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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mo fence="false" stretchy="false">⌊</mml:mo> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo fence="false" stretchy="false">⌋</mml:mo> </mml:mrow> </mml:math> .