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March 25, 2026Annales Henri Poincaré1 citationsOpen Access

Hilbert–Schmidt Norm Estimates for Fermionic Reduced Density Matrices

FVFrançois L. A. ViscontiLMU Klinikum

Key Points

  • The study aims to establish bounds for the Hilbert–Schmidt norm of k-particle reduced density matrices of N-body fermionic states.
  • Proving upper bounds on the Hilbert–Schmidt norm for k-particle reduced density matrices.
  • Generalizing existing results on 2-particle reduced density matrices to higher order density matrices.
  • Analyzing the implications for von Neumann and 2-Rényi entropy.
  • Established an upper bound of C_kN^(k/2) for the Hilbert–Schmidt norm.
  • Confirmed that this scaling matches that of Slater determinant states.
  • Provided lower bounds on von Neumann and 2-Rényi entropy for the reduced density matrices.

Abstract

Abstract We prove that the Hilbert–Schmidt norm of k -particle reduced density matrices of N -body fermionic states is bounded from above by CₖN^k/2 C k N k / 2 - matching the scaling behaviour of Slater determinant states. This generalises a recent result of Christiansen 3 on 2-particle reduced density matrices to higher order density matrices. Moreover, our estimate directly yields a lower bound on the von Neumann entropy and the 2-Rényi entropy of reduced density matrices, thereby providing further insight into conjectures of Carlen–Lieb–Reuvers 2, 8.

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Cite This Study

François L. A. Visconti (2026) studied this question.

synapsesocial.com/papers/69c37be2b34aaaeb1a67eb5chttps://doi.org/10.1007/s00023-026-01686-z
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