We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real N=3 states
ω(z) and solve the case
z<0 where
z is the non-diagonal entry in the density matrix. We also report a surprising result about the behaviour of the output entropy of the diagonal map for arbitrary dimensions
N; showing a bifurcation at N=6.
Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement. Quanta 2020; 9: 22–36.
We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real
N=3 states
ω(z) and solve the case
z<0 where
z is the non-diagonal entry in the density matrix. We also report a surprising result about the behavior of the output entropy of the diagonal map for arbitrary dimensions
N; showing a bifurcation at
N=6.
In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems
A and
B guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system
AB. This means that for any state
ρ of
AB there is a Hermitian operator
Λρ on
HAB=HA⊗HB such that
EF(ρ)=TrΛρρ and
EF(σ)≥TrΛρσ for any state
σ of
AB. We present an explicit example showing that, when
ρ is degenerate, this is not true even in the simplest case when
A and
B are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state
ρ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state
ρ. We use Wootters' formula and the help of Claude Fable 5 to find a state
ρ of the system
AB for which the latter property does not hold.
We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state
ρ (i.e. inequalities of the form
EF(ρ)−EF(σ)≤Cρ∥ρ−σ∥1) with and without restrictions on the support of the state
σ.
Convexity arises naturally in the study of quantum information. As a result, many useful tools from convex analysis can be used to give important results regarding aspects of quantum information. This thesis builds up methods using core concepts from convex analysis, including convex optimization problems, convex roof constructions, and conic programming, to study mathematical problems related to quantum entanglement. In this thesis, I develop a method for solving convex optimization problems that arise in quantum information theory by analyzing the corresponding converse problem. That is, given an element in a convex set, I determine a family of convex functions that are minimized at this point. This method is used find explicit formulas for the relative entropy of entanglement, as well as other important quantities used to quantify entanglement, and allows one to show important relationships between them. I also construct a practical algorithm that can be used to compute these quantities. This thesis also presents a method to compute convex roofs of arbitrary entanglement measures evaluated on highly symmetric bipartite states. I also establish a framework for completely characterizing quantum resource theories that are convex. For resource theories with a simple mathematical structure, this gives rise to a complete set of resource monotones that can be computed in practice using semidefinite programs. This has applications to the study of entanglement transformations.
A universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results on the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of π-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyze continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.