Improved two-qubit quantum state sharing protocol based on entanglement swapping of bell states

Abstract

In a recent paper (Ma, Int J Theor Phys 59: 1844–1853, 2020), a multiparty quantum state sharing protocol based on entanglement swapping of Bell states was presented. However, as we show, when the number of agents is more than three, this protocol is insecure in the sense that the first agent and the last one can gain access to the dealer’s secret state without the others’ cooperation by collusion attack in this protocol. Hence, we propose two improved versions of this protocol in this paper. Security analysis indicates that two improved protocols are secure against the collusion attack and entangled probe attack. The successful ratio analysis shows that two improved protocols can be achieved with probability of 100%.

Appendix

The symbols contained in following two tables are explain as: (1) $\left| {\Phi^{ + } } \right\rangle ,\left| {\Phi^{ - } } \right\rangle ,\left| {\Psi^{ + } } \right\rangle ,\left| {\Psi^{ - } } \right\rangle$ are four different Bell states, i.e., $\left| {\Phi^{ \pm } } \right\rangle = \frac{1}{\sqrt 2 }\left( {\left| {00} \right\rangle \pm \left| {11} \right\rangle } \right)$,$\left| {\Psi^{ \pm } } \right\rangle = \frac{1}{\sqrt 2 }\left( {\left| {01} \right\rangle \pm \left| {10} \right\rangle } \right)$; (2) $I,\sigma_{x} ,\sigma_{y} ,\sigma_{z}$ are four different Pauli operators, and the corresponding matrices are defined as: $I = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right), \, \sigma_{x} = \left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right), \, \sigma_{y} = \left( {\begin{array}{*{20}c} 0 & { - i} \\ i & 0 \\ \end{array} } \right), \, \sigma_{z} = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & { - 1} \\ \end{array} } \right)$.