One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's Dicke. In this paper, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state. The combination of representation theory and entropy estimates could be a guideline to study more general collective noise models.
One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's Dicke. In this paper, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state. The combination of representation theory and entropy estimates could be a guideline to study more general collective noise models.
We propose a conceptually simple method for proving various entropic inequalities by means of spectral estimates. Our approach relies on a two-sided estimate of the relative entropy between two quantum states in terms of certain variance-type quantities. As an application of this estimate, we answer three important problems in quantum information theory and dissipative quantum systems that were left open until now: first, we prove the existence of strictly contractive constants for the strong data processing inequality for quantum channels that satisfy a notion of detailed balance. Second, we prove the existence of the complete modified logarithmic Sobolev constant which controls the exponential entropic convergence of a quantum Markov semigroup towards its equilibrium. Third, we prove a new generalization of the strong subadditivity of the entropy for the relative entropy distance to a von Neumann algebra under a specific gap condition. All the three results obtained are independent of the size of the environment.
States of open quantum systems often decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that finite-dimensional quantum Markov semigroups with GNS detailed balance universally obey complete modified logarithmic Sobolev inequalities (CMLSIs), yielding exponential decay of relative entropy to a subspace of fixed point states. We analyze continuous processes that combine dissipative with Hamiltonian time-evolution, precluding this notion of detailed balance. First, we find counterexamples to CMLSI-like decay for these processes and determine conditions under which it fails. In contrast, we prove that despite its absence at early times, exponential decay re-appears for unital, finite-dimensional quantum Markov semigroups at finite timescales. Finally, we show that when dissipation is much stronger than Hamiltonian time-evolution, the rate of eventual, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone. Dubbed self-restricting noise, this inverse relationship arises when strong damping suppresses effects that would otherwise spread noise beyond its initial subspace.
We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.
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.
In a quantum many-body system, such as atoms, molecules, and atomic nuclei, interparticle interactions generate a complex quantum level scheme that can be analyzed progressively: from the simplest picture of independent quasiparticles moving in a self-consistent mean field to more complete descriptions that include residual interactions and the opening of decay channels above some excitation thresholds. Residual interactions lead to two seemingly contrasting phenomena: the emergence of collective modes (such as rotational and vibrational excitations) associated with specific symmetries, and internal thermalization, which gives rise to features of quantum chaos and causes these collective modes to damp. The picture becomes more intricate once decay channels become energetically accessible, allowing the system to decay irreversibly by means such as particle emission and thus transition to an open quantum regime. In this work, we investigate the competition between internal many-body dynamics and explicit, irreversible decay. These processes can occur on comparable timescales, leading to an initial stage of rapid dynamics followed by equilibration. We show that the decay of the faster components drives the system into an exponentially decaying regime, characterized by a noisy decay current. For this study, we adopt a general, albeit simplified, model of a collective mode embedded in a chaotic background and coupled to open decay channels. The interplay of timescales revealed in this framework appears to reflect universal features of open quantum systems.
One of the greatest challenges in the fields of quantum information processing and quantum technologies is the detailed coherent control over each and every constituent of quantum systems with an ever increasing number of particles. Within this endeavor, harnessing of many-body entanglement against the detrimental effects of the environment is a major pressing issue. Besides being an important concept from a fundamental standpoint, entanglement has been recognized as a crucial resource for quantum speed-ups or performance enhancements over classical methods. Understanding and controlling many-body entanglement in open systems may have strong implications in quantum computing, quantum simulations of many-body systems, secure quantum communication or cryptography, quantum metrology, our understanding of the quantum-to-classical transition, and other important questions of quantum foundations.In this paper we present an overview of recent theoretical and experimental efforts to underpin the dynamics of entanglement under the influence of noise. Entanglement is thus taken as a dynamic quantity on its own, and we survey how it evolves due to the unavoidable interaction of the entangled system with its surroundings. We analyze several scenarios, corresponding to different families of states and environments, which render a very rich diversity of dynamical behaviors.In contrast to single-particle quantities, like populations and coherences, which typically vanish only asymptotically in time, entanglement may disappear at a finite time. In addition, important classes of entanglement display an exponential decay with the number of particles when subject to local noise, which poses yet another threat to the already-challenging scaling of quantum technologies. Other classes, however, turn out to be extremely robust against local noise. Theoretical results and recent experiments regarding the difference between local and global decoherence are summarized. Control and robustness-enhancement techniques, scaling laws, statistical and geometrical aspects of multipartite-entanglement decay are also reviewed; all in order to give a broad picture of entanglement dynamics in open quantum systems addressed to both theorists and experimentalists inside and outside the field of quantum information.