Lieb-Mattis states for robust entangled differential phase sensing

Developing sensors with large particle numbers \$N\$ that can resolve subtle physical effects is a central goal in precision measurement science. Entangled quantum sensors can surpass the standard quantum limit (SQL), where the signal variance scales as \$1/N\$, and approach the Heisenberg limit (HL) with variance scaling as \$1/N\textasciicircum2\$. However, entangled states are typically more sensitive to noise, especially common-mode noise such as magnetic field fluctuations, control phase noise, or vibrations in atomic interferometers.