Jop Briët

Properties of Classical and Quantum Jensen-Shannon Divergence

Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JDα for α > 0), the Jensen divergences of order α, which generalize JD as JD1 = JD. Using a result of Schoenberg, we prove that JDα is the square of a metric for α ∈ (0, 2] , and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order α (QJDα). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD α is a metric space which can be isometrically embedded in a real Hilbert space when α ∈ (0, 2] . In analogy with Burbea and Rao’s generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.

Grothendieck Inequalities for Semidefinite Programs with Rank Constraint

Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constraint is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give two applications: approximating ground states in the n-vector model in statistical mechanics and XOR games in quantum information theory.

On the Existence of 0/1 Polytopes with High Semidefinite Extension Complexity

Rothvoss [1] showed that there exists a 0/1 polytope (a polytope whose vertices are in {0,1} n ) such that any higher-dimensional polytope projecting to it must have 2Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.

Monotonicity testing and shortest-path routing on the cube

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions f : {0, 1}n → Rcan be tested with O(n/e) queries, for any totally ordered range R. More generally, if at least a µ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(eµ(n))). We construct a family of instances of Ω(2n) pairs in n-dimensional hypercubes such that no more than roughly a 1/√n fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [LR01], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≅ n3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanskis conjecture about routing on the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not n1/2-δ-realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths. We also prove a lower bound of Ω(n/e) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.

Violating the Shannon capacity of metric graphs with entanglement

The Shannon capacity of a graph G is the maximum asymptotic rate at which messages can be sent with zero probability of error through a noisy channel with confusability graph G. This extensively studied graph parameter disregards the fact that on atomic scales, nature behaves in line with quantum mechanics. Entanglement, arguably the most counterintuitive feature of the theory, turns out to be a useful resource for communication across noisy channels. Recently [Leung D, Mančinska L, Matthews W, Ozols M, Roy A (2012) Commun Math Phys 311:97–111], two examples of graphs were presented whose Shannon capacity is strictly less than the capacity attainable if the sender and receiver have entangled quantum systems. Here, we give natural, possibly infinite, families of graphs for which the entanglement-assisted capacity exceeds the Shannon capacity.

Entanglement-Assisted Zero-Error Source-Channel Coding

We study the use of quantum entanglement in the zero-error source-channel coding problem. Here, Alice and Bob are connected by a noisy classical one-way channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alices input while using the channel as little as possible. In the zero-error regime, the optimal rates of source codes and channel codes are given by graph parameters known as the Witsenhausen rate and Shannon capacity, respectively. The Lovász theta number, a graph parameter defined by a semidefinite program, gives the best efficiently computable upper bound on the Shannon capacity and it also upper bounds its entanglement-assisted counterpart. At the same time, it was recently shown that the Shannon capacity can be increased if Alice and Bob may use entanglement. Here, we partially extend these results to the source-coding problem and to the more general source-channel coding problem. We prove a lower bound on the rate of entanglement-assisted source-codes in terms of Szegedys number (a strengthening of the theta number). This result implies that the theta number lower bounds the entangled variant of the Witsenhausen rate. We also show that entanglement can allow for an unbounded improvement of the asymptotic rate of both classical source codes and classical source-channel codes. Our separation results use low-degree polynomials due to Barrington, Beigel and Rudich, Hadamard matrices due to Xia and Liu, and a new application of remote state preparation.

Lower Bounds for Approximate LDCs

We study an approximate version of q-query LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A q-query (α,δ)-approximate LDC is a set V of n points in ℝ d so that, for each i ∈ [d] there are Ω(δn) disjoint q-tuples (u 1,…,u q ) in V so that span(u 1,…,u q ) contains a unit vector whose i’th coordinate is at least α. We prove exponential lower bounds of the form \(n \geq 2^{\Omega(\alpha \delta \sqrt{d})}\) for the case q = 2 and, in some cases, stronger bounds (exponential in d).

Locally Decodable Quantum Codes

We study a quantum analogue of locally decodable error-correcting codes. A

Explicit Lower and Upper Bounds on the Entangled Value of Multiplayer XOR Games

q

A generalized Grothendieck inequality and entanglement in XOR games

-query \emph{locally decodable quantum code} encodes