We study the entanglement R\'enyi $\ensuremath{\alpha}$ entropy $(\mathrm{ER}\ensuremath{\alpha}\mathrm{E})$ as the measure of entanglement. Instead of a single quantity in standard entanglement quantification for a quantum state by using the von Neumann entropy for the well-accepted entanglement of formation (EoF), the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ gives a continuous spectrum parametrized by variable $\ensuremath{\alpha}$ as the entanglement measure, and it reduces to the standard EoF in the special case $\ensuremath{\alpha}\ensuremath{\rightarrow}1$. The $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ provides more information in entanglement quantification and can be used, for example, to determine the convertibility of entangled states by local operations and classical communication. A series of results is obtained: (i) we show that the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ of two states, which can be mixed or pure, may be incomparable, in contrast to the fact that there always exists an order for the EoF of two states; (ii) similar to the case of EoF, we study in a fully analytical way the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ for arbitrary two-qubit states, the Werner states, and isotropic states in the general
d dimension; and (iii) we provide a proof of the previous conjecture for the analytical functional form of the EoF of isotropic states in the arbitrary
d dimension.
Abstract In an article in 2010, Kim et al introduced the definition of Rényi- α entanglement for bipartite quantum states and established an analytic formula of Rényi- α entanglement for arbitrary two-qubit states with α ⩾ 1 . They also derived a monogamy of entanglement in multi-qubit systems in terms of Rényi- α entanglement for α ⩾ 2 Kim et al (2010 J. Phys. A: Math. Theor . 43 445305). We find the proofs of theorems 2 and 3 contain some errors and we also present an improved derivation to overcome this flaw. The alternative derivation shows that the main conclusions remain valid despite the invalidity of the proofs.
As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
In this paper, we investigate the characterization of multipartite entanglement based on the unified
(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure
Eq,sA∣B(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures
Aq,sk(ρ) and
Gq,sk(ρ) for quantifying
k-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters
q and
s are in certain ranges,
Aq,sk(ρ) and
Gq,sk(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures
Aq,sk(ρ) and
Gq,sk(ρ) with respect to the parameters
q and
s for fixed quantum states. In addition, for fixed
q and
s, we also give their order relations for any two pure states.
As two of the most important entanglement measures--the entanglement of formation and the entanglement of distillation--have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures--the relative entropy of entanglement and the geometric measure of entanglement--are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.