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arXiv:2603.20786

The typicality of symmetry-induced entanglement

Christian Boudreault, Nicolas Levasseur

quant-phcond-mat.otherSubmitted Mar 21, 2026• Announced Mar 24, 2026• 0 citations• Revised Apr 27, 2026 (v2)
v1 · Mar 21, 2026v2 · Apr 27, 2026
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Abstract

In the presence of a globally conserved charge $N$, a natural question is whether a given separable state can be separated into charge-conserving components. We dub this problem the Symmetric Separability Problem (SSP). On random states, the SSP is answered negatively with probability one for almost all $N$. Using a witness to the failure of symmetric separability, namely the number entanglement (NE) introduced in arXiv:2110.09388, we show that most symmetric and separable states are actually far from being symmetrically separable, with the NE featuring Gaussian concentration around a strictly positive mean value. We discuss some consequences of our results for quantum tasks in the presence of a superselection rule or in the absence of a common reference frame. Progress is made on the question of the size of the separable space constrained by $N$. We also touch upon the question of the complexity of SSP, and multiparty entanglement.

Comments
Main text: 9 pages, 3 figures. Appendix: 12 pages, 3 figures

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In the presence of a globally conserved charge NNN, a natural question is whether a given separable state can be separated into charge-conserving components. We dub this problem the Symmetric Separability Problem (SSP). On random states, the SSP is answered negatively with probability one for almost all NNN. Using a witness to the failure of symmetric separability, namely the number entanglement (NE) introduced in arXiv:2110.09388, we show that most symmetric and separable states are actually far from being symmetrically separable, with the NE featuring Gaussian concentration around a strictly positive mean value. We discuss some consequences of our results for quantum tasks in the presence of a superselection rule or in the absence of a common reference frame. Progress is made on the question of the size of the separable space constrained by NNN. We also touch upon the question of the complexity of SSP, and multiparty entanglement.
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Sufficient conditions for inseparability of mixed states include (logarithmic) negativity. Here we explore systems with a globally conserved charge, such as a particle number. We argue that separable states contain entanglement in fixed charge sectors as long as the state can not be separated into symmetric classical components. As a witness of symmetric inseparability we study the entropy change due to a subsystem's charge measurement, ΔSm\Delta S_mΔSm​. Whenever ΔSm>0\Delta S_m>0ΔSm​>0, there exist inseparable charge sectors, having finite negativity, even when the full state is either separable or has vanishing negativity. We study the scaling of ΔSm\Delta S_mΔSm​ in thermal 1D systems combining high temperature expansion and conformal field theory. We discuss routes towards experimental observation.
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For a given Hamiltonian HHH on a multipartite quantum system, one is interested in finding the energy E0{E}_{0}E0​ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}$ of HHH among all product states. For several concrete model Hamiltonians, we investigate the difference ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}\ensuremath{-}{E}_{0}$, called the separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable of a given state is sufficiently distant from the average value, the state is almost surely entangled.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.