Secure quantum weak oblivious transfer against individual measurements

In quantum weak oblivious transfer, Alice sends Bob two bits and Bob can learn one of the bits at his choice. It was found that the security of such a protocol is bounded by 2PAlice∗+PBob∗≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2P_{\mathrm{Alice}}^{*}+P_{\mathrm{Bob}}^{*}\ge 2$$\end{document}, where PAlice∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_{\mathrm{Alice}}^{*}$$\end{document} is the probability with which Alice can guess Bob’s choice, and PBob∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{\mathrm{Bob}}^{*}$$\end{document} is the probability with which Bob can guess both of Alice’s bits given that he learns one of the bits with certainty. Here we propose a protocol and show that as long as Alice is restricted to individual measurements, then both PAlice∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{\mathrm{Alice}}^{*}$$\end{document} and PBob∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{\mathrm{Bob}}^{*}$$\end{document} can be made arbitrarily close to 1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/2$$\end{document}, so that maximal violation of the security bound can be reached. Even with some limited collective attacks, the security bound can still be violated. Therefore, although our protocol still cannot break the bound in principle when Alice has unlimited cheating power, it is sufficient for achieving secure quantum weak oblivious transfer in practice.