On the Complexity of Pure-State Consistency of Local Density Matrices

In this work we investigate the computational complexity of the pure consistency of local density matrices (\(\mathsf{PureCLDM}\)) and pure \(N\)-representability (\(\mathsf{Pure}\)-\(N\)-\(\mathsf{Representability}\)) problems. In these problems the input is a set of reduced density matrices and the task is to determine whether there exists a global \emph{pure} state consistent with these reduced density matrices. While mixed \(\mathsf{CLDM}\), i.e. where the global state can be mixed, was proven to be \(\mathsf{QMA}\)-complete by Broadbent and Grilo [JoC 2022], almost nothing was known about the complexity of the pure version. Before our work the best upper and lower bounds were \(\mathsf{QMA}(2)\) and \(\mathsf{QMA}\). Our contribution to the understanding of these problems is twofold. Firstly, we define a pure state analogue of the complexity class \(\mathsf{QMA}^+\) of Aharanov and Regev [FOCS 2003], which we call \(\mathsf{PureSuperQMA}\). We prove that both \(\mathsf{Pure}\)-\(N\)-\(\mathsf{Representability}\) and \(\mathsf{PureCLDM}\) are complete for this new class. Along the way we supplement Broadbent and Grilo by proving hardness for 2-qubit reduced density matrices and showing that mixed \(N\)-\(\mathsf{Representability}\) is \(\mathsf{QMA}\) complete. Secondly, we improve the upper bound on \(\mathsf{PureCLDM}\). Using methods from algebraic geometry, we prove that \(\mathsf{PureSuperQMA} \subseteq \mathsf{PSPACE}\). Our methods, and the \(\mathsf{PSPACE}\) upper bound, are also valid for \(\mathsf{PureCLDM}\) with exponential or even perfect precision, hence \(\mathsf{precisePureCLDM}\) is not \(\mathsf{preciseQMA}(2) = \mathsf{NEXP}\)-complete, unless \(\mathsf{PSPACE} = \mathsf{NEXP}\). We view this as evidence for a negative answer to the longstanding open question whether \(\mathsf{PureCLDM}\) is \(\mathsf{QMA}(2)\)-complete.