A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks

We consider two fundamental tasks in quantum information theory, data compression with quantum side information as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. We show that these characterizations - in contrast to earlier results - enable us to derive tight second order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree up to logarithmic terms with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our techniques also naturally yields bounds on operational quantities for finite block lengths.Comment: See also arXiv:1208.1400, which independently derives part of our result: the second order asymptotics for binary hypothesis testin

The operational meaning of min- and max-entropy

We show that the conditional min-entropy Hmin(A|B) of a bipartite state rho_AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of rho_AB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy Hmax(A|B) to the maximum fidelity of rho_AB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B

The Operational Meaning of Min- and Max-Entropy

In this paper, we show that the conditional min-entropy $H_{min}(A vert B)$ of a bipartite state $rho_{A B}$ is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the $B$-part of $rho_{A B}$ are allowed. In the special case where $A$ is classical, this overlap corresponds to the probability of guessing $A$ given $B$. In a similar vein, we connect the conditional max-entropy $H_{max}(A vert B)$ to the maximum fidelity of $rho_{AB}$ with a product state that is completely mixed on $A$. In the case where $A$ is classical, this corresponds to the security of $A$ when used as a secret key in the presence of an - adversary holding $B$. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing $A$ given $B$ is a lower bound on the number of uniform secret bits that can be extracted from $A$ relative to an adversary holding $B$

Catalysis in Quantum Information Theory

Catalysts open up new reaction pathways which can speed up chemical reactions while not consuming the catalyst. A similar phenomenon has been discovered in quantum information science, where physical transformations become possible by utilizing a (quantum) degree of freedom that remains unchanged throughout the process. In this review, we present a comprehensive overview of the concept of catalysis in quantum information science and discuss its applications in various physical contexts.Comment: Review paper; Comments and suggestions welcome

Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels

This thesis combines two parallel research directions: an exploration into the continuity properties of certain entropic quantities, and an investigation into a simple class of physical systems whose time evolution is given by the repeated application of a quantum channel. In the first part of the thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions; that is, for real-valued functions which are decreasing in the majorization pre-order. Continuity bounds provide a quantitative measure of robustness, addressing the following question: If there is some uncertainty or error in the input, how much uncertainty is there in the output? Our technique uses a particular relationship between majorization and the trace distance between quantum states (or total variation distance, in the case of probability distributions). Namely, the majorization pre-order attains a maximum and a minimum over ε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance ε, we obtain the path of ``majorization flow’’. An analysis of the derivatives of Schur concave functions along this path immediately yields tight continuity bounds for such functions. In this way, we find a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy, and the necessary and sufficient conditions for its saturation, in a universal framework which extends to the other functions, including the Rényi and Tsallis entropies. In particular, we prove a novel uniform continuity bound for the α-Rényi entropy with α > 1 with much improved dependence on the dimension of the underlying system and the parameter α compared to previously known bounds. We show that this framework can also be used to provide continuity bounds for other Schur concave functions, such as the number of connected components of a certain random graph model as a function of the underlying probability distribution, and the number of distinct realizations of a random variable in some fixed number of independent trials as a function of the underlying probability mass function. The former has been used in modeling the spread of epidemics, while the latter has been studied in the context of estimating measures of biodiversity from observations; in these contexts, our continuity bounds provide quantitative estimates of robustness to noise or data collection errors. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes, i.e. environmental systems, one at a time. The state of the system after each interaction is related to the state of the system before the interaction by the so-called reduced dynamics, which is described by the action of a quantum channel. When each probe and the way it interacts with the system is identical, the reduced dynamics at each step is identical. In this scenario, under the additional assumption that the reduced dynamics satisfies a faithfulness property, we characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference, after finitely many steps. In this case, the reduced dynamics is said to be eventually entanglement-breaking. This investigation helps improve our understanding of which kinds of noisy time evolution destroy entanglement. When the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction. The quantities of interest are then studied as random variables on the space of outcomes of the energy measurements of the probes, providing a deeper insight into the interrelations between energy and entropy in this setting.Cantab Capital Institute for the Mathematics of Informatio

Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma

For quantum spin systems in any spatial dimension with a local, translation-invariant Hamiltonian, we prove that asymptotic state convertibility from a quantum state to another one by a thermodynamically feasible class of quantum dynamics, called thermal operations, is completely characterized by the Kullback-Leibler (KL) divergence rate, if the state is translation-invariant and spatially ergodic. Our proof consists of two parts and is phrased in terms of a branch of the quantum information theory called the resource theory. First, we prove that any states, for which the min and max Rényi divergences collapse approximately to a single value, can be approximately reversibly converted into one another by thermal operations with the aid of a small source of quantum coherence. Second, we prove that these divergences collapse asymptotically to the KL divergence rate for any translation-invariant ergodic state. We show this via a generalization of the quantum Stein's lemma for quantum hypothesis testing beyond independent and identically distributed (i.i.d.) situations. Our result implies that the KL divergence rate serves as a thermodynamic potential that provides a complete characterization of thermodynamic convertibility of ergodic states of quantum many-body systems in the thermodynamic limit, including out-of-equilibrium and fully quantum situations