General framework for verifying pure quantum states in the adversarial scenario

Bipartite and multipartite entangled states are of central interest in quantum information processing and foundational studies. Efficient verification of these states, especially in the adversarial scenario, is a key to various applications, including quantum computation, quantum simulation, and quantum networks. However, little is known about this topic in the adversarial scenario. Here we initiate a systematic study of pure-state verification in the adversarial scenario. In particular, we introduce a general method for determining the minimal number of tests required by a given strategy to achieve a given precision. In the case of homogeneous strategies, we can even derive an analytical formula. Furthermore, we propose a general recipe to verifying pure quantum states in the adversarial scenario by virtue of protocols for the nonadversarial scenario. Thanks to this recipe, the resource cost for verifying an arbitrary pure state in the adversarial scenario is comparable to the counterpart for the nonadversarial scenario, and the overhead is at most three times for high-precision verification. Our recipe can readily be applied to efficiently verify bipartite pure states, stabilizer states, hypergraph states, weighted graph states, and Dicke states in the adversarial scenario, even if only local projective measurements are accessible. This paper is an extended version of the companion paper Zhu and Hayashi, Phys. Rev. Lett. 123, 260504 (2019).

Figure 1

The region $R_{N,\mathrm{\Omega }}$ composed of $(p_{\rho },f_{\rho })$ as defined in Eq. (22). This region is the convex hull of points $({\eta }_{\mathbf{k}}\left(\mathit{\lambda }\right),{\zeta }_{\mathbf{k}}\left(\mathit{\lambda }\right))$ for $\mathbf{k}\in {\mathcal{S}}_N$, which are highlighted as red dots. Here, $\mathrm{\Omega }$ has three distinct eigenvalues, namely, 1, 0.4, and 0.2.

Figure 2

Variations of $\zeta (N,\delta ,\lambda )$ and $F(N,\delta ,\lambda )$ with $\delta$ and $\lambda$ for $N=2$ (left plots) and $N=10$ (right plots).

Figure 3

Minimum numbers of tests required to verify a pure state with five different homogeneous strategies. Here, $\epsilon =0.01$ in the upper plot and $\epsilon =0.1$ in the lower plot. In each plot, the red curve represents the approximate formula $(1-\delta )/\left(\epsilon \delta \right)$ when $\lambda =0$ [cf. Eq. (51)]. The four lines represent the approximate formula $(F+\lambda \epsilon ){log}_{10}\delta /\left(\lambda \epsilon {log}_{10}\lambda \right)$ [cf. Eq. (76)].

Figure 4

Variation of $N(\epsilon ,\delta ,\lambda )$ with $\lambda$ and $\delta$. Here, $\epsilon =0.01$ in the upper plot and $\epsilon =0.1$ in the lower plot. The four curves in each plot represent the approximate formula $ln\delta /(\lambda \epsilon ln\lambda )$ [cf. Eqs. (78) and (80)].

Figure 5

Optimal homogeneous strategy in the limit $\delta \to 0$. Here ${\lambda }_{*}\left(\epsilon \right)$ denotes the value of $\lambda$ that minimizes $\mathcal{N}(\epsilon ,\lambda )$ defined in Eq. (82), which determines the number of required tests. ${\overline{\mathcal{N}}}_{*}\left(\epsilon \right)$ denotes the number of required tests normalized with respect to the benchmark, as defined in Eq. (88).

Figure 6

Single-copy verification in the adversarial scenario and nonadversarial scenario. The target state can be verified within infidelity $\epsilon$ and significance level $\delta$ in the adversarial (nonadversarial) scenario using a single test if the value of $\delta$ lies above the blue solid curve (red dashed line) [cf. Eqs. (98) and (5)].

Figure 7

The optimal probability $p_{*}(\nu ,\tau )$ for performing the trivial test (upper plot), the prefactor $h_{*}(\nu ,\tau )$ (middle plot), and the overhead $\nu h_{*}(\nu ,\tau )$ (lower plot) for high-precision QSV in the adversarial scenario. In the legend, $\beta =1-\nu$.

Figure 8

The optimal probability $p_{*}\left(\nu \right)$ for performing the trivial test in high-precision QSV and a pretty-good approximation $p_0\left(\nu \right)=\nu /\mathrm{e}$ (upper plot). Variations of $\nu h_{*}\left(\nu \right)$ and its upper bound $\nu h(p_0\left(\nu \right),\nu )$ with $\nu$ (lower plot). The black solid curve in the lower plot represents the first upper bound for $\nu h(p_0\left(\nu \right),\nu )$ presented in Eq. (148).

Figure 9

Upper bound on the ratio of $N(\epsilon ,\delta ,{\mathrm{\Omega }}_p)$ over $N_{\mathrm{NA}}(\epsilon ,\delta ,\mathrm{\Omega })$ according to the first bound in Eq. (152) with $\delta =\epsilon$, where $p=\nu /\mathrm{e}$ or $p_{*}(\nu ,\tau )\le p\le p_{*}\left(\nu \right)$. This ratio characterizes the overhead of QSV in the adversarial scenario.

Figure 10

Qualitative comparison among various approaches for estimating or verifying quantum states with respect to the efficiency and the strength of assumptions. Thanks to the recipe proposed in Sec. 9, QSV in the adversarial scenario can achieve nearly the same efficiency as QSV in the nonadversarial scenario, although the underlying assumptions are much weaker.