No cardinality bound for squashed entanglement

Abstract:Squashed entanglement is an additive bipartite entanglement measure. For a state $\rho_{AB}$, it is defined as the infimum of all conditional mutual information $I(A:B\mid E)$ evaluated on extensions $\rho_{ABE}$. It was unresolved whether there is a bound on the dimension of the conditioning system, $E$, needed for this optimization. We show that no such cardinality bound is possible. Explicitly, we find that the squashed entanglement for a partially dephased Bell pair on two qubits cannot be achieved for any finite dimensional extension $E$. A crucial ingredient of this proof is a direct sum step, which in a sense, symmetrizes two purifying systems while keeping the conditional mutual information unchanged and the coherence no worse.

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2609.14031 [quant-ph]

(or arXiv:2609.14031v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.14031 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.