Many-body entanglement quantification in arbitrary states through a few partial-transpose moments

The paper in detail

For a bipartite state with partial-transpose eigenvalues \(\lambda_i\), the logarithmic negativity is \(\mathcal E=\ln\sum_i|\lambda_i|\) and the moments are \(p_k=\sum_i\lambda_i^k\). Even moments see only magnitudes, as does \(\mathcal E\). Writing \(q_i=\lambda_i^2/p_2\), which form a probability distribution, and \(\mu_k=p_{2k}/p_2^k\), \[\mathcal E=\tfrac12\ln p_2+\ln S,\qquad S=\sum_iq_i^{1/2}=\int_0^1x^{-1/2}\,d\nu,\] where \(\nu\) places weight \(q_i\) at \(q_i\). Given \(\mu_1,\dots,\mu_r\) with \(r=K/2\), the smallest \(\int x^{-1/2}d\sigma\) over all probability measures on \([0,1]\) with those moments gives a valid lower bound \(\mathcal E_K\).

Theorem 1 finds the minimizer: with \(\ell=\lfloor K/4\rfloor\), it is the \(\ell\)-point Gauss rule of the given moments when \(r\) is even, and the Gauss–Radau rule with a fixed node at \(x=1\) when \(r\) is odd, or \(\nu\) itself if it has at most \(\ell\) atoms. For \(K=6\) the free node and its weight are \[a=\frac{\mu_2-\mu_3}{1-\mu_2},\qquad w_a=\frac{(1-\mu_2)^2}{1-2\mu_2+\mu_3},\] and \(\mathcal E_6=\tfrac12\ln p_2+\ln[w_aa^{-1/2}+1-w_a]\); for \(K=8\) the two nodes are roots of a quadratic in \(\mu_2,\mu_3,\mu_4\). Normalization \(p_1=1\) alone gives \(\mathcal E\ge0\), so the certified value is \(\max\{\mathcal E_K,0\}\).

Each additional moment adds a constraint, so \(\mathcal E_4\le\mathcal E_6\le\mathcal E_8\le\dots\le\mathcal E\), and Proposition 1 shows that the improvement is strict until the bound is exact. Theorem 2 gives the exact order: \(\mathcal E_K=\mathcal E\) if and only if there are at most \(\lfloor K/4\rfloor\) distinct nonzero magnitudes, so \(K_{\rm sat}=4s\). Because a measure on a compact interval is fixed by its moments, the bounds converge to \(\mathcal E\) for every state, regardless of dimension.

For the Werner state at \(p=1/2\), \(\mathcal E_4=0.170\), \(\mathcal E_6=0.174\) and \(\mathcal E_8=\mathcal E=0.223\). For two five-site blocks inside a 12-site transverse-field Ising chain at \(h=0.85\), \(\mathcal E=0.542\) against \(\mathcal E_4=0.245\), \(\mathcal E_6=0.290\) and \(\mathcal E_8=0.380\), with several hundred distinct magnitudes; XXZ and Lipkin–Meshkov–Glick ground states behave similarly, DMRG extends the analysis to 36 sites, and in a thermal state each \(\mathcal E_K\) reaches zero below the temperature at which entanglement disappears. With noisy moments, \(K=6\) data are admissible only if \(\mu_2^2<\mu_3<\mu_2\); in a simple noise model the error levels off at the exact gap \(\mathcal E-\mathcal E_6\) once enough shots are taken.

Transferable methods

1. Hermite interpolation certificate for moment problems

   To lower-bound the integral of a function whose derivatives alternate in sign, given finitely many moments, build a polynomial that touches the function at quadrature nodes and stays below it. Its integral is fixed by the moments, and the quadrature rule attains it, so the bound is both valid and optimal.

   For \(h(x)=x^{-1/2}\), \(h^{(m)}(x)=(-1)^m\binom{2m}{m}\frac{m!}{4^m}x^{-m-1/2}\): even derivatives are positive and odd ones negative on \((0,1]\). The Hermite interpolant \(P\) of degree \(2\ell-1\) matching \(h\) and \(h^\prime\) at the Gauss nodes satisfies \[h(x)-P(x)=\frac{h^{(2\ell)}(\xi)}{(2\ell)!}\prod_{j=1}^{\ell}(x-x_j)^2\ge0,\] and \(\int P\,d\sigma\) depends only on the given moments. With one more moment, an extra interpolation point \(c\) adds a factor \((x-c)\) multiplied by a negative odd derivative, which is nonnegative on \([0,1]\) only for \(c=1\): the Radau node is forced by the sign.

   Editorial check: for the Werner state at \(p=1/2\), whose partial transpose has eigenvalue \(3/8\) three times and \(-1/8\) once, the closed forms give \(\mathcal E_4=0.1703\), \(\mathcal E_6=0.1743\) and \(\mathcal E_8=0.2231=\ln(5/4)=\mathcal E\), matching the paper. For the pure state \(\tfrac12\ket{00}+\tfrac{\sqrt3}2\ket{11}\), with three magnitudes, they give 0.470, 0.518 and 0.604, all below \(\mathcal E=0.624\), consistent with saturation only at \(K=12\).

   How to reuse it

   Any spectral sum \(\sum_if(\lambda_i)\) of an unknown operator can be bounded optimally from power sums when \(f\), after a change of variables, has derivatives of fixed alternating sign: entropies from Rényi moments, realignment norms, fidelities. Quadrature gives both the bound and the extremal spectrum, which shows what the data cannot distinguish.

2. Saturation by counting spectral levels

   The bound becomes exact exactly when the unknown measure has no more atoms than the quadrature can place. Counting distinct eigenvalue magnitudes therefore predicts in advance how many moments are needed.

   The \(\ell\)-point Gauss rule reproduces moments up to order \(2\ell\). If the true measure has \(s\le\ell\) atoms it is itself the unique minimizer, so \(\mathcal E_K=\mathcal E\); uniqueness of the minimizer shows that more than \(\lfloor K/4\rfloor\) distinct magnitudes leave a strict gap. Hence \(K_{\rm sat}=4s\). For an entangled pure stabilizer state the Schmidt spectrum is flat, all nonzero \(|\lambda_i|\) are equal, \(s=1\), and \[\mathcal E_4=\tfrac12\ln\frac{p_2^3}{p_4}\] is already exact.

   Editorial check: a Bell pair has partial-transpose eigenvalues \(\pm\tfrac12\), so \(p_2=1\), \(p_4=1/4\) and \(\mathcal E_4=\tfrac12\ln4=\ln2=\mathcal E\). For random two-qutrit mixed states the ordering \(\mathcal E_4\le\mathcal E_6\le\mathcal E_8\le\mathcal E\) held in every sample, with \(\mathcal E_4\) and \(\mathcal E_6\) negative even for entangled states, which is why the certified value is \(\max\{\mathcal E_K,0\}\).

   How to reuse it

   Before spending effort on higher moments, estimate how many distinct levels the relevant spectrum has: few levels mean exactness at low order, a long tail of small values means slow convergence. The same atom-counting logic applies to any estimator built on Gauss quadrature.

Source: Setup (Eqs. 1–7); Theorem 1 and closed forms (Eqs. 8–13); Proposition 1, Theorems 2–3, Corollary 1; End Matter Appendices A–D. Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Moment data taken to detect entanglement can now be turned into a certified amount at no extra experimental cost, with closed forms up to \(K=8\). The saturation criterion says in advance when a few moments suffice: entangled pure stabilizer states are quantified exactly by \(p_2\) and \(p_4\). In spin-chain examples the eighth moment recovers about half of the gap left by \(\mathcal E_4\).

Limitations. Only even moments are used; the odd ones, which carry the signs of the eigenvalues and include the cheap third moment, are left aside. Estimated moments can fall outside the admissible region, and the paper gives point estimates rather than lower bounds at a stated confidence level. For critical states with many small eigenvalues convergence is slow, with \(K_{\rm sat}\) above \(10^3\), far beyond present experiments.

Description based on the full text of the linked arXiv version, read 2026-10-28; equations follow the paper’s notation.