Refining transposition bounds from traceless quantum perturbations

Abstract

The transpose map is the canonical nonphysical map in quantum information that is central to entanglement tests and to converse bounds for quantum capacity. It is a longstanding conjecture that transposition satisfies a dimension-independent strict submultiplicativity for physically relevant channel differences. We answer this affirmatively by proving that for every quantum channel $T$ on $\mathsf{L}({\mathbb{C}}^d)$, $\Vert \Theta \circ (\mathrm{id}-T){\Vert }_{\diamond }\le \frac{1}{\sqrt{2}}\Vert \Theta {\Vert }_{\diamond }\Vert \mathrm{id}-T{\Vert }_{\diamond }$. The same bound holds for all Hermiticity-preserving trace-annihilating maps. Because a trace-preserving channel produces a traceless deviation from the identity, transposition cannot act as strongly on such physically generated errors as it can on arbitrary perturbations. Thus, although transposition is extremal in the unrestricted diamond-norm geometry, physical constraints mandate a universal margin in the submultiplicative bound. As one immediate consequence, the finite-error Holevo--Werner converse extends from decoding errors $\epsilon <1/2$ to $\epsilon <1/\sqrt{2}$. The same tracelessness mechanism also yields rank-sensitive continuity refinements for negativity and logarithmic negativity, demonstrating its potential for broader application in quantum information.

arXiv comment9 pages