the Gorini–Kossakowski– Sudarshan–Lindblad master equation with time-dependent coefficients. For two types of interaction with coherent control, three types of objectives are considered: 1) maximizing the Hilbert–Schmidt overlap between the final and target density matrices; 2) minimizing the Hilbert–Schmidt distance between these matrices; 3) steering the overlap to a given value. For the first problem, we develop the Krotov type methods directly in terms of density matrices with or without regularization for piecewise continuous controls with constaints and find the cases where the methods produce (either exactly or with some precision) zero controls which satisfy the Pontryagin maximum principle and produce the overlap’s values close to their upper bounds. For the problems 2) and 3), we find cases when the dual annealing method steers the objectives close to zero and produces a non-zero control.
This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for generic
N-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system's state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage's duration can be significantly less than the unmodified first stage's duration, but at the cost of performing optimization in the class of piecewise constant controls. Bibliography: 72 titles.