No dimension limit for systems used in squashed entanglement measures
No cardinality bound for squashed entanglement
Information Theory
Summary
Squashed entanglement is a way to measure quantum connections between two systems using an extra helper system. People wondered if this helper system could always be chosen to have a limited size. The authors found that this is not possible: to get the exact measurement in some cases, the helper system must be infinitely large. They demonstrated this using a certain two-qubit quantum state and a special mathematical technique to keep the measurement unchanged while symmetrizing parts of the system.
What this means in practice
- •For quantum communication engineers: Design quantum communication protocols knowing there is no fixed dimension limit for auxiliary systems in squashed entanglement calculations.
- •For quantum cryptography developers: Assess security assumptions in entanglement-based cryptographic schemes with awareness that some entanglement measures need infinite-dimensional helpers.
A theory result. No direct application yet.
Authors
Rabsan Galib Ahmed, Graeme Smith
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.