Curator's Take
This article uncovers a universal rule for how entanglement grows when many typical quantum states are coherently added, showing that only an exponentially large number of components can push a sub‑maximally entangled superposition into the maximally entangled regime. By linking the Rényi‑2 entropy density to a simple logarithmic boost ΔS(m)=ln m, the work clarifies why random circuit outputs and tensor‑network states often saturate entanglement quickly—a fact that informs both benchmarking of quantum processors and the design of variational ansätze. The results also bridge Haar‑random physics with more structured families such as Gaussian, matrix‑product, and stabilizer states, offering a concrete metric for when approximations to truly random behavior become reliable.
— Mark Eatherly
Summary
We investigate universal entanglement properties inherent to superpositions of randomized states. We find that an $m$-fold superposition of typical states may be classified into two distinct entanglement classes via the 2nd Rényi entropy density $s_2$. The maximally entangled regime is defined by $s_2 \sim \ln (2)$, for which superposition adds no additional entanglement. The sub-maximally entangled regime, $s_2<\ln 2$, instead constrains the reduced density matrices of independent components to be orthogonal in the thermodynamic limit, which fixes the entanglement of the superposition to a logarithmic enhancement $ΔS(m)=\ln (m)$. As a consequence, an exponentially large number of superpositions is required to transition from the sub-maximally entangled class to maximal entanglement. We explicitly calculate $s_2$ and the logarithmic enhancement, and demonstrate orthogonality for two canonical examples of the sub-maximally entangled regime (superpositions of pure Gaussian states and of random matrix-product states). We also examine the entanglement of superpositions of random stabilizer states, and discuss their relaxation to the Haar limit.