Application of Ramsey theory to localization of set of product states via multicopies

TL;DR

This paper uses Ramsey theory to estimate the minimum number of quantum state copies needed for LOCC distinguishability of orthogonal product states, providing improved bounds and asymptotic results.

Contribution

It introduces a novel connection between Ramsey theory and quantum state distinguishability, deriving tighter bounds on the number of copies required.

Findings

Proves $f_2(N) \,\leq\, \lceil N/6 \rceil + 2$, improving previous bounds.

Establishes that $f_r(N) \,\leq\, \lceil \epsilon N \rceil$ for large $N$ and any $\\epsilon > 0$.

Provides asymptotic bounds for the resource requirements in LOCC state discrimination.

Abstract

It is well known that any $N$ orthogonal pure states can always be perfectly distinguished under local operation and classical communications (LOCC) if $(N-1)$ copies of the state are available [Phys. Rev. Lett. 85, 4972 (2000)]. It is important to reduce the number of quantum state copies that ensures the LOCC distinguishability in terms of resource saving and nonlocality strength characterization. Denote $f_r(N)$ the least number of copies needed to LOCC distinguish any $N$ orthogonal $r$-partite product states. This work will be devoted to the estimation of the upper bound of $f_r(N)$. In fact, we first relate this problem with Ramsey theory, a branch of combinatorics dedicated to studying the conditions under which orders must appear. Subsequently, we prove $f_2(N)\leq \lceil\frac{N}{6}\rceil+2$, which is better than $f_2(N)\leq \lceil\frac{N}{4}\rceil$ obtained in [Eur. Phys. J. Plus 136, 1172 (2021)] when $N>24$. We further exhibit that for arbitrary $\epsilon>0$, $f_r(N)\leq\lceil\epsilon N\rceil$ always holds for sufficiently large $N$.