Self-restricting Noise in Quantum Dynamics

States of open quantum systems usually decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that a finite-dimensional quantum Markov semigroup with detailed balance induces exponential decay toward a subspace of invariant or fully decayed states, under what are called modified logarithmic Sobolev inequalities. We analyze continuous processes that combine coherent and stochastic processes, breaking detailed balance. We find counterexamples to analogous decay bounds for these processes. Through analogs of the quantum Zeno effect, noise can suppress interactions that would spread it. Faster decay of a subsystem may thereby slow overall decay. Hence the relationship between the strength of noise on a part and induced decay on the whole system is often non-monotonic. We observe this interplay numerically and its discrete analog experimentally on IBM Q systems. Our main results then explain and generalize the phenomenon theoretically. In contrast, we also lower bound decay rates above any given timescale by combining estimates for simpler, effective processes across times.Comment: 40 pages, 7 figures. Update removes some content on discrete compositions of channels, revises and corrects some mathematical content, and generalizes some main theorem

On the nature and decay of quantum relative entropy

Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features. An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty. We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity. Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels

Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy

The relative entropy of a quantum density matrix to a subalgebraic restriction appears throughout quantum information. For subalgebra restrictions given by commuting conditional expectations in tracial settings, strong subadditivity shows that the sum of relative entropies to each is at least as large as the relative entropy to the intersection subalgebra. When conditional expectations do not commute, an inequality known as quasi-factorization or approximate tensorization replaces strong subadditivity. Multiplicative or strong quasi-factorization yields relative entropy decay estimates known as modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents. In this work, we show multiplicative comparisons between subalgebra-relative entropy and its perturbation by a quantum channel with corresponding fixed point subalgebra. Following, we obtain a strong quasi-factorization inequality with constant scaling logarithmically in subalgebra index. For conditional expectations that nearly commute and are not too close to a set with larger intersection algebra, the shown quasi-factorization is asymptotically tight in that the constant approaches one. We apply quasi-factorization to uncertainty relations between incompatible bases and to conditional expectations arising from graphs.Comment: 29 pages, 1 figure; updated to reflect recent developments and resulting improvement

Stability of logarithmic Sobolev inequalities under a noncommutative change of measure

We generalize Holley-Stroock's perturbation argument from commutative to quantum Markov semigroups. As a consequence, results on (complete) modified logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for self-adjoint quantum Markov process can be used to prove estimates on the exponential convergence in relative entropy of quantum Markov systems which preserve a fixed state. This leads to estimates for the decay to equilibrium for coupled systems and to estimates for mixed state preparation times using Lindblad operators. Our techniques also apply to discrete time settings, where we show that the strong data processing inequality constant of a quantum channel can be controlled by that of a corresponding unital channel.Comment: 26 page

Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups

Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as specific norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this article, we introduce a class of continuous-time quantum channels that we called transferred channels, which are built through representation theory from a classical Markov kernel defined on a compact group. We study two subclasses of such kernels: H\"ormander systems on compact Lie-groups and Markov chains on finite groups. Examples of transferred channels include the depolarizing channel, the dephasing channel, and collective decoherence channels acting on $d$ qubits. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We then extend tools developed in earlier work by Gao, Junge and LaRacuente to transfer estimates of the classical Markov kernel to the transferred channels and study in this way different non-commutative functional inequalities. The main contribution of this article is the application of this transference principle to the estimation of various capacities as well as estimation of entanglement breaking times, defined as the first time for which the channel becomes entanglement breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an important scenario that has been overlooked so far because of a lack of techniques.Comment: 35 pages, 2 figures. Close to published versio

Entropy Uncertainty Relations and Strong Sub-additivity of Quantum Channels

We prove an entropic uncertainty relation for two quantum channels, extending the work of Frank and Lieb for quantum measurements. This is obtained via a generalized strong super-additivity (SSA) of quantum entropy. Motivated by Petz's algebraic SSA inequality, we also obtain a generalized SSA for quantum relative entropy. As a special case, it gives an improved data processing inequality.Comment: 33 pages. Comments are welcome