Results 1 
7 of
7
From LowDistortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking
"... Quantum uncertainty relations are at the heart of many quantum cryptographic protocols performing classically impossible tasks. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking [12]. A locking scheme can be viewed as a cryptog ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
Quantum uncertainty relations are at the heart of many quantum cryptographic protocols performing classically impossible tasks. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking [12]. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random nbit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message (the message is “locked”). Furthermore, knowing the key, it is possible to recover (or “unlock”) the message. In this paper, we make the following contributions by exploiting a connection between uncertainty relations and lowdistortion embeddings of ℓ2 into ℓ1.
Almost Euclidean subspaces of ℓN 1 via expander codes
, 2009
"... We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ R N of dimension (1 − o(1))N such that for every x ∈ X, (log N) −O(log log log N) √ N ‖x‖2 � ‖x‖1 � √ N ‖x‖2. If we are allowed to use N 1 / log log N � N o(1) random bits and dim(X) � (1 − η)N for any ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ R N of dimension (1 − o(1))N such that for every x ∈ X, (log N) −O(log log log N) √ N ‖x‖2 � ‖x‖1 � √ N ‖x‖2. If we are allowed to use N 1 / log log N � N o(1) random bits and dim(X) � (1 − η)N for any fixed constant η, the lower bound can be further improved to (log N) −O(1) √ N‖x‖2. Through known connections between such Euclidean sections of ℓ1 and compressed sensing matrices, our result also gives explicit compressed sensing matrices for low compression factors for which basis pursuit is guaranteed to recover sparse signals. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of errorcorrecting codes.
Uncertainty relations for multiple measurements with applications
, 2012
"... ar ..."
(Show Context)
Electronic Colloquium on Computational Complexity, Report No. 126 (2006) Uncertainty Principles, Extractors, and Explicit Embeddings of L2 into
, 2006
"... The area of geometric functional analysis1 is concerned with studying the properties of geometric (normed) spaces. A typical question in the area is: for two spaces X and Y, equipped with norms ‖·‖X and ‖·‖Y, under which conditions is there an embedding F: X → Y such that for any p,q ∈ X, we have ‖p ..."
Abstract
 Add to MetaCart
The area of geometric functional analysis1 is concerned with studying the properties of geometric (normed) spaces. A typical question in the area is: for two spaces X and Y, equipped with norms ‖·‖X and ‖·‖Y, under which conditions is there an embedding F: X → Y such that for any p,q ∈ X, we have ‖p − q‖X ≤ ‖F(p) − F(q)‖Y ≤ C‖p − q‖Y for some constant2 C ≥ 1? A ubiquitous tool