Geometry of Faithful Entanglement

A typical concept in quantum state analysis is based on the idea that states in the vicinity of some pure entangled state share the same properties, implying that states with a high fidelity must be entangled. States whose entanglement can be detected in this way are also called faithful. We prove a structural result on the corresponding fidelity-based entanglement witnesses, resulting in a simple condition for faithfulness of a two-party state. For the simplest case of two qubits faithfulness can directly be decided and for higher dimensions accurate analytical criteria are given. Finally, our results show that faithful entanglement is, in a certain sense, useful entanglement; moreover, they establish connections to computational complexity and simplify several results in entanglement theory.