Related papers: Effective Capacity and Randomness of Closed Sets
Quantum capacity, as the ultimate transmission rate of quantum communication, is characterized by regularized coherent information. In this work, we reformulate approximations of the quantum capacity by operator space norms and give both…
We develop a theory of capacities associated with local Muckenhoupt weights. Fundamental properties of local Muckenhoupt weights will be revisited. Weak type boundedness of nonlinear potential and capacitary strong type inequalities…
Reimann and Slaman initiated the study of sequences that are Martin-L\"of random with respect to a continuous measure, establishing fundamental facts about NCR, the collection of sequences that are not Martin-L\"of random with respect to…
We develop a synthesis of Turing's paradigm of computation and von Neumann's quantum logic to serve as a model for quantum computation with recursion, such that potentially non-terminating computation can take place, as in a quantum Turing…
Let $X$ be a Banach space and $\mu$ a probability measure. A set $K \subseteq L^1(\mu,X)$ is said to be a $\delta\mathcal{S}$-set if it is uniformly integrable and for every $\delta>0$ there is a weakly compact set $W \subseteq X$ such that…
The aim of this work is to study the physical properties of a one-way quantum computer in an effective low-energy cluster state. We calculate the optimal working conditions as a function of the temperature and of the system parameters. The…
We consider the problem of optimal probing of states of a channel by transmitter and receiver for maximizing rate of reliable communication. The channel is discrete memoryless (DMC) with i.i.d. states. The encoder takes probing actions…
Bounds analogous to entropic uncertainty relations allow one to design practical tests to detect quantum entanglement by a collective measurement performed on several copies of the state analyzed. This approach, initially worked out for…
"How much c.e. sets could cover a given set?" in this paper we are going to answer this question. Also, in this approach some old concepts come into a new arrangement. The major goal of this article is to introduce an appropriate definition…
New partial results are obtained related to the following old problem of Erd\"os: for any infinite set $X$ of real numbers to show that there is always a measurable (or, equivalently, closed) subset of reals of positive Lebesgue measure…
We investigate the existence of a maximiser among open, bounded, convex sets in $\R^d,\,d\ge 3$ for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if $d=2$), with constraints involving Lebesgue measure or…
An effective quantum number determining with high accuracy the levels ordering in arbitrary centrally symmetric potentials for any space dimensionality is introduced and calculated by means of certain universal methods based on the known…
Let $R$ be a (commutative Noetherian) local ring of prime characteristic that is $F$-pure. This paper studies a certain finite set ${\mathcal I}$ of radical ideals of $R$ that is naturally defined by the injective envelope of the simple…
Exchangeability -- in which the distribution of an infinite sequence is invariant to reorderings of its elements -- implies the existence of a simple conditional independence structure that may be leveraged in the design of statistical…
Effectively inseparable pairs and their properties play an important role in the meta-mathematics of arithmetic and incompleteness. Different notions are introduced and shown in the literature to be equivalent to effective inseparability.…
We analyze the properties of weakly compact sets in Lipschitz free spaces. Prior research has established that, for a complete metric space $M$, weakly precompact sets in the Lipschitz free space $\mathcal F(M)$ are tight. In this paper, we…
The quantum volume test is a full-system benchmark for quantum computers that is sensitive to qubit number, fidelity, connectivity, and other quantities believed to be important in building useful devices. The test was designed to produce a…
We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map, which is nontrivial because NCP maps may lack a full…
We investigate the control resources needed to effect arbitrary quantum dynamics. We show that the ability to perform measurements on a quantum system, combined with the ability to feed back the measurement results via coherent control,…
For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure.…