Experimental certification of more than one bit of quantum randomness in the two inputs and two outputs scenario

The NPA hierarchy is used to formulate SDPs which after being solved provide an upper bound on the probability of guessing the value of a random number by an adversary [36]. Thus, the values obtained using this method are suitable for certification of the generated randomness from quantum devices. The technique is limited only to optimizing functions that are linear expressions of probabilities. Let us consider the set of all conditional probability distributions $\{P(a,b|x,y)\}$, where a and b are the outcomes of the measurements performed by Alice and Bob, when their measurement settings are x and y, respectively. The general form of a Bell expression, which is a linear functional of the conditional probability distributions, is

$$\begin{equation} \sum_{a,b,x,y} c_{a,b,x,y} P\left(a,b|x,y\right), \end{equation} \tag{ 1 }$$

where the coefficients $\{c_{a,b,x,y} \}$ are real numbers. One of the proposed protocols from [28] used the following Bell operator as a randomness privacy certificate:

$$\begin{equation} C\left(0,1\right) + C\left(0,2\right) + C\left(1,0\right) + C\left(1,1\right) - C\left(1,2\right), \end{equation} \tag{ 2 }$$

where the correlators are defined as follows:

$$\begin{equation} \begin{aligned} C\left(x,y\right) &\equiv P\left(0,0|x,y\right) + P\left(1,1|x,y\right) - P\left(0,1|x,y\right) - P\left(1,0|x,y\right). \end{aligned} \end{equation} \tag{ 3 }$$

One may note that (2) consists of a well-known CHSH expression [21] plus an additional term. The maximal value allowed in quantum mechanics, i.e. the Tsirelson bound, of (2) is $1 + 2 \sqrt{2}$. When the Tsirelson bound is achieved, then the quantum state and all measurement operators are uniquely determined [40], and for the pair of settings x = 0, y = 0, the measurement results are uniformly distributed, ${\forall}{a,b} P(a,b|0,0) = 0.25$. A full analysis of the protocol of randomness generation using (2) including randomness accumulation and extraction aspect has been presented in [14].

In a recent work [41] SDP was used to produce an approximation of the matrix logarithm function, resulting in a numerical method for efficient optimization of expressions on the quantum relative entropy [42]. Furthermore, this method has been used to determine the lower bounds of the conditional von Neumann entropy certified in the device-independent approach [43] using the extended NPA [44] with the NCPOL2SDPA tool [45].

Let Q_A , Q_B , and Q_E be the Hilbert spaces of devices of Alice, Bob, and adversary, respectively, and $\rho_{Q_A, Q_B, Q_E}$ their shared tri-partite quantum system. The numerical technique can be applied to calculate a lower bound on the conditional von Neumann entropy

$$\begin{equation} H\left(Q_A, Q_B|x = x^{*},y = y^{*}, Q_E\right)_{\rho_{Q_A, Q_B, Q_E}}. \end{equation} \tag{ 4 }$$

Let $\{\{M_{a|x} \}_a \}_x$ and $\{\{N_{b|y} \}_b \}_y$ denote operators of the positive operator valued measurements performed by Alice and Bob, respectively. This employs the Gauss–Radau quadrature rule to lower bound (4). Let w_i and t_i be the nodes and weights defined by this quadrature. A lower bound on (4) can be obtained from [46]:

$$\begin{equation} \sum_{i} c_i \sum_{a,b = 0,1} \inf_{\substack{Z_{a,b} \in B\left(Q_E\right), \\ \text{cond}\left(P\right)}} \left( 1 + F\left[M_{a|x^{*}}, N_{b|y^{*}}, Z_{a,b}, t_i\right] \right), \end{equation} \tag{ 5 }$$

where $F[M_{a|x^{*}}, N_{b|y^{*}}, Z_{a,b}, t_i]$ is defined as $\mathrm{Tr} \left[ \rho_{Q_A, Q_B, Q_E} \left( O_1 + O_2 \right) \right]$. In (5) $\text{cond}(P)$ expresses that the probability distribution $P(a,b|x,y) \equiv \mathrm{Tr}[\rho_{Q_A, Q_B} M_{a|x^{*}} \otimes N_{b|y^{*}}]$ satisfies a certain, specified by the protocol, set of linear constraints. The operators O₁ and O₂ are defined by

$$\begin{equation} O_1 \equiv M_{a|x^{*}} \otimes N_{b|y^{*}} \otimes \left( Z_{a,b} + Z_{a,b}^{\dagger} + \left(1 - t_i\right) Z_{a,b} Z_{a,b}^{\dagger} \right), \end{equation} \tag{ 6a }$$

$$\begin{equation} O_2 \equiv t_i \left( \unicode{x1D7D9}_{Q_A Q_B} \otimes Z_{a,b} Z_{a,b}^{\dagger} \right). \end{equation} \tag{ 6b }$$

c_i are coefficients calculated from the Gauss–Radau quadrature as $c_i \equiv w_i / (t_i \log(2))$. The index i in the summation (5) takes the values indexing the nodes in the quadrature, omitting the last one.

To certify the randomness, we employed the recently announced two families of Bell expressions [29]. First of them is, parametrized by $\delta \in (0, \pi/6]$, defined as

$$\begin{equation} I_\delta \equiv C\left(0,0\right) + \frac{1}{\sin{\delta}} \left( C\left(0,1\right) + C\left(1,0\right) \right) - \frac{1}{\cos{2 \delta}} C\left(1,1\right). \end{equation} \tag{ 7 }$$

The members of this family have self-testing properties, use two settings for each party, and can certify two bits of randomness for the measurement settings $x^{*} = y^{*} = 0$.

The second family, parametrized by $\gamma \in [0, \pi/12]$ defines the Bell expressions:

$$\begin{equation} \begin{aligned} J_\gamma \equiv C\left(0,0\right) + \left( 4 \cos^2{\left[ \gamma + \pi / 6 \right]} -1 \right) \left( C\left(0,1\right) + C\left(1,0\right) - C\left(1,1\right) \right). \end{aligned} \end{equation} \tag{ 8 }$$

These Bell expressions also use two settings and have self-testing properties, yet in most cases do not certify two bits of randomness.

The Tsirelson bounds for (7) and (8) are $I_\delta^Q \equiv 2 \cos^3{\delta} / \left( \cos{(2 \delta)} \sin{\delta} \right)$, and $J_\gamma^Q \equiv 8 \cos^3{[\gamma + \pi / 6]}$, respectively. The relative Bell value is defined as $I_\delta^{exp} / I_\delta^Q$ and $J_\gamma^{exp} / J_\gamma^Q$, where $I_\delta^{exp}$ and $J_\gamma^{exp}$ are the values of the Bell expressions (7) and (8) obtained in the experiment, respectively. The relative Bell value attains the value of 1 in the noiseless cases. For correlation-based Bell expressions, like those analyzed in this paper, if η is the relative value of the Bell expression, and the relative value η is attained with the noised state:

$$\begin{equation} \eta \rho_{Q_A,Q_B}^{\mathrm{optimal}} + \left(1-\eta\right) \rho_{Q_A,Q_B}^{\mathrm{white}}, \end{equation} \tag{ 9 }$$

where $\rho_{Q_A, Q_B}^{\mathrm{optimal}}$ and $\rho_{Q_A, Q_B}^{\mathrm{white}}$ are the quantum state providing the Tsirelson bound and the maximally mixed state, respectively.

In real-world applications, one needs to consider the case when the observed quantities, like the values of the certificates, are known only with some limited certainty, due to a finite number of statistics gathered to estimate them. This is to be contrasted with the asymptotic case, valid in the idealized situation of an infinite number of repetitions of the experiment. One of the methods to cover such uncertainties is based on the concept of smooth entropies [47, 48]. In this work we apply the EAT method [6–8], which we recapitulate here in a limited scope; we refer to [14] for a detailed discussion.

In the performed experiments, ith step can be viewed as an application of a completely positive trace-preserving channel $\mathcal{N}_i$ acting on a quantum register $R_{i-1}$, and transforming its state on a quantum register R_i , and, at the same time, preparing states of classical registers A_i , B_i , X_i , and Y_i , which store the measurement results a and b together with the measurement settings x and y of the ith round of the experiments. The following form of the Markov chain condition holds: $I(A^{i-1} B^{i-1}:X_i Y_i | X^{i-1} Y^{i-1} E)$, where the superscript notation means a vector containing the values of given registers in subsequent steps up to the step specified in the superscript value, and E is an arbitrary quantum system possibly entangled with the registers $\{R_i \}$. A collection of such channels in the framework of EAT is called EAT channels. For the measurement settings x and y and results a and b we define the score function U of the Bell expression (1) as

$$\begin{equation} U\left(a,b,x,y\right) \equiv \sum_{a,b,x,y} c_{a,b,x,y} \chi\left(a,b,x,y\right) / P\left(x,y\right), \end{equation} \tag{ 10 }$$

where $\chi(\cdot,\cdot,\cdot,\cdot)$ is the indicator of a given measurement event, and $P(x,y)$ is the probability of the given pair of settings.

For given EAT channels and fixed i, a joint quantum state σ_RE on the register $R_{i-1}$ and the space E is called feasible if $\text{cond}(P)$ holds for $a = A_i$, $b = B_i$, $x = X_i$ and $y = Y_i$. For a given EAT channel, their min-tradeoff function f is any real function affine in $P \equiv \{P(a,b|x,y) \}$ such that for all i for any feasible joint quantum state σ_RE on the register $R_{i-1}$ and the space E it holds $f[P] \unicode{x2A7D} H(Q_A, Q_B|x,y, Q_E)$. These definitions using the EAT theorem allow to provision of explicit warranties on the quality of the generated random numbers [49]. We provide such analysis of our experiments in section 3.3.

Ultraviolet light centered at a wavelength of 390 nm is focused onto two 2 mm thick β barium borate nonlinear crystals placed in interferometric configuration to produce photon pairs emitted into two spatial modes (a) and (b) through the second order degenerate type-I spontaneous parametric down-conversion process. The spatial, spectral, and temporal distinguishability between the down-converted photons is carefully removed by coupling to single-mode fiber, passed through narrow-bandwidth interference filters (F) and quartz wedges respectively. We have realized these quantum protocols by using polarization entangled pairs of photons $| \, \phi+ \rangle = | \, HH \rangle + | \, VV \rangle$.

The measurements for Alice are performed by a half-wave plate (HWP) oriented $\theta_{A0}$ or $\theta_{A1}$, and the measurements for Bob are performed by an HWP-oriented $\theta_{B0}$ or $\theta_{B1}$. The polarization measurement was performed using PBS and single-photon detectors (D) placed at the two output modes of the PBS. Our detectors are actively quenched Si-avalanche photodiodes. All single-detection events were registered using a VHDL-programmed multichannel coincidence logic unit, with a time coincidence window of 1.7 ns.

We performed the experiment for the Bell expressions (7) at a low rate (approximately 675 two-photon coincidences per second). At these low rates, the multi-photon pair emissions are small, and accidental events can be neglected. We benchmark the state preparation by measuring the average visibility in the diagonal polarization basis of 99.07. Each of the measurement runs was taken 180 times with each run with a collection time of 250 s. We have also performed a state tomography to estimate the fidelity of the state and obtained $99.63 \pm 0.04$.

For the Bell expressions (8), the rate was around 780 two-photon coincidences per second with 160 measurements of 250 s and the visibility in the diagonal polarization basis of $99.13\%$. We were able to increase the rate slightly while maintaining visibility above 99%. The fidelity of the state obtained by state tomography is $99.75 \pm 0.02$. For these two experiments, the total average number of events is around 120 million.

We have also performed an experiment for the Bell expressions (7) at a higher rate of 2000 two-photon coincidences per second, the visibility was on average 98.73 in diagonal polarization basis. For this measurement, we have opted to work at a higher rate to send more information per second. However, increasing the rate has the effect of increasing the multi-photon pair emission and therefore reducing visibility. Each of the measurement runs was taken 100 times with each run with a collection time of 250 s. For this experiment, the total average number of events is around 200 million.

To reduce experimental errors in the measurements, we used computer-controlled high-precision motorized rotation stages to set the orientation of wave-plates with repeatability precision 0.02^∘. The error was estimated for each of the experiments by taking the standard deviation of the measurements.

The experimental setup is illustrated in figure 1. The angles of the HWPs for the experiments are provided in the appendix.

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To calculate the guessing probability, we reflected the experimental results by imposing on the maximization a constraint that the value of the Bell expression is equal to the one observed in the experiment.

To calculate the conditional von Neumann entropy, we have considered two different sets of constraints for the certification of the von Neumann entropy using the optimization (5). The standard approach [5] is to impose a constraint that the value of the relevant Bell expression is equal to the one from the experiment. A more involved method [28, 50–52] is to constrain the optimization with more than one parameter. The purpose of this is to increase the amount of the certified randomness, at a price of more demanding error analysis for finite data sets, and complicated numerical calculations. We imposed a constraint that each of the correlators $C(0,0)$, $C(0,1)$, $C(1,0$), and $C(1,1)$ are equal to those from the experiment. Note that the latter constraints are stronger than the former one, as the Bell expressions (7) and (8) are functions of correlators.

To be more precise, we have further relaxed the above constraints. We formulated the single parameter constraint in a form that the value of the relevant Bell expression is not smaller than the one from the experiment. The constraints for more parameters we formulated in a manner that each of the correlators $C(0,0)$, $C(0,1)$ and $C(1,0)$ are not smaller than the one obtained in the experiment, and $C(1,1)$ is not greater than the one from the experiment. It is easy to see that the minimization of the conditional von Neumann entropy with equalities as a constraint will be lower bound by the minimization with inequalities. The reason behind this relaxation is that this improves the stability of the numerical optimization, as the feasible region has a wider interior than with equality constraints. Similarly, for the guessing probability calculations, we relaxed the equality with an inequality imposing a constraint of the optimization that the value of the Bell is not smaller than the one obtained in the experiment. The relaxation of the equality constraints by replacing them with inequalities does not weaken the security proofs. Indeed, the proofs employ the lower bounds on the entropies, and relaxing the constraints of minimization procedures will only decrease the resulting values. Thus the conclusions we draw about the certified randomness are even more cautious than if we had not used these relaxed constraints.

As mentioned, the method [43] requires specifying the number of nodes in the quadrature. We calculated both variants of constraints with 6 nodes, and the optimization with the correlation constraints also with 8 nodes. To improve the certification of entropy, one can increase the number of nodes, but this comes with the price of a longer optimization time.

3.2.1.  I_δ Bell expressions and high relative violation

Firstly, we performed the experiment for the Bell expressions (7). We concentrated on the quality of the source, at the cost of the generated events rate. The experiment has been performed for $\delta = 0.45, 0.5, 0.52$. The obtained relative Bell values were 0.994, 0.994, and 0.997, respectively, and the certified randomness is shown in table 1.

Table 1. Randomness certified by Bell expressions (7) for the experiment concentrated on high relative violation of the Bell inequality in the asymptotic case.

	von Neumann entropy
δ	Cor. (8 Radau)	Cor. (6 Radau)	Bell viol. (6 Radau)	$H_\infty$
0.52	1.88	1.87	1.77	1.50
0.5	1.77	1.76	1.64	1.33
0.45	1.77	1.76	1.61	1.28

We observed 675 events per second, and thus the randomness generation rate for δ = 0.52 is 1270 bits of von Neumann entropy or 1012 bits of min-entropy, per second.

If only finite statistics are taken into account, one should consider also the uncertainty in evaluation e.g. the Bell expression value. In the case of the considered experiment, the values are shown in table 2 with theoretical boundaries for comparison, for $\delta = 0.45, 0.5, 0.52$, respectively. The Gauss–Radau approximation with six nodes showed that this Bell violation allows certifying 1.54, 1.58, and 1.72 bits of von Neumann entropy, respectively, thus slightly less than the asymptotic case of table 1.

Table 2. The experimental values of the Bell expression (7) for the experiment concentrated on high relative violation of the Bell inequality.

δ	$I_{\delta}^{\textrm{Classique}}$	$I_{\delta}^{\textrm{Quantique}}$	$I_{\delta}^{\textrm{Experimental}}$
0.52	5	5.2	$5.179 \pm 0.006$
0.5	5.022	5.218	$5.187 \pm 0.006$
0.45	5.207	5.4	$5.366 \pm 0.007$

3.2.2.  J_γ Bell expressions

Secondly, we investigated the Bell expressions (8). We considered the value $\gamma = 0, \pi / 24, \pi / 12$. The obtained relative Bell values are 0.996, 0.993, and 0.994, respectively. The certified randomness is shown in table 3.

Table 3. Randomness certified in the experiment by Bell expressions (8) in the asymptotic case.

	von Neumann entropy
γ	Cor. (8 Radau)	Cor. (6 Radau)	Bell viol. (6 Radau)	$H_\infty$
0	1.81	1.80	1.72	1.43
$\frac{\pi}{24}$	1.76	1.75	1.68	1.21
$\frac{\pi}{12}$	1.55	1.54	1.39	0.98

The observed event rate was 780 per second, giving the randomness generation rate 1300 bits of von Neumann entropy or 1030 bits of min-entropy, per second.

The violation of Bell's inequality is given in table 4.

Table 4. The experimental values of the Bell expression (8).

γ	$J_{\gamma}^{\textrm{Classique}}$	$J_{\gamma}^{\textrm{Quantique}}$	$J_{\gamma}^{\textrm{Experimental}}$
0	5	5.19	$5.174\pm 0.007$
$\frac{\pi}{24}$	3.55	3.99	$3.968 \pm 0.005$
$\frac{\pi}{12}$	2	2.83	$2.811 \pm 0.003$

3.2.3.  I_δ Bell expressions and high event rate

The third of the performed experiments concerned also the Bell expressions (7). We performed it for values $\delta = 0.5, 0.4, 0.3$ observing the relative Bell values 0.987, 0.991, and 0.991, respectively. We show the certified randomness in table 5.

Table 5. Randomness certified by Bell expressions (7) in the asymptotic case for the experiment concentrated on the high rate of the observed events.

	von Neumann entropy
δ	Cor. (8 Radau)	Cor. (6 Radau)	Bell viol. (6 Radau)	$H_\infty$
0.5	1.50	1.50	1.26	0.89
0.4	1.59	1.58	1.41	1.06
0.3	1.52	1.51	1.21	0.85

The rate of observed events was 2000 per second, so the randomness generation rate, when taking δ = 0.4 is 3180 bits of von Neumann entropy or 2120 bits of min-entropy, per second.

The violation of Bell's inequality is given in table 6.

Table 6. The experimental values of the Bell expression (7) for the experiment concentrated on the high rate of the observed events.

δ	$I_{\delta}^{\textrm{Classique}}$	$I_{\delta}^{\textrm{Quantique}}$	$I_{\delta}^{\textrm{Experimental}}$
0.5	5.02	5.22	$5.15 \pm 0,01$
0.4	5.57	5.76	$5.71 \pm 0,01$
0.3	6.98	7.15	$7.09 \pm 0.01$

Let us now concentrate on the case of the I_δ Bell expression (7) for δ = 0.52 and high relative violation, as presented in section 3.2.1. Recall that the Bell value was in that case $I_{0.52}^{\textrm{Experimental}} = 5.179$ with the standard deviation $\sigma_{I,0.52} = 0.006$, whereas the Tsirelson bound is 5.1967. Let us assume that close to the observed value the error distribution is near to Gaussian.

For the confidence interval of one standard deviation, we perform numerical optimization with the constraint that the Bell value is equal at least $w_1 = I_{0.52}^{\textrm{Experimental}} - \sigma_{I,0.52}$. We note that this constraint is weaker than the one stating that the Bell value is within the range

$$\begin{equation} \left[ I_{0.52}^{\textrm{Experimental}} - \sigma_{I,0.52}, I_{0.52}^{\textrm{Experimental}} + \sigma_{I,0.52} \right], \end{equation} \tag{ 11 }$$

so in fact the actual confidence interval can be expected to be higher than $68\%$. The certified von Neumann entropy for the violation equals at least w₁ calculated at level 2 of NPA with 6 noded of the quadrature is $r_1 = 1.7162$. To calculate the properties of the min-tradeoff function for EAT channels we calculated the certified entropy in the neighborhood of the violation value w₁, viz. at points $w_1 - \Delta$ and $w_1 + \Delta$ for $\Delta = 0.001$, and we obtained the values $r_{1-} = 1.7060$ and $r_{1+} = 1.7265$, respectively. This means, by the convexity of the bounding function, that the line tangent to the von Neumann entropy lower bound is of the form $g_1(x) \equiv r_1 + b_1 \cdot (x - w_1)$ with a certain, specified but unknown precisely, value of b₁ in the set $\left[ \frac{r_1 - r_{1-}}{\Delta}, \frac{r_{1+} - r_1}{\Delta} \right] \approx [10.239, 10.259]$.

Now, for the EAT theorem, we need to specify the range of the min-tradeoff function, see lemma III.1 in [14]. The minimal value of I_0.52 allowed in quantum mechanics is equal to the negation of the Tsirelson bound, viz. $-I_{0.52}^Q$. The algebraic bound on $I_{0.52}^{Alg}$ is obtained with $C(0,0) = C(1,0) = C(0,1) = -C(1,1) = 1$ in (7) and is equal 7.0005. Thus the range of g₁ is within the set

$$\begin{equation} \left[ r_1 + \frac{r_{1+} - r_1}{\Delta} \cdot \left(-I_{0.52}^Q - w_1\right), r_1 + \frac{r_{1+} - r_1}{\Delta} \cdot \left(I_{0.52}^{Alg} - w_1\right) \right], \end{equation} \tag{ 12 }$$

which has a diameter equal

$$\begin{equation} d_1 \equiv \frac{r_{1+} - r_1}{\Delta} \cdot \left(I_{0.52}^{Alg} + I_{0.52}^Q\right) \approx 125.13. \end{equation} \tag{ 13 }$$

A crucial parameter describing spot-checking protocols that use EAT is the probability of a test round, denoted usually by γ. We follow the EAT formulation of theorem II.1 in [14] and define:

$$\begin{equation} \epsilon_V\left(\beta, \gamma, d\right) \equiv \frac{\beta \cdot \ln\left(2\right)}{2} \cdot \left[ \log_2\left(2 \cdot 4^2 + 1\right) + \sqrt{d^2 / \gamma + 2} \right]^2, \end{equation} \tag{ 14a }$$

$$\begin{equation} \epsilon_K\left(\beta, d\right) \equiv \frac{\beta^2 \cdot \left( 2^{\beta \cdot \left(\log_2\left(4\right) + d\right)} \right)}{6 \left(1-\beta\right)^3 \cdot \ln\left(2\right)} \cdot \left[\ln\left(2\right) \cdot \left(\log_2\left(4\right) + d\right) + 2\right]^3, \end{equation} \tag{ 14b }$$

$$\begin{equation} \epsilon_\Omega\left(\beta, p_\Omega, \epsilon_S\right) \equiv \left(1 - \log_2\left(p_\Omega \cdot \epsilon_S\right)\right) / \beta. \end{equation} \tag{ 14c }$$

The value d correspond to the difference $\text{Max}[g] - \text{Min}[g]$ in lemma III.1. We used a slightly modified formula for _K compared to [14], viz. we first used the identity ${\Large \forall}_x \ln(2^x) = x \ln2$, and then concavity of logarithm, to split the expression into a sum of two logarithms, to avoid numerical round-off errors. The entropy consumption is $2 n^{\textrm{test}}$ bits for choosing settings for test rounds and $h_2(\gamma)$ bits for selecting which rounds are used for testing, where h₂ is the binary entropy.

In the experiment, we performed $n^{\textrm{test}} \approx 120\,000\,000$ rounds testing the value of the score function. In the experiment, we did not perform the actual series of generation rounds, and thus our work aims to test the feasibility of the novel certificates in near future implementation. Let n denote a hypothetical number of generation rounds performed by a device of similar quality as the one presented in this paper. For the sake of comparison with an approach involving a high rate of rounds with low violation of CHSH inequality, we juxtapose our results with the ones given in [15].

The parameter describing the quality of the raw randomness obtained in a random numbers generator is the soundness error ε_S reflecting the probability of distinguishing the generated sequence from the uniform one. In [15] the value $\epsilon_S = 3.09 \times 10^{-12}$ was reported, with the total time of running the experiment was 19.2 h, taking as the input $6.778 \times 10^8$ bits and returning the output containing $9.350 \times 10^8$ bits of certified randomness. This resulted in a net gain of $2.57 \times 10^8$ bits or a net gain rate of $3718.2\,\mathrm{bit}\,\mathrm{s}^{-1}$. The ratio between entropy consumption to production is 0.72.

In our experiment, the n^test rounds were performed in $180\,000$ s. We consider hypothetical generation rounds performed with the same device, intertwined with the test rounds occurring with probability given by some value of γ which we establish below, where we consider the parameters range providing the same soundness error as in [15] and compare the net gain rates.

First, let us consider the confidence interval of one standard deviation. In theorem II.1 of [14] we use $t = r_1$, $p_\Omega = 0.68$, $d = d_1$ and assume the same event rate as in the test rounds. We consider a wide range of parameters β and γ, as shown in figure 2. For instance $\beta = 10^{-7}$ and γ = 0.01 would provide net gain rate of $1033.3\,\mathrm{bit}\,\mathrm{s}^{-1}$ when about $1.2 \times 10^{10}$ rounds would be needed; the ratio of entropy consumption to production is 0.06.

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For a confidence interval of three standard deviations, the Bell value is equal to at least $w_3 = I_{0.52}^{\textrm{Experimental}} - 3 \sigma_{I,0.52} \approx 5.161$, certifying at least $r_3 = 1.5943$ bits of von Neumann entropy. The tangent line is $g_3(x) \equiv r_3 + b_3 \cdot (x - w_3)$ with $b_3 \in [10.09, 10.102]$. The diameter of the range set of g₃ is $d_3 \approx 123.22$. The optimal net gain rate in that case would be about $1013.4\ \mathrm{bit}\,\mathrm{s}^{-1}$; the ratio of entropy consumption to production is about 0.07. Even though it is lower than the net gain rate obtained in the case with one standard deviation confidence interval, this case provides a success probability of the protocol much higher (0.997) than the other one (0.68).

For the higher rate experiment from section 3.2.3 with δ = 0.5 we have $n^{\textrm{test}} \approx 2000\,000\,000$ obtained in $100\,000$ s. For three standard deviations confidence interval, we get the randomness lower bound $w_h = 1.1743$, and diameter $d_h = 121.34$, resulting in a net gain rate of about $1951.5\,\mathrm{bit}\,\mathrm{s}^{-1}$; the ratio of entropy consumption to production is 0.09. We show the parameter dependence in figure 3.

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