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 $\mathsf{L}(\mathbb{C}^d)$, $\|\Theta\circ(\mathrm{id}-T)\|_\diamond \le \frac{1}{\sqrt{2}}\|\Theta\|_\diamond \|\mathrm{id}-T\|_\diamond$. 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 $\varepsilon<1/2$ to $\varepsilon<1/\sqrt{2}$. 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:2602.17748 [quant-ph]

(or arXiv:2602.17748v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2602.17748 Focus to learn more arXiv-issued DOI via DataCite

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