We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful 2-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae’94). Using our hypercontractive inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with t -size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an \((r-\varepsilon)\) -approximation of this value requires \(\Omega (n^{1-2/t})\) quantum space, where r is the alphabet size. We next present a lower bound for locally decodable codes ( \(\mathsf {LDC}\) ) \(\mathbb {Z}_r^n\rightarrow \mathbb {Z}_r^N\) over large alphabets with recoverability probability at least \(1/r + \varepsilon\) . Using hypercontractivity, we give an exponential lower bound \(N= 2^{\Omega (\varepsilon ^4 n/r^4)}\) for 2-query (possibly non-linear) \(\mathsf {LDC}\) s over \(\mathbb {Z}_r\) and using the non-commutative Khintchine inequality we prove an improved lower bound of \(N= 2^{\Omega (\varepsilon ^2 n/r^2)}\) .
We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful 2-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae’94). Using our hypercontractive inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with t -size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an ((r-\varepsilon)) -approximation of this value requires (\Omega (n^{1-2/t})) quantum space, where r is the alphabet size. We next present a lower bound for locally decodable codes ( (\mathsf {LDC}) ) (\mathbb {Z}_r^n\rightarrow \mathbb {Z}_r^N) over large alphabets with recoverability probability at least (1/r + \varepsilon) . Using hypercontractivity, we give an exponential lower bound (N= 2^{\Omega (\varepsilon ^4 n/r^4)}) for 2-query (possibly non-linear) (\mathsf {LDC}) s over (\mathbb {Z}_r) and using the non-commutative Khintchine inequality we prove an improved lower bound of (N= 2^{\Omega (\varepsilon ^2 n/r^2)}) .
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m < 0.7 n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first "non-quantum" proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.
Suppose Alice and Bob receive strings X=(X1,...,Xn) and Y=(Y1,...,Yn) each uniformly random in [s]n, but so that X and Y are correlated. For each symbol i, we have that Yi=Xiwith probability 1-ε and otherwise Yiis chosen independently and uniformly from [s]. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from [s]k, but without communicating. How well can they do? The trivial strategy of outputting the first k symbols yields an agreement probability of (1-ε+ε/s)k. In a recent work by Bogdanov and Mossel, it was shown that in the binary case where s=2 and k=k(ε) is large enough then it is possible to extract k bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability (kε)-1/2·2-kε/(2(1-ε/2))using a random construction based on Hamming balls, and this is optimal up to lower order terms. In this paper, we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular, we show no strategy can achieve agreement probability better than (1-ε)k(1+δ(s))kwhere δ(s)→ 0 as s→∞. We also show that Hamming ball-based constructions have much lower agreement probability rate than the trivial algorithm as s→∞. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.
Suppose Alice and Bob receive strings X=(X1,...,Xn) and Y=(Y1,...,Yn) each uniformly random in [s]n but so that X and Y are correlated . For each symbol i, we have that Yi=Xi with probability $1-\eps$ and otherwise Yi is chosen independently and uniformly from [s]. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from [s]k but without communicating. How well can they do? The trivial strategy of outputting the first k symbols yields an agreement probability of $(1 - \eps + \eps/s)^k$. In a recent work by Bogdanov and Mossel it was shown that in the binary case where s=2 and $k = k(\eps)$ is large enough then it is possible to extract k bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability $(k\eps)^{-1/2} \cdot 2^{-k\eps/(2(1 - \eps/2))}$ using a random construction based on Hamming balls, and this is optimal up to lower order terms. In the current paper we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular we show no strategy can achieve agreement probability better than $(1-\eps)^k (1+δ(s))^k$ where δ(s)→0 as s→∞. We also show that Hamming ball based constructions have {\em much lower} agreement probability rate than the trivial algorithm as s→∞. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.
We study the computational complexity of approximating the 2-to-q norm of linear operators (defined as |A|2->q = maxv≠ 0|Av|q/|v|2) for q > 2, as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following: For any constant even integer q ≥ 4, a graph G is a small-set expander if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to the 2->q norm will refute the Small-Set Expansion Conjecture --- a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n2/q) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set-Expansion. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2->4 norm of the projector to low degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique-Games considered by prior works. This improves on the previous upper bound of exp(logO(1) n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require ω(1) rounds. We show reductions between computing the 2->4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2->4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2->4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp(√n poly log(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2->4 norm.
Abstract We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated Lp L p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated Lp L p norms are not uniformly convex for 1≤p≤2 1 ≤ p ≤ 2 . As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.
Suppose Alice holds a uniformly random string X ∈ {0, 1}N and Bob holds a noisy version Y of X where each bit of X is flipped independently with probability e ∈ [0, 1/2]. Alice and Bob would like to extract a common random string of min-entropy at least k. In this work, we establish the communication versus success probability trade-off for this problem by giving a protocol and a matching lower bound (under the restriction that the string to be agreed upon is determined by Alice's input X). Specifically, we prove that in order for Alice and Bob to agree on a common string with probability 2-γk (γk ≥ 1), the optimal communication (up to o(k) terms, and achievable for large N) is precisely [EQUATION], where C:= 4e(1 - e). In particular, the optimal communication to achieve Ω(1) agreement probability approaches 4e(1 - e)k.
We also consider the case when Y is the output of the binary erasure channel on X, where each bit of Y equals the corresponding bit of X with probability 1 - e and is otherwise erased (that is, replaced by a '?'). In this case, the communication required becomes [EQUATION]. In particular, the optimal communication to achieve Ω(1) agreement probability approaches ek, and with no communication the optimal agreement probability approaches [EQUATION].
Our protocols are based on covering codes and extend the approach of (Bogdanov and Mossel, 2011) for the zero-communication case. Our lower bounds rely on hypercontractive inequalities. For the model of bit-flips, our argument extends the approach of (Bogdanov and Mossel, 2011) by allowing communication; for the erasure model, to the best of our knowledge the needed hypercontractivity statement was not studied before, and it was established (given our application) by (Nair and Wang 2015). We also obtain information complexity lower bounds for these tasks, and together with our protocol, they shed light on the recently popular most informative Boolean function conjecture of Courtade and Kumar.