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

Abstract

Can the entanglement of an arbitrary many-body state be quantified without reconstructing the state? We answer the question affirmatively by showing how the measure of logarithmic negativity can be estimated using a finite number of even partial transpose (PT) moments of a many-body state. Considering the squared PT spectrum as a probability measure, we solve exactly the task of a truncated Hausdorff moment problem with Gauss and Gauss-Radau quadrature. We obtain a hierarchy of lower bounds on the logarithmic negativity, which improves with increasing order of moments. Convergence to the exact value of the measure can be achieved for every state of finite dimension. We illustrate the bounds for ground states of three different interacting spin models, as well as for a thermal state. An error analysis demonstrates the efficacy of our scheme towards handling the limitation of finite statistics encountered in actual experiments with randomized measurements. Moreover, an example of density matrix renormalization group based numerical estimation is provided for handling system sizes beyond the scope of exact diagonalization.

The paper asks how much logarithmic negativity can be certified from finitely many even partial-transpose (PT) moments, without state tomography. It maps the normalized squared PT eigenvalues to a probability measure on $[0,1]$, turning quantification into a truncated Hausdorff moment problem: minimize the integral of $h(x)=x^{-1/2}$ subject to the measured moments. For data $p_2,p_4,\ldots,p_K$, the minimum is attained uniquely by Gauss or Gauss–Radau quadrature. Hermite-interpolation polynomials below $h$ certify optimality. This yields lower bounds $\mathcal E_K\le\mathcal E$, starting with the known expression $\mathcal E_4=\tfrac12\ln(p_2^3/p_4)$. Each additional even moment strictly improves the raw bound until exact saturation at $K=4s$, where $s$ counts distinct nonzero PT eigenvalue magnitudes. Pure stabilizer states saturate at $K=4$. Closed forms are provided through $K=8$, with a quadrature algorithm for higher orders. Numerical illustrations cover two-qubit states, three spin models, thermal states, and fixed ten-site windows extracted using DMRG. For one 12-spin Ising example, $\mathcal E=0.542$, while $\mathcal E_4=0.245$ and $\mathcal E_8=0.380$. Optimality applies to the relaxed measure problem, not all physical-spectrum constraints: multiplicities, dimension, odd moments, and trace normalization are omitted, although nonnegativity can be imposed afterward. Generic spectra may require very high orders for saturation; no general convergence-rate estimate is given. Measurement costs increase with moment order and subsystem size. The finite-statistics study uses independent Gaussian relative noise rather than protocol-specific variances, and confidence-certified bounds remain future work.