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<p>We prove that the Hilbert–Schmidt norm of <i>k</i>-particle reduced density matrices of <i>N</i>-body fermionic states is bounded from above by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C_kN^{k/2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>k</mi> </msub> <msup> <mi>N</mi> <mrow> <mi>k</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> - matching the scaling behaviour of Slater determinant states. This generalises a recent result of Christiansen [<CitationRef CitationID="CR3">3</CitationRef>] 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 [<CitationRef CitationID="CR2">2</CitationRef>, <CitationRef CitationID="CR8">8</CitationRef>].</p>

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Hilbert–Schmidt Norm Estimates for Fermionic Reduced Density Matrices

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  • 发表日期：

    2026年3月23日

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  • François L. A. Visconti

摘要

We prove that the Hilbert–Schmidt norm of k-particle reduced density matrices of N-body fermionic states is bounded from above by \(C_kN^{k/2}\) 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].