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A Sharp Rank-r Partial-Trace Inequality

Elazar Gershuni

Published Jul 26, 2026• 0 citations
Publisher

Abstract

Let $\Phi:M_d(\C)\to M_d(\C)$ be a completely positive map with a transpose-symmetric Kraus representation. Its Choi operator determines the rank-two constant \[ \beta_2(\Phi) =\frac12\sup_{\substack{\norm z=1\\ \SR(z)\le2}} \ip z{J(\Phi)z}. \] We prove that, for every $r\ge1$, \[ \Phi\circ\tau+(r-1)\beta_2(\Phi) \left(\Delta-\frac1r\id\right) \] is $r$-positive, where $\tau$ is transposition and $\Delta(X)=\Tr(X)I$. The proof turns the negative spectrum of a partially transposed rank-one operator into the edges of the complete graph $K_r$. Regrouping by vertices costs its degree $r-1$, rather than its $\binom r2$ edges. Transpose symmetry protects one direction and is exactly what produces the term $-\id/r$. For $\Phi=\Lambda_U\otimes\Lambda_V$, with $\Lambda_E(X)=\Tr(X)I-X^T$, a sharp double-skew singular-value estimate gives $\beta_2\le1$, with equality when both local dimensions are at least two. A map identity then yields, for every operator $C$ of rank at most $r$ on $U\otimes V$, \[ \norm{\Tr_U C}_2^2+\norm{\Tr_V C}_2^2 \le r\norm C_2^2+\frac1r\abs{\Tr C}^2, \] with dimension-uniform coefficients that are sharp whenever one local dimension is at least $r$ and the other is at least two. A second family, built from the edges of an arbitrary graph, has the same local constant and attains the universal coefficient $r-1$ whenever the graph contains an $r$-clique. The Choi lifting chain through the partial-trace and graph results, together with its sharpness witnesses, is formalized in Lean~4.

DOI
10.5281/zenodo.21643087

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