Efficient Verification of Pure Quantum States in the Adversarial Scenario

Efficient verification of pure quantum states in the adversarial scenario is crucial to many applications in quantum information processing, such as blind measurement-based quantum computation and quantum networks. However, little is known about this topic so far. Here, we establish a general framework for verifying pure quantum states in the adversarial scenario and clarify the resource cost. Moreover, we propose a simple and general recipe to constructing efficient verification protocols for the adversarial scenario from protocols for the nonadversarial scenario. With this recipe, arbitrary pure states can be verified in the adversarial scenario with almost the same efficiency as in the nonadversarial scenario. Many important quantum states can be verified in the adversarial scenario using local projective measurements with unprecedented high efficiencies.

Figure 1

Number of tests $N(\epsilon ,\delta ,\lambda )$ required to verify a pure state within infidelity $\epsilon$ and significance level $\delta$ in the adversarial scenario using a homogeneous strategy characterized by $\lambda$. For comparison, the approximate formula $(\ln \delta )/(\lambda \epsilon \ln \lambda )$ is plotted as curves.

Figure 2

Overhead of QSV in the adversarial scenario compared with the nonadversarial scenario. Each curve represents an upper bound for the ratio $N(\epsilon ,\delta ,{\mathrm{\Omega }}_p)/N_{\mathrm{NA}}(\epsilon ,\delta ,\mathrm{\Omega })$, where $p=\nu /e$ and $\nu$ is the spectral gap of $\mathrm{\Omega }$. The same bound holds if $p_{*}(\nu ,\tau )\le p\le p_{*}(\nu )$; cf. Eq. (26).