We give a criterion for the existence of a global supporting affine functional for the entanglement of formation at an entangled two-qubit state. It exists if and only if the spin-flip bilinear form is nondegenerate on the support of the state, or equivalently, if the last concurrence root associated with the support is nonzero. In this case the entanglement of formation satisfies a two-sided Lipschitz bound relative to the reference state. In the degenerate case the state has rank two or three, and its projective support is tangent to the product-state quadric. By suitably choosing a perturbation outside the original support space, the decrease in the entanglement of formation is, to leading order, proportional to the square root of the mixing parameter. We give an explicit expression for the leading coefficient. Such a square-root decrease violates the one-sided Lipschitz bound, and therefore no globally supported affine functional exists at this state.
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.
Quantum entanglement does not necessarily imply Einstein-Podolsky-Rosen steering. We identify a \emph{boundary mechanism} that closes this gap when an entangled state meets the boundary of the trusted state space in a nondegenerate way. The mechanism is local: a projective assemblage that approaches a boundary contact with first-order tangential coherence but only second-order inward defect cannot be reproduced by any finite-measure local-hidden-state model. In finite dimensions this yields a support-kernel criterion for an arbitrary fixed steering cut: if a trusted conditional state is rank-deficient and its first-order block couples its support to its kernel, then the state is both NPT (negative partial transpose) and projectively steerable. The two-qubit product-null sector provides the minimal transparent realization. A product vector in the kernel gives the pure boundary contact, and a single coherence entry simultaneously serves as the tangential displacement, the NPT minor, and the steering obstruction. Hence, on this boundary, entanglement collapses to two-way projective steering -- including the genuinely mixed rank-three branch and the rank-two case as a low-dimensional instance. We also provide a filtered standard form, an explicit rank-three Cholesky parametrization, and a compact boundary certificate.
In this paper we investigate two different entanglement measures in the case of mixed states of two qubits. We prove that the negativity of a state can never exceed its concurrence and is always larger than √[(1 − C ) 2 + C 2 ] − (1 − C ), where C is the concurrence of the state. Furthermore, we derive an explicit expression for the states for which the upper or lower bound is satisfied. Finally we show that similar results hold if the relative entropy of entanglement and the entanglement of formation are compared.
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems.
The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state
ρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality
S(ρ)−S(σ)≤Cρ(1−F(ρ,σ)) valid for any state
σ, where
Cρ is a constant depending on the rank of
ρ.
The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state
ρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality
EF(ρ)−EF(σ)≤Cρ(1−Trρσ) valid for any state
σ, where
Cρ is a constant depending on the Schmidt rank of
ρ.
In both cases the optimal constant
Cρ is equal to the optimal constant
Kd in the log-Sobolev inequality for the complete graph with
d vertices: in the first case
d=rankρ, in the second one
d=rankρA=rankρB.
The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
We study the entanglement-breaking (EB) space, such that the entanglement of formation (EOF) of a bipartite quantum state is additive when its range is an EB subspace. We systematically construct the EB spaces in the Hilbert space ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, and the two-dimensional EB space in ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. We characterize the expression of two-qubit states of rank two with nonadditive EOF, if they exist. We further apply our results to construct EB spaces of arbitrarily given dimensions. We show that the example in Phys. Rev. Lett. 89, 027901 (2002) is a special case of our results. We further work out the entanglement cost of a qubit-qutrit state in terms of the two-atom system of the Tavis-Cummings model.
We prove exactly that the squared entanglement of formation, which quantifies the bipartite entanglement, obeys a general monogamy inequality in an arbitrary multiqubit mixed state. Based on this kind of exotic monogamy relation, we are able to construct two sets of useful entanglement indicators: the first one can detect all genuine multiqubit entangled states even in the case of the two-qubit concurrence and n-tangles being zero, while the second one can be calculated via quantum discord and applied to multipartite entanglement dynamics. Moreover, we give a computable and nontrivial lower bound for multiqubit entanglement of formation.