In finite pair-correlated fermion systems (atomic nuclei, mesoscopic superconducting grains), the BCS mean-field state breaks particle-number (gauge) symmetry and, as a two-mode reduced state, is classical: it carries neither concurrence nor quantum discord between a time-reversed mode pair. Zeng, Wu and Xing (Sci. Rep. 4, 06377, 2014) showed that particle-number-projected BCS (PBCS) and the exact solution synchronously reestablish both correlations, and that while the restored concurrence remains almost unchanged with particle number after an initial drop following its maximum at p=1p=1, the two-mode quantum discord grows approximately linearly with the number of pairs pp (an 'extensive' quantity, e.g. D≃2p/Ω+(p/Ω)log⁡2(p/Ω)D\simeq 2p/\Omega+(p/\Omega)\log_2(p/\Omega) for degeneracy Ω≫1\Omega\gg1, p≪Ωp\ll\Omega) and is remarkably insensitive to the pairing strength. These results were established only for two-level and single-degenerate-level pairing Hamiltonians. The open problem is to determine whether this extensive, interaction-robust discord is a general property of pair-correlated finite fermion systems: compute it for multi-level pairing Hamiltonians with nonuniform level spacings and realistic shell structure, quantify the accuracy and convergence of PBCS relative to exact diagonalization for the two-mode correlations as functions of Ω\Omega, pp and the interaction matrix elements, establish whether generalized (POVM) measurements alter the discord scaling, and provide experimentally executable protocols to reconstruct the projected two-mode reduced density matrices and verify the scaling in families of nuclei or mesoscopic superconducting grains.

Background

In the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity and nuclear pairing, the interacting fermions are replaced by a product of independently occupied 'quasiparticle' modes. This mean-field state is not an eigenstate of the particle-number operator: it breaks the global U(1) gauge symmetry of the underlying Hamiltonian. In finite systems — small superconducting grains treated in the canonical ensemble (Mastellone, Falci and Fazio, Phys. Rev. Lett. 80, 4542 (1998)) and atomic nuclei — this symmetry breaking is a real limitation, and it is repaired by projection: the particle-number-projected BCS (PBCS) state is obtained by applying a fixed-particle-number projector to the BCS state. A second, independent lens on pairing is quantum information theory. For a two-mode (two-qubit) reduced density matrix ρ\rho, the concurrence measures entanglement, while the quantum discord (Ollivier and Zurek, Phys. Rev. Lett. 88, 017901 (2001)) measures the part of the mutual information that cannot be extracted by any local measurement; discord can be nonzero even for unentangled (separable) states. For the special 'X-states' (density matrices nonzero only on the diagonal and anti-diagonal), which include the two-mode reduced states of BCS and PBCS, both quantities have closed analytic expressions (Ali, Rau and Alber, Phys. Rev. A 81, 042105 (2010); Chen et al., Phys. Rev. A 84, 042313 (2011)). Earlier work quantified entanglement in exactly solvable pairing models: the local concurrence and condensation energy of the BCS model (Dunning, Links and Zhou, Phys. Rev. Lett. 94, 227002 (2005)) and finite-size scaling of entanglement in the two-level BCS model (Dusuel and Vidal, Phys. Rev. A 71, 060304 (2005)). Against this background, Zeng, Wu and Xing (Sci. Rep. 4, 06377 (2014)) asked what gauge-symmetry breaking does to quantum correlations between modes. They found that the BCS two-mode state is not only unentangled but purely classical (zero discord), and that number projection reestablishes both concurrence and discord. Their central discovery is a sharp dissociation of the two correlation measures as the system grows: for a pairing Hamiltonian with Ω\Omega-fold degenerate levels and pp pairs, the restored concurrence jumps to its maximum at p=1p=1, then decreases by near half and keeps almost unchanged with particle number, while the discord grows approximately linearly with pp — in the single-level analytic limit, total correlation I(ρ)≃2p/ΩI(\rho)\simeq 2p/\Omega, classical correlation C(ρ)≃−(p/Ω)log⁡2(p/Ω)C(\rho)\simeq -(p/\Omega)\log_2(p/\Omega), discord D(ρ)≃2p/Ω+(p/Ω)log⁡2(p/Ω)D(\rho)\simeq 2p/\Omega+(p/\Omega)\log_2(p/\Omega) for Ω≫1\Omega\gg1, p≪Ωp\ll\Omega — and the discord is nearly independent of the pairing strength in the strong-coupling regime. Numerically, the PBCS two-mode correlations converge to the exact diagonalization results as the degeneracy grows, and remain close to them for nonuniform interaction matrix elements Gkk′G_{kk'} at small degeneracy. All of this was established for the two-level model and the single-degenerate-level model only. The authors conjecture that the extensive-discord behavior 'should imply profound universality for any many-body system consisting of pairwise correlated particles', anticipate (without computing) that replacing von Neumann measurements with general positive-operator-valued measures (POVMs) in the discord optimization 'will remain essentially the same' for the X-states at hand, and observe that the predicted scalings are testable experimentally, e.g. along chains of even-even isotones (nuclei with equal neutron number) sharing the same valence-neutron shell. None of these extensions was carried out in the source.

Problem Statement

Determine whether the quantumness-reestablishment phenomenon found by Zeng, Wu and Xing for the two-level and single-degenerate-level pairing models is a general property of pair-correlated finite fermion systems, and whether it is experimentally observable. Concretely: (a) for multi-level pairing Hamiltonians with arbitrary level spacings, nonuniform interaction matrix elements Gkk′G_{kk'} and realistic nuclear or mesoscopic shell structures, compute the two-mode reduced density matrices of the time-reversed mode pairs (k,kˉ)(k,\bar{k}) after particle-number restoration (PBCS and exact diagonalization or equivalent exact methods) and establish whether the two-mode quantum discord is again extensive in the pair number pp — growing approximately linearly with pp at fixed level structure, the asymptotics D(ρ)≃2p/Ω+(p/Ω)log⁡2(p/Ω)D(\rho)\simeq 2p/\Omega+(p/\Omega)\log_2(p/\Omega) for Ω≫1\Omega\gg1, p≪Ωp\ll\Omega being the single-degenerate-level analytic limit — while the concurrence remains approximately pair-number independent, and whether the near-independence of the discord from the pairing strength survives; (b) quantify, as functions of the degeneracy Ω\Omega, pair number pp and the matrix elements Gkk′G_{kk'}, the convergence of the PBCS two-mode correlations (concurrence and discord) to the exact ones, extending the numerical convergence observations of the source; (c) determine whether extremizing the discord over generalized POVMs instead of von Neumann (projective) measurements changes the quantitative discord scaling for the two-mode states arising in these models, testing the source's conjecture that it does not; and (d) supply experimentally executable protocols — with predicted observables, required measurements and error analyses — for reconstructing the projected two-mode reduced density matrices and verifying the predicted scalings in families of finite nuclei (e.g. even-even isotone chains with common valence shells) or mesoscopic superconducting grains. Deliverables (a)-(d) jointly serve one objective — establishing the generality and observability of the extensive, interaction-insensitive quantum discord as a signature of symmetry restoration in pair-correlated finite systems — and a complete affirmative resolution requires all four. The objective bundles several independently refutable sub-claims of equal standing, each falsifiable on its own by exact computation on an explicit multi-level pairing system: the restored two-mode discord of a time-reversed mode pair is not extensive in pp; the restored concurrence is not approximately pair-number independent; the near-independence of the discord from the pairing strength fails; the PBCS two-mode correlations do not converge to the exact ones; or extremizing over generalized POVMs changes the discord scaling, contradicting the source's conjecture. Refuting any one of these resolves that sub-claim — fixing a boundary on the source's claims — while leaving the other sub-claims open; only (a)-(d) established affirmatively together complete the task.

References

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