Continuity and Stability of Partial Entropic Sums

Extensions of Fannes' inequality with partial sums of the Tsallis entropy are obtained for both the classical and quantum cases. The definition of kth partial sum under the prescribed order of terms is given. Basic properties of introduced entropic measures and some applications are discussed. The derived estimates provide a complete characterization of the continuity and stability properties in the refined scale. The results are also reformulated in terms of Uhlmann's partial fidelities.Comment: 9 pages, no figures. Some explanatory and technical improvements are made. The bibliography is extended. Detected errors and typos are correcte

Zero-error channel capacity and simulation assisted by non-local correlations

Shannon's theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly, and in the presence of various resources that are all classes of non-signalling correlations: Shared randomness, shared entanglement and arbitrary non-signalling correlations. Specifically, when the channel being simulated is noiseless, this reduces to the zero-error capacity of the channel, assisted by the various classes of non-signalling correlations. When the resource channel is noiseless, it results in the "reverse" problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, 'one-shot' separations between the power of the different assisting correlations are exhibited. The most striking result of this kind is that entanglement can assist in zero-error communication, in stark contrast to the standard setting of communicaton with asymptotically vanishing error in which entanglement does not help at all. In the asymptotic case, shared randomness is shown to be just as powerful as arbitrary non-signalling correlations for noisy channel simulation, which is not true for the asymptotic zero-error capacities. For assistance by arbitrary non-signalling correlations, linear programming formulas for capacity and simulation are derived, the former being equal (for channels with non-zero unassisted capacity) to the feedback-assisted zero-error capacity originally derived by Shannon to upper bound the unassisted zero-error capacity. Finally, a kind of reversibility between non-signalling-assisted capacity and simulation is observed, mirroring the famous "reverse Shannon theorem".Comment: 18 pages, 1 figure. Small changes to text in v2. Removed an unnecessarily strong requirement in the premise of Theorem 1

Multi-partite analysis of average-subsystem entropies

So-called average subsystem entropies are defined by first taking partial traces over some pure state to define density matrices, then calculating the subsystem entropies, and finally averaging over the pure states to define the average subsystem entropies. These quantities are standard tools in quantum information theory, most typically applied in bipartite systems. We shall first present some extensions to the usual bipartite analysis, (including a calculation of the average tangle, and a bound on the average concurrence), follow this with some useful results for tripartite systems, and finally extend the discussion to arbitrary multi-partite systems. A particularly nice feature of tri-partite and multi-partite analyses is that this framework allows one to introduce an "environment" for small subsystems to couple to.Comment: Minor changes. 1 reference added. Published versio

Relations for certain symmetric norms and anti-norms before and after partial trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.Comment: 11 pages, no figures. A typo error in Eq. (5.15) is corrected. Minor improvements. J. Stat. Phys. (in press

Resource Theories as Quantale Modules

We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The methods we address relate to questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of the ubiquity of such questions, methods for answering them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, concrete methods are generalized to the abstract language to find under what conditions they are applicable. Then, one can determine whether potentially novel contexts satisfy these conditions. Here, we mainly focus on the first part of this two-stage process. The thesis starts with a more thorough introduction to resource theories and our perspective on them in chapter 1. Chapter 2 then provides a selection of mathematical ideas that we make heavy use of in the rest of the manuscript. In chapter 3, we present two variants of the abstract framework, whose relations to existing ones are summarized in table 1.1. The first one, universally combinable resource theories, offers a structure in which resources, desired tasks, and resource manipulations may all be viewed as "generalized resources". Blurring these distinctions, whenever appropriate, is a simplification that lets us understand the abstract results in elementary terms. It offers a slightly distinct point of view on resource theories from the traditional one, in which resources and their manipulations are considered independently. In this sense, the second framework in terms of quantale modules follows the traditional conception. Using these, we make contributions towards the task of generalizing concrete methods in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource, summarized in its generalized form in theorems 4.21 and 4.22. Among other applications, this construction may be used to extend measures from a subset of resources to a larger domain—such as from states to channels and other processes. Another construction allows the translation of resource measures between resource theories. A particularly useful version thereof is the translation of measures of distinguishability to other resource theories, which we study in detail. Special cases include resource robustness and weight measures as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on

Certifying randomness in quantum state collapse

The unpredictable process of state collapse caused by quantum measurements makes the generation of quantum randomness possible. In this paper, we explore the quantitive connection between the randomness generation and the state collapse and provide a randomness verification protocol under the assumptions: (I) independence between the source and the measurement devices and (II) the L\"{u}ders' rule for collapsing state. Without involving heavy mathematical machinery, the amount of genereted quantum randomness can be directly estimated with the disturbance effect originating from the state collapse. In the protocol, we can employ general measurements that are not fully trusted. Equipped with trusted projection measurements, we can further optimize the randomness generation performance. Our protocol also shows a high efficiency and yields a higher randomness generation rate than the one based on uncertainty relation. We expect our results to provide new insights for understanding and generating quantum randomnes