Entanglement witness via quantum-memory-assisted entropic uncertainty relation

By virtue of the quantum-memory-assisted entropic uncertainty relation (EUR), we analyze entanglement witness via the efficiencies of the estimates proposed by Berta (2010 Nat. Phys. 6 659) and Pati (2012 Phys. Rev. A 86 042105). The results show that, without a structured reservoir, the entanglement regions witnessed by these EUR estimates are only determined by the chosen estimated setup, and have no correlation with the explicit form of the initial state. On the other hand, with the structured reservoirs, the time regions during which the entanglement can be witnessed, and the corresponding entanglement regions closely depend on the choice of the estimated setup, the initial state and the state purity $p$. Concretely, for a pure state with $p=1$, the entanglement can be witnessed by both estimates, while for mixed states with $p=0.78$, it can only be witnessed using the Pati estimate. What is more, we find that the time regions incorporating the Pati estimate become discontinuous for the initial state with $|{\varphi }^{\prime }⟩=(|01⟩+|10⟩)/\sqrt{2}$, and the corresponding entanglement regions remain the same; however the entanglement can only be witnessed once by utilizing the Berta estimate. (letter)