General Entanglement Diagnostics

Abstract

We develop a balance diagnostic for bipartite pure states of arbitrary Schmidt rank n built from the elementary symmetric polynomials of the Schmidt spectrum. The natural altitude is hn = √ e2, half of the I-concurrence CI = p 2(1 − Tr ρ2) in any dimension; the classical AM ≥ GM ≥ HM hierarchy becomes Maclaurin’s and Newton’s inequalities on symmetric means, and the concurrence bound CI ≤ CI,max = p 2(n − 1)/n is exactly Maclaurin’s first inequality, saturated at the maximally entangled state. Via Newton’s identities the symmetric polynomials are interchangeable with the replica moments Tr ρk, so the balance hierarchy is a repackaging of the data used to compute entanglement (R´enyi) entropies. We give two exactly solvable 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 the master formula Tr ρ2 = (1−q)(1+qn) (1+q)(1−qn) , q = e−βϵ. Finally we generalize the distinction between the maximal-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−1 2 βϵ, vanishing only at infinite temperature. The two-qubit results (concurrence = 2 √ xy, offset 1 2 tanh(βϵ/2)) are recovered as the n = 2 case. Lifting the model to the eternal BTZ black hole, the single-interval replica moments are Tr ρn = L−c 6 (n−1/n) and the symmetric-polynomial generating function becomes a Fredholm determinant; 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 the linearized Einstein equation. For a single interval in the planar-BTZ state we then evaluate this relative entropy in closed form, S(ρ ∥ ρ0) = c 3  u2 6 − ln(sinh u/u)  with u = πℓ/β: the first law is the cancellation of its O(u2) part, positivity follows from an elementary inequality on the zeros of sinh, and the leading c 540u4 is the quantum Fisher information, matched to the bulk canonical energy in the entanglement wedge. Reconstructing this relative entropy from the gravitational symplectic flux gives a convergent series in even zeta values resumming to the Ryu–Takayanagi area, and the two-interval case is the holographic mutual information c 3 ln[η/(1−η)], with a Ryu–Takayanagi phase transition at cross-ratio η = 1 2 , while the free-fermion counterpart, computed from the resolvent, is smooth in η — the twointerval measure’s theory dependence made explicit. Finally we extend the framework beyond qudits: the diagnostic depends only on the Schmidt rank (so asymmetric local dimensions add nothing), the spectral machinery is universal for all density operators (with h measuring mixedness rather than entanglement off the pure-state locus), and Gaussian continuous-variable states lift mode by mode, each two-mode-squeezed pair being the oscillator thermofield double with q = tanh2 r and Tr ρ2 = sech 2r.