Protocols and quantum circuits for implementing entanglement concentration in cat state, GHZ-like state and nine families of 4-qubit entangled states

Abstract

Three entanglement concentration protocols (ECPs) are proposed. The first ECP and a modified version of that are shown to be useful for the creation of maximally entangled cat and GHZ-like states from their non-maximally entangled counterparts. The last two ECPs are designed for the creation of maximally entangled $(n+1)$-qubit state $\frac{1}{\sqrt{2}}\left(|\Psi_{0}\rangle|0\rangle+|\Psi_{1}\rangle|1\rangle\right)$ from the partially entangled $(n+1)$-qubit normalized state $\alpha|\Psi_{0}\rangle|0\rangle+\beta|\Psi_{1}\rangle|1\rangle$, where $\langle\Psi_{1}|\Psi_{0}\rangle=0$ and $|\alpha|\neq\frac{1}{\sqrt{2}}$. It is also shown that W, GHZ, GHZ-like, Bell and cat states and specific states from the 9 SLOCC-nonequivalent families of 4-qubit entangled states can be expressed as $\frac{1}{\sqrt{2}}\left(|\Psi_{0}\rangle|0\rangle+|\Psi_{1}\rangle|1\rangle\right)$ and consequently the last two ECPs proposed here are applicable to all these states. Quantum circuits for implementation of the proposed ECPs are provided and it is shown that the proposed ECPs can be realized using linear optics. Efficiency of the ECPs are studied using a recently introduced quantitative measure (Phys. Rev. A $\textbf{85}$, 012307 (2012)). Limitations of the measure are also reported.