We develop a balance diagnostic for bipartite pure states of arbitrary Schmidt rank n builtfrom the elementary symmetric polynomials of the Schmidt spectrum. The natural altitudeis hn =√e2, half of the I-concurrence CI =p2(1 − Tr ρ2) in any dimension; the classicalAM ≥ GM ≥ HM hierarchy becomes Maclaurin’s and Newton’s inequalities on symmetricmeans, and the concurrence bound CI ≤ CI,max =p2(n − 1)/n is exactly Maclaurin’s firstinequality, saturated at the maximally entangled state. Via Newton’s identities the symmetricpolynomials are interchangeable with the replica moments Tr ρk, so the balance hierarchy is arepackaging of the data used to compute entanglement (R´enyi) entropies. We give two exactlysolvable models—the n-level qudit and the oscillator thermofield doubles—in which the purity,concurrence, and entropy are closed-form functions of the gap-to-temperature ratio βϵ, with themaster formula Tr ρ2 = (1−q)(1+qn)(1+q)(1−qn) , q = e−βϵ. Finally we generalize the distinction between themaximal-entanglement point and the first-law (linearized-Einstein) point to the probability simplex:the two have distinct loci whose relative-entropy offset is D(u ∥ a) = ln(Zn/n) + n−12 βϵ,vanishing only at infinite temperature. The two-qubit results (concurrence = 2√xy, offset12 tanh(βϵ/2)) are recovered, matching the companion two-qubit analysis Aut26, as the n = 2case. Lifting the model to the eternal BTZ black hole, the single-interval replica moments areTr ρn = L−c6 (n−1/n) and the symmetric-polynomial generating function becomes a Fredholmdeterminant; the concurrence then saturates and loses discriminating power in the continuum,so the surviving proximity measure is the (UV-finite) relative entropy whose first law is thelinearized Einstein equation. Finally we extend the framework beyond qudits: the diagnosticdepends only on the Schmidt rank (so asymmetric local dimensions add nothing), the spectralmachinery is universal for all density operators (with h measuring mixedness rather thanentanglement off the pure-state locus), and Gaussian continuous-variable states lift mode bymode, each two-mode-squeezed pair being the oscillator thermofield double with q = tanh2 rand Tr ρ2 = sech 2r.
Alfredo Sepulveda-Jimenez (Sat,) studied this question.