A classical interpretation of the Scrooge distribution

The Scrooge distribution is a probability distribution over the set of pure states of a quantum system. Specifically, it is the distribution that, upon measurement, gives up the least information about the identity of the pure state, compared with all other distributions having the same density matrix. The Scrooge distribution has normally been regarded as a purely quantum mechanical concept, with no natural classical interpretation. In this paper we offer a classical interpretation of the Scrooge distribution viewed as a probability distribution over the probability simplex. We begin by considering a real-amplitude version of the Scrooge distribution, for which we find that there is a non-trivial but natural classical interpretation. The transition to the complex-amplitude case requires a step that is not particularly natural but that may shed light on the relation between quantum mechanics and classical probability theory.Comment: 17 pages; for a special issue of Entropy: Quantum Communication--Celebrating the Silver Jubilee of Teleportatio

It'll probably work out: improved list-decoding through random operations

In this work, we introduce a framework to study the effect of random operations on the combinatorial list-decodability of a code. The operations we consider correspond to row and column operations on the matrix obtained from the code by stacking the codewords together as columns. This captures many natural transformations on codes, such as puncturing, folding, and taking subcodes; we show that many such operations can improve the list-decoding properties of a code. There are two main points to this. First, our goal is to advance our (combinatorial) understanding of list-decodability, by understanding what structure (or lack thereof) is necessary to obtain it. Second, we use our more general results to obtain a few interesting corollaries for list decoding: (1) We show the existence of binary codes that are combinatorially list-decodable from $1/2-\epsilon$ fraction of errors with optimal rate $\Omega(\epsilon^2)$ that can be encoded in linear time. (2) We show that any code with $\Omega(1)$ relative distance, when randomly folded, is combinatorially list-decodable $1-\epsilon$ fraction of errors with high probability. This formalizes the intuition for why the folding operation has been successful in obtaining codes with optimal list decoding parameters; previously, all arguments used algebraic methods and worked only with specific codes. (3) We show that any code which is list-decodable with suboptimal list sizes has many subcodes which have near-optimal list sizes, while retaining the error correcting capabilities of the original code. This generalizes recent results where subspace evasive sets have been used to reduce list sizes of codes that achieve list decoding capacity