General Entanglement Diagnostics

Abstract

A balance diagnostic for bipartite pure states of arbitrary Schmidt rank is built from the elementary symmetric polynomials of the Schmidt spectrum, where the altitude e2\sqrt{e_2} e2 coincides with half the I-concurrence, the AM–GM–HM chain generalizes to Maclaurin's and Newton's inequalities, and Newton's identities tie the hierarchy to the replica moments that compute entanglement entropy. Exactly solvable qudit and oscillator thermofield doubles give purity, concurrence, and entropy in closed form, and separate the maximal-entanglement point from the first-law point by an explicit relative-entropy offset. Lifted to the eternal BTZ black hole, the concurrence saturates and loses discriminating power, leaving relative entropy as the geometric measure — evaluated in closed form, matched to the bulk canonical energy, reconstructed as a rigid even-zeta symplectic-flux series resumming to the Ryu–Takayanagi area, and extended to the two-interval mutual information and its RT phase transition.