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arXiv:2607.06474

Typical Entanglement of Superpositions

Damien Quinn, Joshuah T. Heath, Graham Kells

quant-phSubmitted Jul 7, 2026• Announced Jul 8, 2026• 0 citations
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Abstract

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.

Comments
13 pages, 5 figures

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We investigate universal entanglement properties inherent to superpositions of randomized states. We find that an mmm-fold superposition of typical states may be classified into two distinct entanglement classes via the 2nd Rényi entropy density s2s_2s2​. The maximally entangled regime is defined by s2∼ln⁡(2)s_2 \sim \ln (2)s2​∼ln(2), for which superposition adds no additional entanglement. The sub-maximally entangled regime, s2<ln⁡2s_2<\ln 2s2​<ln2, 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)ΔS(m)=\ln (m)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 s2s_2s2​ 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.
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We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-NNN expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index qc=2q_c=2qc​=2: for q>2q>2q>2 the magic density is 1/(q−1)1/(q-1)1/(q−1), controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer qqq as coefficient integrals over SU(2q)\mathrm{SU}(2q)SU(2q), with dimension independent of NNN, and evaluate the q=2q=2q=2 case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding SYK2\mathrm{SYK}_2SYK2​ ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction κ<κ⋆G≃0.7613κ<κ_\star^{\rm G}\simeq0.7613κ<κ⋆G​≃0.7613 of their modes in the thermodynamic limit, remaining magical even beyond the 2/32/32/3 threshold for Haar-random states in the full Hilbert space.