Additive Extensions of a Quantum Channel

We study extensions of a quantum channel whose one-way capacities are described by a single-letter formula. This provides a simple technique for generating powerful upper bounds on the capacities of a general quantum channel. We apply this technique to two qubit channels of particular interest--the depolarizing channel and the channel with independent phase and amplitude noise. Our study of the latter demonstrates that the key rate of BB84 with one-way post-processing and quantum bit error rate q cannot exceed H(1/2-2q(1-q)) - H(2q(1-q)).Comment: 6 pages, one figur

Strong laws of large numbers for sub-linear expectations

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.Comment: 10 page

Simple test for quantum channel capacity

Basing on states and channels isomorphism we point out that semidefinite programming can be used as a quick test for nonzero one-way quantum channel capacity. This can be achieved by search of symmetric extensions of states isomorphic to a given quantum channel. With this method we provide examples of quantum channels that can lead to high entanglement transmission but still have zero one-way capacity, in particular, regions of symmetric extendibility for isotropic states in arbitrary dimensions are presented. Further we derive {\it a new entanglement parameter} based on (normalised) relative entropy distance to the set of states that have symmetric extensions and show explicitly the symmetric extension of isotropic states being the nearest to singlets in the set of symmetrically extendible states. The suitable regularisation of the parameter provides a new upper bound on one-way distillable entanglement.Comment: 6 pages, no figures, RevTeX4. Signifficantly corrected version. Claim on continuity of channel capacities removed due to flaw in the corresponding proof. Changes and corrections performed in the part proposing a new upper bound on one-way distillable etanglement which happens to be not one-way entanglement monoton

Extending travel-time based models for dynamic network loading and assignment, to achieve adherence to first-in-first-out and link capacities

AbstractAn important class of models for macroscopic dynamic network loading (DNL) and dynamic traffic assignment (DTA) is based on treating link travel times as a function of link occupancy. However, these models suffer from some problems or deficiencies namely (a) the link outflows can violate first-in-first-out (FIFO), (b) the link outflows can exceed the link outflow capacities, (c) the link inflows can exceed the link inflow capacities, and (d) the link occupancies can exceed the link occupancy capacities. In this paper we introduce methods to overcome each of these problems.To remove problems (a) and (b) we extend the link travel-time model to better reflect behaviour when traffic flow is varying over time. To remove problems (c) and (d) we introduce more substantial changes in the model, to introduce capacities, spillback and queues compatible with the model. These extensions strengthen the realism, behavioural basis and usability of the link travel-time model and the DNL and DTA models that are based on it. They have no obvious adverse implications or side effects and require little additional computational effort. The original model is a special case of the new/extended model: the above extensions are activated if and only if any of the problems (a)–(d) arise, otherwise the new model reduces to the original model

Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

On the Extension of Pseudo-Boolean Functions for the Aggregation of Interacting Criteria

The paper presents an analysis on the use of integrals defined for non-additive measures (or capacities) as the Choquet and the \Sipos{} integral, and the multilinear model, all seen as extensions of pseudo-Boolean functions, and used as a means to model interaction between criteria in a multicriteria decision making problem. The emphasis is put on the use, besides classical comparative information, of information about difference of attractiveness between acts, and on the existence, for each point of view, of a ``neutral level'', allowing to introduce the absolute notion of attractive or repulsive act. It is shown that in this case, the Sipos integral is a suitable solution, although not unique. Properties of the Sipos integral as a new way of aggregating criteria are shown, with emphasis on the interaction among criteria.

Superadditivity in trade-off capacities of quantum channels

In this article, we investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading classical communication, quantum communication and entanglement. Understanding such additivity property is important if we want to optimally use a quantum channel for general communication purpose. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity, and it is largely unknown if this could lead to an superadditive double or triple resource capacity. For example, if a channel has an additive classical and quantum capacity, can the classical-quantum capacity be superadditive? In this work, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple capacity region.Comment: 15 pages. v2: typo correcte