Quantum state targeting

We introduce a primitive for quantum cryptography that we term “state targeting.” We show that increasing one’s probability of success in this task above a minimum amount implies an unavoidable increase in the probability of a particular kind of failure. This is analogous to the unavoidable disturbance to a quantum state that results from gaining information about its identity, and can be shown to be a purely quantum effect. We solve various optimization problems for state targeting that are useful for the security analysis of two-party cryptographic tasks implemented between remote antagonistic parties. Although we focus on weak coin flipping, the results are significant for other two-party protocols, such as strong coin flipping, partially binding and concealing bit commitment, and bit escrow. Furthermore, the results have significance not only for the traditional notion of security in cryptography, that of restricting a cheater’s ability to bias the outcome of the protocol, but also for a different notion of security that arises only in the quantum context, that of cheat sensitivity. Finally, our analysis leads to some interesting secondary results, namely, a generalization of Uhlmann’s theorem and an operational interpretation of the fidelity between two mixed states.

Figure 1

A slice of the Bloch sphere, showing the two possible target states $⟨{\psi }_0\mid$ and $\mid {\psi }_1⟩$. The state $\mid {\psi }_M⟩$ is the one that Alice must submit to achieve her maximal control. If she submits the state ${\rho }_H$, then she can achieve some nontrivial control at the cost of some disturbance by steering to a particular convex decomposition of the state prior to her announcement. The pair of states $\mid \varphi ⟨,\mid {\varphi }^{\prime }⟩$ indicate the elements of a convex decomposition that Alice might use where the target state is $\mid {\psi }_0⟩$. In general the optimal state to submit in order to minimize her disturbance for a given control lies between ${\rho }_H$ and $\mid {\psi }_M⟩$.

Figure 2

The control-disturbance trade-off for two pure states, with ${\mathbf{\mid }⟨{\psi }_0\mid {\psi }_1⟩\mathbf{\mid }}^2=1∕2$. Curve (a) is for the case where Alice does not have the option to decline being tested, considered in Sec. 3B, whereas curve (b) is for the case where she does, considered in Sec. 5B. For the former, the disturbance is nonzero for every control above 1∕2, while for the latter it is only nonzero after the point indicated in the diagram.

Figure 3

A section of the Bloch sphere showing the state ${\rho }^{\mathrm{opt}}$ that yields the largest disturbance-free control when the possible target states are $\mid {\psi }_0⟩$ and $\mid {\psi }_1⟩$.

Figure 4

A pair of nonorthogonal classical probability distributions.