No cardinality bound for squashed entanglement

• URL: http://arxiv.org/abs/2609.14031v1
• Date: Sat, 12 Sep 2026 16:32:11 GMT
• Title: No cardinality bound for squashed entanglement
• Abstract summary: We show that squashed entanglement for a partially dephased Bell pair on two qubits cannot be achieved for any finite dimensional extension $E$.<n>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.
• Score: 2.277447144331876
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Squashed entanglement is an additive bipartite entanglement measure. For a state $ρ_{AB}$, it is defined as the infimum of all conditional mutual information $I(A:B\mid E)$ evaluated on extensions $ρ_{ABE}$. 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.

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