Entangled states in the role of witnesses

Abstract

Quantum entanglement lies at the heart of quantum mechanics and quantum information processing. In this work, we show a new framework where entangled states play the role of witnesses. We extend the notion of entanglement witnesses, developing a hierarchy of witnesses for classes of observables. This hierarchy captures the fact that entangled states act as witnesses for detecting entanglement witnesses and separable states act as witnesses for the set of non-block-positive Hermitian operators. Indeed, more hierarchies of witnesses exist. We introduce the concept of finer and optimal entangled states. These definitions not only give an unambiguous and non-numeric quantification of entanglement and a new perspective on edge states but also answer the open question of what the remainder of the best separable approximation of a density matrix. Furthermore, we classify all entangled states into disjoint families with optimal entangled states at its heart. This implies that we can focus only on the study of a typical family with optimal entangled states at its core when we investigate entangled states. Our framework also assembles many seemingly different findings with simple arguments that do not require lengthy calculations.

Common Coherence Witnesses and Common Coherent States

“…A framework which assembles hierarchies of “witnesses” has been proposed in [ 42 ]. In this framework, a coherence witness can witness coherent states, and on the other hand, a coherent state can also act as a “high-level-witness ” of coherence witnesses which witnesses coherence witnesses.…”

Entanglement-Separability Boundary Within a Quantum State

“…They showed that an arbitrary quantum state ρ can always be written in a form as ρ = λρ BSA +(1−λ)ρ OptE , where ρ BSA is a separable state and the weight λ of the separable part is maximal. Later, the form was proven to be unique [7,8]. The separable state ρ BSA is called the BSA of ρ, and the convex decomposition is called the BSA decomposition (also called Lewenstein-Sanpera decomposition).…”

“…The separable state ρ BSA is called the BSA of ρ, and the convex decomposition is called the BSA decomposition (also called Lewenstein-Sanpera decomposition). Recently, Wang showed a framework where entangled states play the role of high-level witnesses [8,9]. Instead of using a numerical value [10], Wang characterized the entanglement of an entangled state ρ using a set of entanglement witnesses for detecting the entangled state D ρ = {W |tr(W ρ) < 0}, where W is the entanglement witness of ρ.…”

“…It is determined that ρ is optimal if there is no other entangled state which is finer. It was showed that the optimal entangled state just corresponds to the remainder of the BSA of a density matrix [8].…”

“…The purely entangled structure and the purely separable structure of a quantum state.-Since the optimal entangled state doesn't include any separable state, here we call it the purely entangled structure of a quantum state or, we call it the purely entangled state if we cannot subtract any projector onto a product vector from itself. Given a separable state (higher-level witness [8]) σ, define D σ = {Θ|tr(Θσ) < 0, Θ = Θ † }; that is the set of operators "witnessed" by σ. For our purpose, we restrict the not-block-positive Hermitian operator Θ in the "generally-normalized" scope with −I ≤ Θ ≤ I and the operator norm Θ ∞ = 1, where I is the identity matrix.…”