Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are ``as random as a coin toss.'' Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power
ep(U) of the basic two-particle unitary building block,
U, of the circuit that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally, we show, both analytically and numerically, how local averaging over random realizations of the single-particle unitaries
ui and
vi such that the building block is ${U}^{\ensuremath{'}}=({u}_{1}\ensuremath{\bigotimes}{u}_{2})U({v}_{1}\ensuremath{\bigotimes}{v}_{2})$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power
ep(U). Finally, we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map, which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.
We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.
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.
Understanding the far-from-equilibrium dynamics of dissipative quantum systems, where dissipation and decoherence coexist with unitary dynamics, is an enormous challenge with immense rewards. Often, the only realistic approach is to forgo a detailed microscopic description and search for signatures of universal behavior shared by collections of many distinct, yet sufficiently similar, complex systems. Quantum chaos provides a powerful statistical framework for addressing this question, relying on symmetries to obtain information not accessible otherwise. This thesis examines how to reconcile chaos with dissipation, proceeding along two complementary lines. In Part I, we apply non-Hermitian random matrix theory to open quantum systems with Markovian dissipation and discuss the relaxation timescales and steady states of three representative examples of increasing physical relevance: single-particle Lindbladians and Kraus maps, open free fermions, and dissipative Sachdev-Ye-Kitaev (SYK) models. In Part II, we investigate the symmetries, correlations, and universality of many-body open quantum systems, classifying several models of dissipative quantum matter. From a theoretical viewpoint, this thesis lays out a generic framework for the study of the universal properties of realistic, chaotic, and dissipative quantum systems. From a practical viewpoint, it provides the concrete building blocks of dynamical dissipative evolution constrained by symmetry, with potential technological impact on the fabrication of complex quantum structures. (Full abstract in the thesis.)
Abstract Bang–bang dynamical decoupling protects an open quantum system from decoherence due to its interaction with the surrounding bath/environment. In its standard form, this is achieved by strongly kicking the system with cycles of unitary operations, which average out the interaction Hamiltonian. In this paper, we generalize the notion of dynamical decoupling to repeated kicks with a quantum channel, which is applied to the bath. We derive necessary and sufficient conditions on the employed quantum channel and find that bath dynamical decoupling works if and only if the kick is ergodic. Furthermore, we study in which circumstances completely positive trace-preserving (CPTP) kicks on a mono-partite quantum system induce quantum Zeno dynamics with its Hamiltonian cancelled out. This does not require the ergodicity of the kicks, and the absence of decoherence-free subsystems is both necessary and sufficient. While the standard unitary dynamical decoupling is essentially the same as the quantum Zeno dynamics, our investigation implies that this is no longer true in the case of CPTP kicks. To derive our results, we prove some spectral properties of ergodic quantum channels, that might be of independent interest. Our approach establishes an enhanced and unified mathematical understanding of several recent experimental demonstrations and might form the basis of new dynamical decoupling schemes that harness environmental noise degrees of freedom.