Abstract We consider a new type of numerical range motivated by recent applications in quantum computing. We term the object of interest local C ‐numerical rangeWloc(C, A) of A. It is obtained by replacing the special unitary group in the definition of the C ‐numerical range by the so‐called local subgroup of SU (2N ), i.e. by the N ‐fold tensor product SU (2) ⊗ · · · ⊗ SU(2) of unitary (2 × 2)‐matrices. First, it is shown that the local C ‐numerical range has rather unusual geometric properties compared to the ordinary one, e.g. it is in general neither star‐shaped nor simply connected. Then two numerical algorithms, a Newton and a conjugate gradient method on the Lie group SU (2) ⊗ · · · ⊗ SU (2), are demonstrated to maximize the real part of Wloc(C, A) which also gives a Euclidean measure of the so‐called pure‐state entanglement in quantum computing. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
This article shows how C-numerical-range related new strucures may arise from practical problems in quantum control – and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function . In quantum control of n qubits one may be interested (i) in having U∈SU(2 n ) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e., to the n-fold tensor product . Interestingly, the latter then leads to a novel entity, the local C-numerical range W loc(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying article on Relative C-Numerical Ranges for Application in Quantum Control and Quantum Information [Dirr, G., Helmke, U., Kleinsteuber, M. and Schulte-Herbrüggen, T., 2008 Dirr, G, Helmke, U, Kleinsteuber, M and Schulte-Herbrüggen, T. 2008. Relative C-numerical ranges for application in quantum control and quantum information. Linear and Multilinear Algebra, 56: 27–51. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], Linear and Multilinear Algebra, 56, 27–51]. We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. (3) We conclude by connecting the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient-flow algorithms.
Motivated by applications in quantum computing and quantum control, we consider the task of maximizing the trace function Re(tr (CUAU)) on the Lie group of N fold tensor products of special unitary (2×2)-transformations. In order to approach this highly non-convex optimization problem, a new object, the local C-numerical range Wloc(C, A), is introduced and its geometry is studied. We first present examples that illustrate the rather complex geometric structure of the local C-numerical range. It is shown that, in contrast to the ordinary C-numerical range, Wloc(C, A) is in general neither star-shaped nor simply connected. The equivalence of finding bounds on the size of Wloc(C, A) and maximizing the above trace function is derived. We then describe and analyse two intrinsic optimization methods to tackle these problems: (a) a gradient flow with an Euler step discretization scheme and (b) a Jacobi-type algorithm. Explicit step size selections are given for which the gradient algorithm converges to the set of critical points. Finally, numerical experiments are presented.
Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is sufficiently general for setting up gradient flows on (sub)manifolds, Lie (sub)groups, and (reductive) homogeneous spaces. Relevant convergence conditions are discussed, in particular for gradient flows on compact and analytic manifolds. This is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SUloc(2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SUloc(2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.
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.
The interplay between the quantum state space and a specific set of measurements can be effectively captured by examining the set of jointly attainable expectation values. This set is commonly referred to as the (convex) joint numerical range. In this work, we explore geometric properties of this construct for measurements represented by tensor products of Pauli observables, also known as Pauli strings. The structure of pairwise commutation and anticommutation relations among a set of Pauli strings determines a graph
G, sometimes also called the frustration graph. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range. Such an outer approximation can be very practical since ellipsoids can be handled analytically even in high dimensions. We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O. Gühne, arXiv:2207.02197], and answer an open question in [M. B. Hastings and R. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graph parameter that we call
β(G). Besides, we develop this approach in different directions, such as comparison with graph-theoretic approaches in other fields, applications in quantum information theory, numerical methods, properties of the new graph parameter, etc. Our approach suggests many open questions that we discuss briefly at the end.