Parity superselection obstructs monogamy of mutual information in free fermions

TL;DR

This paper demonstrates that superselection rules in free fermions violate the expected monogamy of mutual information, revealing the importance of operator algebra choices in quantum information measures.

Contribution

It proves that parity superselection rules cause violations of monogamy of mutual information in free fermions, providing exact identities and numerical evidence for this phenomenon.

Findings

Violations of monogamy of mutual information in free fermions with spin factorization.

Parity superselection enforces non-positivity of the tripartite information $I_3$ at all fillings.

Factorization contributions dominate genuine interactions in the $I_3$ measure, explaining deviations.

Abstract

We prove that free fermions in the spin (tensor product) factorization violate monogamy of mutual information: $I_3^{\mathrm{spin}} > 0$ for adjacent strips of width $w \le 6$ at all Fermi momenta, and for all $w$ at $z = k_F w < z^* \approx 1.329$. Many-body computation at $w=6$ via the $G$-matrix formula maps the scaling-limit function $I_3^{\mathrm{spin}}(z)$: it has a minimum of $0.100$ at $z \approx 1.5$, numerically establishing the conjecture $I_3^{\mathrm{spin}} > 0$ for all $z$ and $w$. The proof rests on an exact identity: the fermionic and spin reduced density matrices of disjoint regions $A$, $D$ separated by $B$ differ by the parity insertion $(-1)^{N_B}$ in the partial trace. A Perron--Frobenius argument proves element-wise coherence damping; for free fermions, an independent Gaussian bound gives the entropy ordering $\Delta S_{AD} \ge 0$. Exact diagonalization confirms this for interacting fermions. DMRG on the $t$-$V$ chain shows that the factorization contribution exceeds the genuine interaction contribution to $I_3$ by a factor of 8, accounting for $\sim 80\%$ of the deviation in spin-basis numerics. Strong repulsion ($K \lesssim 0.7$) restores monogamy. Conversely, $Z_2$ parity superselection enforces $I_3 \le 0$ at all fillings (proved for $w \le 3$), with the ratio of parity entropy to quantum excess approaching $2\ln 2/(3\ln(4/3)) = 1.606$. Any use of $I_3$ as a diagnostic for holographic duality, quantum chaos, or Fermi surface topology must specify the operator algebra; without this, the sign of $I_3$ is ambiguous.