We study the entanglement R{e}nyi $alpha$-entropy (ER$alpha $E) as the measure of entanglement. Instead of a single quantity in standard entanglement quantification for a quantum state by using the von Neumann entropy for the well-accepted entangleme
nt of formation (EoF), the ER$alpha $E gives a continuous spectrum parametrized by variable $alpha $ as the entanglement measure, and it reduces to the standard EoF in the special case $alpha rightarrow 1$. The ER$alpha $E provides more information in entanglement quantification, and can be used such as in determining the convertibility of entangled states by local operations and classical communication. A series of new results are obtained: (i) we can show that ER$alpha $E of two states, which can be mixed or pure, may be incomparable, in contrast to the fact that there always exists an order for EoF of two states; (ii) similar as the case of EoF, we study in a fully analytical way the ER$alpha $E for arbitrary two-qubit states, the Werner states and isotropic states in general d-dimension; (iii) we provide a proof of the previous conjecture for the analytical functional form of EoF of isotropic states in arbitrary d-dimension.
We propose a quantum method to judge whether two spatially separated clocks have been synchronized within a specific accuracy $sigma$. If the measurement result of the experiment is obviously a nonzero value, the time difference between two clocks is
smaller than $sigma$; otherwise the difference is beyond $sigma$. On sharing the 2$N$-qubit bipartite maximally entangled state in this scheme, the accuracy of judgement can be enhanced to $sigmasim{pi}/{(omega(N+1))}$. This criterion is consistent with Heisenberg scaling that can be considered as beating standard quantum limit, moreover, the unbiased estimation condition is not necessary.
