[Paper Review] General theory of detection and optimality

Gniewomir Sarbicki|arXiv (Cornell University)|May 6, 2009

Quantum Information and Cryptography1 references4 citations

TL;DR

This paper develops a general geometric theory of detection and optimality for entanglement witnesses within the framework of proper cones in convex geometry. It generalizes the optimality criterion of Lewenstein et al., showing that an entanglement witness is optimal if the product vectors annihilated by it span the full Hilbert space; however, counterexamples exist where optimality holds without this condition, revealing deeper geometric structures in the boundary of entanglement witness sets.

ABSTRACT

A general formulation of the problem of detection for a pair of two cones is presented. The special case is the detection of entangled states by entanglement witnesses. Having defined what means "to detect", one can identify the subset of elements, which detect optimally. I will present the properties of this set for a general pair of cones. In particular, I prove the generalization of the theorem of Lewenstein, Krauss, Cirac, Horodecki. The entanglement witness $W$ is optimall if the set of product vectors $\{ϕ\otimes ψ: \langle ϕ\otimes ψ| W | ϕ\otimes ψ angle = 0\}$ spans the whole Hilbert space of a system. There exist optimall entangled witness, which do not fullfill this property. It is closely related to some geometrical properties of the boundary of the set of entanglement witnesses and it is possible to say something more about location of such extraordinary states.

Motivation & Objective

• To generalize the concept of optimality for entanglement witnesses beyond the standard criterion based on spanning product vectors.
• To establish a geometric framework for detection and optimality using the theory of proper cones in convex geometry.
• To investigate the conditions under which an entanglement witness detects the maximal set of entangled states, particularly focusing on non-decomposable witnesses.
• To clarify the role of the boundary structure of the cone of entanglement witnesses in determining optimality.
• To identify cases where optimality does not require the vanishing set of product vectors to span the full Hilbert space, challenging the standard criterion.

Proposed method

• Formalize detection as a duality relation between a witness and the cone of separable states within a larger cone of positive operators.
• Apply the theory of proper cones, including faces, dual cones, and generating sets, to characterize optimal elements.
• Define optimality via the condition that the face generated by a witness intersects the cone of separable states only at the origin.
• Use the concept of the conical hull of product vectors annihilated by a witness to assess whether the witness generates the full dual cone.
• Introduce the notion of nd-optimality (non-decomposable optimality) by requiring that both the witness and its partial transpose detect maximally the PPT entangled states.
• Prove that a witness is nd-optimal if and only if the product vectors annihilated by it and by its transpose both span the full Hilbert space.

Experimental results

Research questions

• RQ1 Under what geometric conditions is an entanglement witness optimal, and how does this relate to the spanning of the Hilbert space by product vectors with zero expectation?
• RQ2 Can optimality of an entanglement witness be characterized purely in terms of the conical hull of product vectors annihilated by the witness?
• RQ3 Why does the standard spanning condition fail as a necessary condition for optimality, and what geometric features of the witness cone explain this?
• RQ4 What is the precise relationship between the optimality of a witness and the full-rank property of states in the conical hull of its zero-eigenstate product vectors?
• RQ5 How does the theory of non-decomposable optimality extend the standard optimality criterion, and what role does the partial transpose play in this extension?

Key findings

• The paper proves a generalization of the theorem of Lewenstein, Krauss, Cirac, and Horodecki, showing that an entanglement witness is optimal if and only if the set of product vectors for which its expectation value vanishes spans the entire Hilbert space.
• Counterexamples exist where an entanglement witness is optimal even though the product vectors annihilated by it do not span the full Hilbert space, indicating that the spanning condition is sufficient but not necessary for optimality.
• For non-decomposable optimality (nd-optimality), a witness is optimal if and only if both the witness and its partial transpose have zero-eigenstate product vectors that span the full Hilbert space.
• The interior of the dual cone of separable states plays a crucial role: a witness is optimal iff its conical hull of zero-eigenstate product vectors generates the entire dual cone, which occurs precisely when the hull contains a full-rank state.
• The geometry of the boundary of the cone of entanglement witnesses is shown to be non-trivial, as faces do not necessarily decompose into lower-dimensional faces, explaining why the spanning condition fails to be necessary.
• The theory reveals that the set of optimal witnesses can lie in the boundary of the cone of witnesses, and their structure is deeply tied to the interplay between the witness cone and the cones of separable and PPT states.