Fig. 1: The schematic diagram of sampling reduced density matrix and the characters of Heisenberg ladder.

a A path integral configuration of a reduced density matrix. The length of imaginary time is β, and the time boundary of B/A is periodic/open. b Lattice of the Spin-1/2 Heisenberg ladder from an overhead view, with spins in A and B depicted by red and blue circles, respectively. c Its phase diagram by denoting \({J}_{{{{\rm{leg}}}}}=\cos \theta\) and \({J}_{{{{\rm{rung}}}}}=\sin \theta\).