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

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 (Walgate et al. 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 \(f_2(N)\le \lceil \frac{N}{6}\rceil +2\), which is better than \(f_2(N)\le \lceil \frac{N}{4}\rceil\) obtained in Shu (Eur. Phys. J. Plus 136, 1172 (2021)) when \(N>32\). We further exhibit that for arbitrary \(\epsilon >0\), \(f_r(N)\le \lceil \epsilon N\rceil\) always holds for sufficiently large N.