Almost all even-particle pure states are determined by their half-body marginals

Abstract Determining whether the original global state is uniquely determined by its local marginals is a prerequisite for some efficient tools for characterizing quantum states. In this paper, we show that almost all generic pure states of even N-particle with equal local dimension are uniquely determined among all pure states (UDP) by four of their half-body marginals. Furthermore, we give a graphical description of the marginals for determining genuinely multipartite entangled states, which leads to several lower bounds on the number of required marginals. Finally, we present a construction of N-qudit states obtained from certain combinatorial structures that cannot be UDP by their k-body marginals for some k ⩾ ⌊ N / 2 ⌋ .