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.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.

quant-pharXiv:2607.06474Jul 8, 20260 citations

Entanglement of subspaces in terms of entanglement of superpositions

Gilad Gour, Aidan Roy

We investigate upper and lower bounds on the entropy of entanglement of a superposition of bipartite states as a function of the individual states in the superposition. In particular, we extend the results by Gour [Phys. Rev. A 76, 052320 (2007)] to superpositions of several states rather than just two. We then investigate the entanglement in a subspace as a function of its basis states: we find upper bounds for the largest entanglement in a subspace and demonstrate that no such lower bound for the smallest entanglement exists. Finally, we consider entanglement of superpositions using measures of entanglement other than the entropy of entanglement.We investigate upper and lower bounds on the entropy of entanglement of a superposition of bipartite states as a function of the individual states in the superposition. In particular, we extend the results by Gour [Phys. Rev. A 76, 052320 (2007)] to superpositions of several states rather than just two. We then investigate the entanglement in a subspace as a function of its basis states: we find upper bounds for the largest entanglement in a subspace and demonstrate that no such lower bound for the smallest entanglement exists. Finally, we consider entanglement of superpositions using measures of entanglement other than the entropy of entanglement.

arXiv:0711.1344Physical Review AJan 30, 200819 citations

Superposition, Entanglement and Other Oddities

Hans Luth

Graduate texts in physicsOct 22, 20120 citations

Reexamination of entanglement of superpositions

Gilad Gour

We find tight lower and upper bounds on the entanglement of a superposition of two bipartite states in terms of the entanglement of the two states constituting the superposition. Our upper bound is dramatically tighter than the one presented by Linden et al. [Phys. Rev. Lett. 97, 100502 (2006)] and our lower bound can be used to provide lower bounds on different measures of entanglement such as the entanglement of formation and the entanglement of subspaces. We also find that in the case in which the two states are one-sided orthogonal, the entanglement of the superposition state can be expressed explicitly in terms of the entanglement of the two states in the superposition.We find tight lower and upper bounds on the entanglement of a superposition of two bipartite states in terms of the entanglement of the two states constituting the superposition. Our upper bound is dramatically tighter than the one presented by Linden et al. [Phys. Rev. Lett. 97, 100502 (2006)] and our lower bound can be used to provide lower bounds on different measures of entanglement such as the entanglement of formation and the entanglement of subspaces. We also find that in the case in which the two states are one-sided orthogonal, the entanglement of the superposition state can be expressed explicitly in terms of the entanglement of the two states in the superposition.

arXiv:0707.1521Physical Review ANov 21, 200731 citations

Quantum Superposition and Entanglement

Belal Ehsan Baaquie, L. C. Kwek

Springer eBooksJan 1, 20230 citations

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.