Semi-device-independent randomness certification using $n\to 1$ quantum random access codes

We study random-number-expansion protocols based on the $n\to 1$ quantum random access codes (QRACs). We consider them in the semi-device-independent scenario where the inner workings of the devices are unknown to us but we can certify the dimensions of the systems being communicated. This approach does not require the use of entanglement and makes the physical realization of these protocols much easier than in the standard device-independent scenario. In our work, we propose a protocol for randomness expansion, compute min-entropy for the semi-device-independent protocol, and investigate $n\to 1$ QRACs with a view to their use in randomness-expansion protocols. We also calculate the dependence of the effectiveness of the randomness generation on $n$ and find it optimal for $n=3$, and provide the explanation for this fact.

Figure 1

Semi-device-independent random-number-expansion protocol.

Figure 2

The relationship between the average efficiency of the $2\to 1$ QRAC measured by $T_2$ and the min-entropy bound.

Figure 3

The relationship between the average efficiency of the $3\to 1$ QRAC measured by $T_3$ and the min-entropy bound. We see that the positive amount of randomness can be generated as soon as $T_3>6.65$, and the maximal value of the min-entropy is 0.3425 with $T_3=6.928203$.