Generalized measurements on qubits in quantum randomness certification and expansion

TL;DR

The paper addresses certifying quantum randomness when devices are untrusted by leveraging generalized measurements (POVMs) on qubits to surpass the traditional one-bit-per-qubit limit. It combines an elegant Bell operator-based DI approach with prepare-and-measure games, the Entropy Accumulation Theorem, and SDP/NPA techniques to bound min-entropy against classical and quantum adversaries. Experimentally, it reports >1 bit of min-entropy per qubit: 1.21 bits from a single POVM on a qubit in an entangled setup and 1.07 bits from a prepare-and-measure scenario, supported by finite-data analysis. The results demonstrate that POVMs can substantially boost randomness generation rates and security for both DI and MDI protocols, with wide implications for quantum cryptography and metrology, and point toward extensions to higher-dimensional systems.

Abstract

Quantum mechanics has greatly impacted our understanding of the microscopic nature. One of the key concepts of this theory is generalized measurements, which have proven useful in various quantum information processing tasks. However, despite their significance, they have not yet been shown empirically to provide an advantage in quantum randomness certification and expansion protocols. This investigation explores scenarios where generalized measurements can yield more than one bit of certified randomness with a single qubit system measurement on untrusted devices and against a quantum adversary. We compare the robustness of several protocols to exhibit the advantage of exploiting generalized measurements. In our analysis of experimental data, we were able to obtain $1.21$ bits of min-entropy from a measurement taken on one qubit of an entangled state. We also obtained $1.07$ bits of min-entropy from an experiment with quantum state preparation and generalized measurement on a single qubit. We also provide finite data analysis for a protocol using generalized measurements and the Entropy Accumulation Theorem. Our exploration demonstrates the potential of generalized measurements to improve the certification of quantum sources of randomness and enhance the security of quantum cryptographic protocols and other areas of quantum information.

Figures (5)

• Figure 1: Illustration of the relaxations of the quantum set of probability distributions under the constraint that the probability distributions have correlators exactly equal to those obtained in the experiment, referred to as equality constraining. This set is denoted as $\mathcal{Q}_{=}$. This set is contained in a relaxed set $\mathcal{Q}_{2+ABE, =}$ obtained at the level $2+ABE$ of NPA with the same equality constraining. Further relaxation is obtained if the equalities are replaced with inequalities, denoted as $\mathcal{Q}_{2+ABE, \geq}$. Any set satisfying the equality constraining satisfies the inequalities.
• Figure 2: The min-entropy certified by given experimental values of modified elegant Bell operator \ref{['eq:certificate_Entangled']} with $k = 2$. For the Bell value obtained in the experiment smania2020experimental, equal to $6.8907$, the certified min-entropy is $1.55$ bits secure against a classical adversary. In the perfect case, 2 bits of min-entropy are certified.
• Figure 3: The dependence of the guessing probability on the coefficient $k$ in the additional term of the prepare-and-measure protocol. The optimal value of $k$ is $2$, which gives a guessing probability of $0.54565$. The guessing probability is calculated for the Bell expression \ref{['eq:certificate_PnM']} using the data from the experiment smania2020experimental. The values are obtained using level $3$ of MLP.
• Figure 4: Demonstration of the robustness of min-entropy certification with the prepare-and-measure protocol when an additional term is added with the coefficient $k=2$, which is shown to be optimal. The values are obtained using level $3$ of MLP. The maximal value is $\log_2 (1/3) \approx 1.58$. The experiment tavakoli2020self attained a value of the certificate \ref{['eq:certificate_PnM']} equal to $0.785361$ giving $0.87$ bits of min-entropy. The critical value of the certificate that allows certifying more than 1 bit of min-entropy is $0.787$.
• Figure 5: (color online) Robustness of randomness certification protocols using various certificates. The black dots refer to the relative certificate values obtained in experiments from smania2020experimentaltavakoli2020self, viz.$\eta^{ent} = 0.9962$ and $\eta^{pnm} = 0.9834$ for Bell non-locality certificate \ref{['eq:certificate_Entangled']} with $k = 1$, and prepare-and-measure certificate \ref{['eq:certificate_PnM']} with $k = 1$, respectively. The other experiment by Liu et al.liu2019experimental exploiting elegant Bell operator achieved $\eta^{liu} = {6.8138} / {4 \sqrt{3}} \approx 0.9835$ was not able to exhibit any advantage of POVMs, as for this relative certificate value for Bell non-locality protocols, the efficiency of that operator and CHSH with PMs are similar; it also lacked certification of the POVM form (cf. smania2020experimental) and thus was not device-independent.