Comparing the three W-like states with the W state

Abstract

In [Phy. Rev. Lett., 98, 260501 (2007)] and previous papers, the W-like states were used in many quantum communication schemes, distillation, teleportation and superdense coding. So far, no one has compared the W-like states with the W state. We investigate the properties of three W-like states named as $|\vartheta ^{\prime }\rangle $, $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $. We point out that $|\vartheta ^{\prime }\rangle $ (resp. $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $) has the maximal von Neumann entanglement entropy (vNEE) $S(\rho _{A})=\ln 2$ (resp. $S(\rho _{B})=\ln 2$ and $S(\rho _{C})=\ln 2$) and the maximal tangle $\tau _{A(BC)}=1$ (resp. $\tau _{B(AC)}=1$ and $\tau _{C(AB)}=1$) while the W state does not. We also indicate that if some one of three qubits is traced out, then the remaining two qubits of $|\vartheta ^{\prime }\rangle $ (resp. $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $) are more entangled than the two qubits of the W state by several entanglement measures. It means that the three W-like states have higher robustness against some particle loss than the W state. We show that $|\vartheta ^{\prime }\rangle $, $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $ can be suitable for perfect teleportation and superdense coding but the W state cannot.