Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Matrix Discrepancy from Quantum Communication   We develop a novel connection between discrepancy minimization and (quantum)
communication complexity. As an application, we resolve a substantial special
case of the Matrix Spencer conjecture. In particular, we show that for every
collection of symmetric $n \times n$ matrices $A_1,\ldots,A_n$ with $\|A_i\|
\leq 1$ and $\|A_i\|_F \leq n^{1/4}$ there exist signs $x \in \{ \pm 1\}^n$
such that the maximum eigenvalue of $\sum_{i \leq n} x_i A_i$ is at most
$O(\sqrt n)$. We give a polynomial-time algorithm based on partial coloring and
semidefinite programming to find such $x$.
  Our techniques open a new avenue to use tools from communication complexity
and information theory to study discrepancy. The proof of our main result
combines a simple compression scheme for transcripts of repeated (quantum)
communication protocols with quantum state purification, the Holevo bound from
quantum information, and tools from sketching and dimensionality reduction. Our
approach also offers a promising avenue to resolve the Matrix Spencer
conjecture completely -- we show it is implied by a natural conjecture in
quantum communication complexity.

9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org