Exploring pure quantum states with maximally mixed reductions

Abstract

We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for $n$-qubit states, we express the condition that all $k$-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all $k$ vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group ${\mathcal{P}}_{n}$ that is isomorphic to ${\mathbb{Z}}_{2}^{n}$. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed $k$ for fixed $n$. In particular, the ratio $k/n$ can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed $k$-qubit reductions with $k>1$. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though $k>1$ is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.