Regularising data for practical randomness generation

We define a Bell test in the following way: a user has access to two devices ${ \mathcal A }$ and ${ \mathcal B }$. The internal working of those devices is unknown: they are treated as black boxes. The only possible interaction with those black boxes is as follows: upon receiving an input x ∈ {0, 1} (resp. y ∈ {0, 1}), ${ \mathcal A }$ (resp. ${ \mathcal B }$) produces an output a ∈ {0, 1} (resp. b ∈ {0, 1}). We associate the random variables A, B, X and Y to a, b, x and y respectively, and ${P}_{{AB}| {XY}}$ denotes the conditional probabilities of the outputs given the inputs, which we call hereafter a behaviour (the subscript indicating the random variables is sometimes omitted when they are clear from the context). We assume that this behaviour obeys quantum mechanics, i.e. ${P}_{{AB}| {XY}}({ab}| {xy})=\mathrm{Tr}[\rho \,{M}_{a| x}^{A}\otimes {M}_{b| y}^{B}]$, where ρ is a quantum state and ${\{{M}_{a| x}^{{\rm{A}}}\}}_{a}{}_{}$ and ${\{{M}_{b| y}^{{\rm{B}}}\}}_{b}{}_{}$ are positive-operator valued measures. This implies in particular that input x (resp. y) has no influence on output b (resp. a), i.e. ${P}_{{AB}| {XY}}$ is no-signalling [22, 23]. We denote by ${ \mathcal Q }$ the set of all quantum behaviours.

When a Bell test is repeated n times, we write ${\bf{x}}=({x}_{1},\,\ldots ,\,{x}_{n})$ for the sequence of inputs of ${ \mathcal A }$. We define y, a, b, as well as their associated random variables A, B, X, Y, in the same way. We now briefly remind some of the key concepts that we will use later on.

2.1.1. Min-entropy

We quantify the randomness of the outputs produced in a Bell test via the min-entropy. The min-entropy of (A, B) given (X, Y) conditioned on some event λ, according to a distribution P = P_ABXY, is:

$$\begin{eqnarray}&&{H}_{\min }{\left({\bf{A}},{\bf{B}}| {\bf{X}},{\bf{Y}},\lambda \right)}_{P}=-{\mathrm{log}}_{2}\displaystyle \sum _{{\bf{x}},{\bf{y}}}P({\bf{x}},{\bf{y}}| \lambda )\mathop{\max }\limits_{{\bf{a}},{\bf{b}}}P({\bf{a}},{\bf{b}}| {\bf{x}},{\bf{y}},\lambda ).\end{eqnarray} \tag{ 1 }$$

Essentially, the min-entropy quantifies the number of almost-uniform random bits that can be obtained from a source via a randomness extractor. The event λ is typically a function of the specific inputs that were chosen and the specific outputs that were obtained during the Bell experiment, such as a statistical estimate. For a detailed review on the relevance of this quantity, see [24].

We now introduce all the elements that allow us to lower-bound this quantity.

2.1.2. Bell expression

We call a real linear functional in ${P}_{{AB}| {XY}}$ a Bell expression:

$$\begin{eqnarray}&&{ \mathcal I }({P}_{{AB}| {XY}})=\displaystyle \sum _{a,b,x,y}{c}_{{abxy}}{P}_{{AB}| {XY}}({ab}| {xy}).\end{eqnarray} \tag{ 2 }$$

For a given Bell expression, its maximal value over all local deterministic strategies, i.e. all behaviours with a (resp. b) being a deterministic function of x (resp. y), gives rise to the local bound ${I}_{{ \mathcal L }}$. A behaviour is said to be local if it can be written as a convex mixture of deterministic strategies, and the corresponding set is denoted ${ \mathcal L }$. The inequality:

$$\begin{eqnarray}&&\forall {P}_{{AB}| {XY}}\in { \mathcal L },\quad { \mathcal I }({P}_{{AB}| {XY}})\leqslant {I}_{{ \mathcal L }}\end{eqnarray} \tag{ 3 }$$

is referred to as a Bell inequality [25].

However, in quantum physics, non-local behaviours are accessible. For a given ${P}_{{AB}| {XY}}$, it might then happen that this local bound is violated. In this case, we call Bell violation the value that the Bell expression takes, and we denote by ${I}_{{ \mathcal Q }}^{+}$ the maximal value allowed in quantum theory, and by ${I}_{{ \mathcal Q }}^{-}$, respectively, the minimal value:

$$\begin{eqnarray}&&{I}_{{ \mathcal Q }}^{+}=\mathop{\max }\limits_{P\in { \mathcal Q }}{ \mathcal I }(P),\quad {I}_{{ \mathcal Q }}^{-}=\mathop{\min }\limits_{P\in { \mathcal Q }}{ \mathcal I }(P).\end{eqnarray} \tag{ 4 }$$

As is usually done, we call 'local bound' and 'quantum bound' the maximal values of a Bell expression over the local and quantum sets. However, we also introduce the minimal value of a Bell expression over the quantum set, as we need it to express the min-entropy bound given by theorem 1 (see [11, 17] for details).

2.1.3. Observed frequencies

For a given realisation of (A, B, X, Y), we define the observed frequencies as:

$$\begin{eqnarray}&&{\hat{P}}_{{AB}| {XY}}({ab}| {xy})=\displaystyle \frac{{N}_{{abxy}}}{{N}_{{xy}}},\end{eqnarray} \tag{ 5 }$$

where N_abxy (resp. N_xy) is the number of occurrences of the quadruplet (a, b, x, y) (resp. the pair (x, y)) in the n length sequence (a, b, x, y). In the unlikely event that a given pair $({x}^{* },{y}^{* })$ was never input, i.e. ${N}_{{x}^{* }{y}^{* }}=0$, the experiment should be performed again.

2.1.4. Observed Bell violation

For simplicity, we assume that the inputs (x, y) are chosen independently and identically at each round with probability P(X_i = x, Y_i = y) = π_xy. For a given Bell expression, as defined in equation (2), and a given realisation of (A, B, X, Y), we define the observed average Bell violation as:

$$\begin{eqnarray}&&\hat{I}=\displaystyle \sum _{a,b,x,y}{c}_{{abxy}}\displaystyle \frac{{N}_{{abxy}}}{n\cdot {\pi }_{{xy}}}.\end{eqnarray} \tag{ 6 }$$

We point out that, even though $\hat{P}$ and $\hat{I}$ are both estimators, they do not involve the inputs in the same manner. To compute $\hat{P}$, one counts the occurrences of both the quadruplets (a, b, x, y) and the input pairs (x, y), whereas for $\hat{I}$, one only counts the quadruplets (a, b, x, y) and uses directly the input distributions π_xy, instead of the frequencies of each input pair for a given realisation. Both can be computed from a realisation of Bell experiments, as π_xy is chosen by the user (see details hereafter). However, we decide to compute the observed frequencies using N_xy to ensure that ${\hat{P}}_{{AB}| X=x,Y=y}$ is normalised for each (x, y), and can thus be identified as a probability distribution. On the other hand, we decide to compute the observed Bell violation $\hat{I}$ directly using the input distribution, as this is crucial for the derivation of theorem 1 (see [11, 17] for details). Note that, if the behaviours of the devices at each round are independent and identically distributed (i.i.d.) according to some distribution ${P}_{{AB}| {XY}},\hat{I}$ converges towards ${ \mathcal I }({P}_{{AB}| {XY}})$ when n tends to infinity. However, we do not need to make such an assumption to define this quantity.

2.1.5. Distance between distributions

We say that two distributions P_ABXY and ${\tilde{P}}_{{\bf{ABXY}}}$ are -close if their total variation distance is upper bounded by :

$$\begin{eqnarray}&&d(P,\tilde{P})=\displaystyle \frac{1}{2}\displaystyle \sum _{{\bf{a}},{\bf{b}},{\bf{x}},{\bf{y}}}| P({\bf{a}},{\bf{b}},{\bf{x}},{\bf{y}})-\tilde{P}({\bf{a}},{\bf{b}},{\bf{x}},{\bf{y}})| \leqslant \epsilon .\end{eqnarray} \tag{ 7 }$$

2.1.6. Randomness-bounding function

The main ingredient needed to lower-bound the min-entropy is the RB function. We now explain how to compute it via the guessing probability problem. This general form was introduced and extensively explained in [11]. Here, we only briefly present the reasoning that leads to this formulation. For a given Bell expression ${ \mathcal I }$ and a specific value ${I}^{* }$ of ${ \mathcal I }$, finding the lower bound ${H}_{{ \mathcal I }}^{\chi }({I}^{* })$ defined by requirements R.1 and R.2 amounts to solving a minimisation problem over all quantum behaviours P such that ${ \mathcal I }(P)={I}^{* }$. However, the optimisation problem obtained in this way is not easily solvable, due to the presence of the logarithm and to the complicated nature of the quantum set ${ \mathcal Q }$ [19, 20, 26].

This led the authors of [11] to consider instead the following problem. For (α, β) ∈ {0, 1}² and $(\gamma ,\delta )\in \chi \subseteq \{0,1\}{}^{2}$, let $\{{\tilde{P}}^{\alpha \beta \gamma \delta }\}$ be $4\times | \chi |$ variables, where $| \chi |$ is the cardinality of the set χ, that represent unnormalised behaviours. The problem then reads:

$$\begin{eqnarray}\begin{array}{rcl}{G}_{{ \mathcal I }}^{\chi }({I}^{* }) & = & \mathop{\max }\limits_{\{{\tilde{P}}^{\alpha \beta \gamma \delta }\}}\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\alpha ,\beta \in \{0,1\}{}^{2}}{\gamma ,\delta \in \chi }}{\tilde{P}}^{\alpha \beta \gamma \delta }(\alpha \beta | \gamma \delta )\\ & {\rm{s}}.{\rm{t}}. & \displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\alpha ,\beta \in \{0,1\}{}^{2}}{\gamma ,\delta \in \chi }}{ \mathcal I }({\tilde{P}}^{\alpha \beta \gamma \delta })={I}^{* },\\ & & \displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\alpha ,\beta \in \{0,1\}{}^{2}}{\gamma ,\delta \in \chi }}\mathrm{Tr}[{\tilde{P}}^{\alpha \beta \gamma \delta }]=1,\\ & & \forall \ \alpha ,\beta ,\gamma ,\delta ,{\tilde{P}}^{\alpha \beta \gamma \delta }\in {\tilde{{ \mathcal Q }}}_{k},\end{array}\end{eqnarray} \tag{ 8 }$$

where $\mathrm{Tr}[\tilde{P}]={\sum }_{{ab}}\tilde{P}({ab}| {xy})$ is the norm of $\tilde{P}$ (which is independent of (x, y) by no-signalling) and ${\tilde{{ \mathcal Q }}}_{k}$ is the set of unnormalised behaviours that belong to the kth level of the NPA hierarchy [19, 20]. This problem is then a semi-definite programme (SDP), and, as such, can be efficiently solved. Moreover, if we let ${H}_{{ \mathcal I }}^{\chi }=-{\mathrm{log}}_{2}{G}_{{ \mathcal I }}^{\chi },{H}_{{ \mathcal I }}^{\chi }$ satisfies both requirements R.1 and R.2, and is thus a RB function for χ (see [11] for details). It is, however, not necessarily tight, in particular because the NPA hierarchy is merely a relaxation of ${ \mathcal Q }$.

In the case where χ contains only one input pair, the guessing probability problem has a simple interpretation: it is the maximal guessing probability, over all quantum strategies, of an adversary who is bound to keep the Bell violation ${I}^{* }$ unchanged. This problem was introduced in [10, 18], along with another optimisation problem that we now remind. The idea is the following: if we consider the guessing probability as a theoretical measure of the randomness of a behaviour ${P}_{{AB}| {XY}}$, constraining this behaviour to only a Bell violation, that is, constraining only a linear functional of ${P}_{{AB}| {XY}}$ to a fixed value, amounts to discarding some information about the behaviour. It might thus result in an underestimation of the intrinsic randomness contained in ${P}_{{AB}| {XY}}$. On the contrary, the following problem takes into account the complete information about the behaviour to evaluate its randomness:

$$\begin{eqnarray}\begin{array}{rcl}{G}_{{\rm{full}}}^{\chi }(P)\,= & & \mathop{\max }\limits_{\{{\tilde{P}}^{\alpha \beta \gamma \delta }\}}\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\alpha ,\beta \in {\{0,1\}}^{2}{}^{}}{\gamma ,\delta \in \chi }}{\tilde{P}}^{\alpha \beta \gamma \delta }(\alpha \beta | \gamma \delta )\\ & {\rm{s}}.{\rm{t}}. & \displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\alpha ,\beta \in {\{0,1\}}^{2}{}^{}}{\gamma ,\delta \in \chi }}{\tilde{P}}^{\alpha \beta \gamma \delta }=P,\\ & & \forall \ \alpha ,\beta ,\gamma ,\delta ,{\tilde{P}}^{\alpha \beta \gamma \delta }\in {\tilde{{ \mathcal Q }}}_{k}.\end{array}\end{eqnarray} \tag{ 9 }$$

As problem (9) is more constrained than problem (8), it is clear that ${G}_{{\rm{full}}}^{\chi }(P)\leqslant {G}_{{ \mathcal I }}^{\chi }({ \mathcal I }(P))$. One could compare these two problems in the following way: ${G}_{{\rm{full}}}^{\chi }(P)$ is a measure of the randomness of a behaviour P, whereas ${G}_{{ \mathcal I }}^{\chi }({I}^{* })$ is measure of the randomness that can be certified by a Bell expression ${ \mathcal I }$. Yet these two formulations are connected: the dual problem of (9) precisely returns a Bell expression ${{ \mathcal I }}^{* }$ such that ${G}_{{{ \mathcal I }}^{* }}({{ \mathcal I }}^{* }(P))={G}_{{\rm{full}}}(P)$ [10, 18]. When the Bell expression is well chosen, (8) and (9) are thus equivalent.

Let us stress however that these quantities can only be considered as theoretical measures of randomness for theoretical objects such as probability distributions and Bell expressions. In order to obtain practical bounds, one has to develop statistical tools.

With the concepts defined above, we are now able to formulate a probabilistic statement on the min-entropy of the outputs obtained after a sequence of n Bell tests. Most of this section is a reformulation, adapted to our case, of the results first presented in [4], corrected in [12, 17], and extended in [11]. Let us fix a behaviour ${P}_{{\bf{AB}}| {\bf{XY}}}$, an i.i.d. input distribution π_xy, and a Bell expression ${ \mathcal I }$. Then the formal statement reads:

Theorem 1. Let $\{{J}_{m}| m\in [0,M]\}$ be a sequence of $M+1$ Bell violation thresholds, with ${I}_{{ \mathcal L }}={J}_{0}\lt {J}_{1}\,\lt ...\,\lt {J}_{M}={I}_{{ \mathcal Q }}^{+}$. Let ${\lambda }_{m}$ be the event that the estimated Bell violation $\hat{I}$ falls between the thresholds ${J}_{m}$ and ${J}_{m+1}$, and let ${{\mathbb{P}}}_{\tilde{P}}({\lambda }_{m})$ be the probability that this event occurs according to some distribution ${\tilde{P}}_{{\bf{ABXY}}}$. Let and $\epsilon ^{\prime}$ be two positive parameters. Then the true distribution ${P}_{{\bf{ABXY}}}$ is $\epsilon$-close to a distribution ${\tilde{P}}_{{\bf{ABXY}}}$ such that exactly one of these two statements holds:

• 1.  
  ${{\mathbb{P}}}_{\tilde{P}}({\lambda }_{m})\leqslant \epsilon ^{\prime}$,
• 2.  
  ${H}_{\min }{\left({\bf{A}},{\bf{B}}| {\bf{X}},{\bf{Y}},{\lambda }_{m}\right)}_{{\tilde{P}}_{{\bf{ABXY}}}}\geqslant {{nH}}_{{ \mathcal I }}^{\chi }({J}_{m}-\mu )-\gamma ({\bf{x}})\eta -{\mathrm{log}}_{2}\tfrac{1}{\epsilon ^{\prime} }$,

where

$$\begin{eqnarray}&&\mu =\nu \sqrt{\displaystyle \frac{2}{n}\mathrm{ln}\displaystyle \frac{1}{\epsilon }},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\nu =\max \left\{\mathop{\max }\limits_{a,b,x,y}\displaystyle \frac{{c}_{{abxy}}}{{\pi }_{{xy}}}-{I}_{{ \mathcal Q }}^{-},{I}_{{ \mathcal Q }}^{+}-\mathop{\min }\limits_{a,b,x,y}\displaystyle \frac{{c}_{{abxy}}}{{\pi }_{{xy}}}\right\},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\gamma ({\bf{x}})=n-\displaystyle \sum _{j=1}^{n}{{\mathbb{1}}}_{\chi }({x}_{j}),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\eta =\max \{{H}_{{ \mathcal I }}^{\chi }({I}_{{ \mathcal Q }}^{+}),{H}_{{ \mathcal I }}^{\chi }({I}_{{ \mathcal Q }}^{-})\},\end{eqnarray} \tag{ 13 }$$

and ${{\mathbb{1}}}_{\chi }({x}_{j})$ is the indicator function, which returns 1 if ${x}_{j}\in \chi$ and vanishes otherwise.

The above theorem is equivalent to theorem 1 of [11] in the case where one considers a single Bell expression. Its proof thus follows essentially the same steps as that for theorem 1 of [11] and we refer the readers to [11] for details. Taking into account only one Bell expression, however, leads to numerous simplifications in its formulation, due in particular to the monotonicity of ${H}_{{ \mathcal I }}^{\chi }$ over $[{I}_{{ \mathcal L }},{I}_{{ \mathcal Q }}^{+}]$. In this sense, it is closer to the way it is stated in [17]. However, from [11], we keep a few improvements on the parameters, and the possibility to select only a subset of inputs via χ. This enables improvement on the bound in some cases where the inputs have very different output probabilities: if the RB function is significantly better for a subset of inputs χ, this formulation allows to use the RB function for χ only, and corrects the bound via the penalty term γ(x)η. In that case, we have an interest in biasing the input distribution towards χ, in order to reduce the effect of the term γ(x)η and thus produce as much randomness as possible. However, the trade-off between the quality of the RB function and the number of inputs from which randomness is generated depends on the total number of runs of a given protocol.

The bound given in the second statement of the theorem is the figure of merit that we aim at optimising in this work. Indeed, this expression depends on the choice of the Bell expression ${ \mathcal I }$, and we now present a systematic approach to finding a well suited ${ \mathcal I }$.

As previously mentioned, solving the dual problem of (9) provides the Bell expression that is optimal for certifying the randomness of the given behaviour. When given an uncharacterised pair of devices, one could thus first generate some input–output data in order to estimate the corresponding underlying behaviour. This estimate $\hat{P}$ can then be used to obtain a Bell inequality that is presumably better for witnessing the randomness generated from these devices, by computing the dual solution to the guessing probability problem. Unfortunately, as mentioned above, the guessing probability problem is only properly defined over the set of quantum behaviour ${ \mathcal Q }$, or one of its NPA relaxation sets ${{ \mathcal Q }}_{k}$, or over the set of no-signalling behaviours. On the other hand, there is no guarantee that the behaviour built from the observed frequencies $\hat{P}$ belong to any of these sets: $\hat{P}$ is on the contrary almost always signalling, even if the underlying behaviour is not, due to finite statistics. In this case, problem (9) will be infeasible.

We now introduce our method to circumvent this problem, using the tools developed in [21]. The authors provide a set of tools to regularise the estimated behaviour $\hat{P}$ to one of the NPA sets ${{ \mathcal Q }}_{k}$. It consists in minimising a norm-based metric or a statistical distance between $\hat{P}$ and ${{ \mathcal Q }}_{k}$, the desired relaxation set, and taking the unique minimiser as the regularised behaviour ${P}_{{AB}| {XY}}^{\mathrm{reg}}$. In this work, we employ two methods considered therein. The first one corresponds to minimising a statistical distance, namely the conditional Kullback–Leibler (KL) divergence [27, 28], and is defined in the following way:

$$\begin{eqnarray}&&{P}_{\mathrm{ML}}(\hat{P})=\mathop{\mathrm{argmin}}\limits_{P\in {{ \mathcal Q }}_{k}}{D}_{\mathrm{KL}}(\hat{P}| | P),\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray*}&&{D}_{\mathrm{KL}}(\hat{P}| | P)=\displaystyle \sum _{a,b,x,y}\displaystyle \frac{{N}_{{xy}}}{n}\hat{P}(a,b| x,y){\mathrm{log}}_{2}\left(\displaystyle \frac{\hat{P}(a,b| x,y)}{P(a,b| x,y)}\right).\end{eqnarray*}$$

and where ML stands for 'maximal likelihood'.

The second one corresponds to minimising the two-norm distance:

$$\begin{eqnarray}&&{P}_{\mathrm{LS}}(\hat{P})=\mathop{\mathrm{argmin}}\limits_{P\in {{ \mathcal Q }}_{k}}\sqrt{\displaystyle \sum _{a,b,x,y}{\left(\hat{P}(a,b| x,y)-P(a,b| x,y)\right)}^{2}},\end{eqnarray} \tag{ 15 }$$

where 'LS' stands for 'least-squares'. It is important to note that both these minimisations can be efficiently solved (see [21] for details), thus making this approach operationally relevant. A detailed study of these regularisation methods, and, in particular, of their convergence to the underlying distribution, is beyond the scope of this article; we refer the readers to [21] for information on that subject.

We can now define the following regularisation-based protocol for generating randomness from uncharacterised devices:

• (i)  
  Input a number ${N}_{\mathrm{est}}$ of (x, y) drawn from an i.i.d. uniform distribution (they can be public) and obtain the corresponding (a, b) in order to estimate the behaviour.
• (ii)  
  From this set of data, construct the observed frequencies $\hat{P}$ and compute ${P}_{{AB}| {XY}}^{\mathrm{reg}}$, the regularisation of $\hat{P}$ (where ${P}_{{AB}| {XY}}^{\mathrm{reg}}$ can be either ${P}_{\mathrm{ML}}(\hat{P})$ or ${P}_{\mathrm{LS}}(\hat{P})$).
• (iii)  
  Solve the corresponding optimisation problem ${G}_{{full}}^{\chi }({P}_{{AB}| {XY}}^{\mathrm{reg}})$ for different χ and select χ accordingly (see below for further details).
• (iv)  
  Extract the optimal Bell expression ${ \mathcal I }$ from the dual.
• (v)  
  Input a number ${N}_{\mathrm{raw}}$ of (x, y), drawn according to a distribution ${P}_{{XY}}^{\chi }$ (they can be public), obtain the corresponding (a, b), and compute the observed Bell violation $\hat{I}$.
• (vi)  
  Apply theorem 1 to lower-bound the min-entropy of the raw set of data ${({a}_{i},{b}_{i},{x}_{i},{y}_{i})}_{i\in \{1,{N}_{\mathrm{raw}}\}}$.

We now make a few observations on this protocol, which is summarised in figure 1. χ is chosen at step (iii), thanks to ${P}_{{AB}| {XY}}^{\mathrm{reg}}$. Indeed, ${P}_{{AB}| {XY}}^{\mathrm{reg}}$ reveals some information about the underlying behaviour. One might thus intuitively do the following: compute the values of ${G}_{{full}}^{(x,y)}({P}_{{AB}| {XY}}^{\mathrm{reg}})$ for all the inputs, and decide accordingly; if the value is roughly the same for all (x, y), one would choose χ = {0, 1}²; if one input pair $({x}^{* },{y}^{* })$ yields a much lower guessing probability, one would choose $\chi =({x}^{* },{y}^{* })$. However, if ${N}_{\mathrm{raw}}$ is not big enough, χ = {0, 1}² is likely to result in a better min-entropy bound in any case, as our results show.

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The optimised Bell expression ${ \mathcal I }$ obtained in step (iv) may not be unique and the different possible representations of ${ \mathcal I }$ are only artefacts of numerical computations. However, the choice of a representative for ${ \mathcal I }$ matters, since two physically equivalent representations can lead to different statistical estimates [29], and thus to distinct lower bounds on the min-entropy. In order to avoid such effects, we use the unique representation introduced in [29], by setting the signalling part to zero (see [29] for details). Note that, after this step, the regularised distribution no longer plays a role, and the min-entropy bound associated to that Bell inequality is valid independently of which regularisation method was used.

In step (v), we assume that the specific distributions ${P}_{{XY}}^{\chi }$ can be generated using some freely available resource. If this is the case, one might consider that the task of randomness generation is already achievable, and we might then call our primitive 'randomness expansion', rather than 'randomness generation'. However, the input randomness can be public: it needs to be random to anyone beforehand, but it can be accessed by anyone after it is produced. Conversely, the output randomness is private: its value resides in the fact that it is only accessible to the user. We can thus refer to this process as 'private random bits generation'.

In step (vi), in order to apply theorem 1, we need to know the quantum bounds ${I}_{{ \mathcal Q }}^{+}$ and ${I}_{{ \mathcal Q }}^{-}$. We approximate these bounds with the extrema over an NPA set, so that they can be easily computed. Moreover, we only bound the min-entropy of the data generated in step (v). Indeed, it is essential that the set of data used for the estimation be different from the one for which the bound on the min-entropy is derived: the statistical analysis of the data cannot depend on the data itself. This implies that, contrarily to [11], our method requires that part of the data is used only for parameter estimation, and then thrown away.

Finally, note that even though the regularisation method described in [21] is meaningful only when the underlying distribution ${P}_{{\bf{AB}}| {\bf{XY}}}$ is i.i.d., the derivation of the bound on the min-entropy does not rely on this assumption. For this reason, the probabilistic statement that we obtain via our method will still be valid, even if ${P}_{{\bf{AB}}| {\bf{XY}}}$ is not i.i.d. In this case, the Bell expression that we obtain might be inadequate, which might result in a trivial lower bound on the min-entropy (that equals to zero), but it will not result in an overestimation of the min-entropy of the raw data. In this sense, the optimisation method might become irrelevant, but the security analysis will not be compromised.

In order to adjust the parameters of our protocol, we simulate some pairs of devices, by generating for each one a random state ρ and some random measurements $\{{M}_{a| x}^{{\rm{A}}}\}{}_{a}$ and $\{{M}_{b| y}^{{\rm{B}}}\}{}_{b}$. The random states are picked at random in the space of two qubit pure states via their Schmidt decomposition, and the random measurements are generated via their associated rank-1 projectors, picked at random on the Bloch sphere.

We then compute the associated behaviour:

$$\begin{eqnarray}&&{P}_{{AB}| {XY}}({ab}| {xy})=\mathrm{Tr}[\rho \,{M}_{a| x}^{A}\otimes {M}_{b| y}^{B}].\end{eqnarray} \tag{ 16 }$$

To ensure that the obtained behaviours are non-local, we compute their associated values ${{ \mathcal I }}_{\mathrm{CHSH}}({P}_{{AB}| {XY}})$ of the Clauser–Horne–Shimony–Holt (CHSH) inequality [30]:

$$\begin{eqnarray}&&{{ \mathcal I }}_{\mathrm{CHSH}}({P}_{{AB}| {XY}})=\displaystyle \sum _{x,y,a,b}{\left(-1\right)}^{{xy}+a+b}{P}_{{AB}| {XY}}({ab}| {xy}),\end{eqnarray} \tag{ 17 }$$

and discard those for which ${{ \mathcal I }}_{\mathrm{CHSH}}({P}_{{AB}| {XY}})\leqslant 2$. We then construct the corresponding ${N}_{\mathrm{tot}}$-round behaviour using ${P}_{{AB}| {XY}}$ in an i.i.d. way, i.e.

$$\begin{eqnarray}&&{P}_{{\bf{AB}}| {\bf{XY}}}({\bf{ab}}| {\bf{xy}})=\displaystyle \prod _{i=1}^{{N}_{\mathrm{tot}}}{P}_{{AB}| {XY}}({a}_{i}{b}_{i}| {x}_{i}{y}_{i}).\end{eqnarray} \tag{ 18 }$$

We set ${N}_{\mathrm{tot}}={N}_{\mathrm{est}}+{N}_{\mathrm{raw}}={10}^{8}$, in accordance with the state-of-the-art experimental demonstration of device-independent randomness generation [31]. We then conduct a detailed study of four of these random behaviours, to heuristically fix three crucial parameters of our protocol:

• the regularisation method,
• the number of rounds used for the estimation ${N}_{\mathrm{est}}$,
• the inputs subset used to generate randomness χ.

Based on the data we obtained, presented in appendix A, we decided to set:

• ${P}_{{AB}| {XY}}^{\mathrm{reg}}={P}_{{\rm{ML}}}$,
• ${N}_{\mathrm{est}}={10}^{6}$,
• $\chi =\{0,1\}{}^{2}$.

The graphs that corroborate these decisions can be found in appendix A. Before we give the results of several simulations that illustrate the efficiency of our protocol, note that, when one sets ${N}_{\mathrm{tot}}={10}^{8}$, generating randomness from only one input pair (i.e. setting $\chi =({x}^{* },{y}^{* })$) does not usually result in higher min-entropy bounds than when one sets χ = {0, 1}². The same effect can be observed in the simulations carried out by the authors in [11]. It is not surprising: in order to obtain a good min-entropy rate when certifying randomness from only one input pair, one should bias the input distribution towards that pair as much as possible. However, in order to obtain a reliable estimate of the Bell violation, one should evaluate it with many occurrences of each possible input. These two assertions are in an apparent contradiction, and they can both hold simultaneously only if ${N}_{\mathrm{tot}}$ is high enough. It seems that, for most behaviours, ${N}_{\mathrm{tot}}={10}^{8}$ is not sufficient. We however checked that, when ${N}_{\mathrm{tot}}$ is sufficiently big, our method provides better min-entropy bounds for $\chi =({x}^{* },{y}^{* })$ than for $\chi =\{0,1\}{}^{2}$. The corresponding graph can be found in appendix B.

Our figure of merit is the comparison between the min-entropy bound obtained from our protocol, denoted ${H}_{\min }$ in the following, and the one obtained from a direct evaluation of the CHSH inequality, ${H}_{\min }^{\mathrm{CHSH}}$. We generate 50 behaviours at random (in the same way as described above) and run 500 simulations for each of them. To compute the lower bound on ${H}_{\min }$, one should set $n={N}_{\mathrm{raw}}$ in theorem 1, whereas for ${H}_{\min }^{\mathrm{CHSH}},n={N}_{\mathrm{tot}}\gt {N}_{\mathrm{raw}}$, as no estimation is required⁶ .

The parameters of the bound of theorem 1 are set as follows: we fix $\epsilon =\epsilon ^{\prime} ={10}^{-6}$, we divide the interval $[{I}_{{ \mathcal L }},{I}_{{ \mathcal Q }}^{+}]$ in M + 1 = 1000 segments of the same length, and we use the level 2 of the hierarchy defined in [33] (i.e. local level 2 defined in [34]) for the regularisation and the guessing probability problems. We then compute the corresponding min-entropy rate by dividing these values by ${N}_{\mathrm{tot}}$ in both cases. We also computed $-{\mathrm{log}}_{2}({G}_{{\rm{full}}}^{\chi }({P}_{{AB}| {XY}}))$, which corresponds to the maximal achievable min-entropy rate. To show that it is worth sacrificing part of the data for estimation, we then compared these three quantities. The results are presented in figure 2.

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In this figure, we plot the ratios between the min-entropy rates for ${H}_{\min }$ and ${H}_{\min }^{\mathrm{CHSH}}$ for every simulated pairs of devices, as well as the ratios between the maximal achievable rate $-{\mathrm{log}}_{2}({G}_{{\rm{full}}}^{\chi }({P}_{{AB}| {XY}}))$ and ${H}_{\min }^{\mathrm{CHSH}}$. For clarity, we sorted them in ascending order of the latter. We highlighted in grey the areas between the line y = 1, where the amount of randomness given by our protocol is the same as using CHSH inequality, and the curves connecting the optimal ratios. Our protocol is good whenever a point falls in this area. Indeed, it means that, despite the ${N}_{\mathrm{est}}$ bits that were thrown away, we obtain a higher bound on the min-entropy than if we had simply used the CHSH inequality on all the bits.

We observe that our method performs well in 98% of the simulations, in the following sense: when the optimal rate is nearly achieved with the CHSH inequality (i.e. the CHSH inequality gives a bound that is above 95% of the optimal rate), so does our method; when the CHSH inequality does not achieve the optimal rate, our method performs significantly better (with rates up to 1.6 times more) in all but one case.

Before we move on to concluding remarks, let us explain what happened with the last point of our simulations, from which no randomness can be certified via our protocol. The corresponding underlying behaviour has a low CHSH value, and the optimal Bell inequality is such that the gap between the local bound and the quantum bound is very small. This seems to indicate that this behaviour is of the kind presented in [35], i.e. it is almost local, but also close to the border of the quantum set. The authors of [35] proved that, in theory, a lot of randomness could be certified from such behaviours, as can be observed by the corresponding red circle in figure 2. However, those behaviours are not good from a practical point of view: the small gap between the local and the quantum bounds of their associated optimal Bell inequality requires that the confidence interval on the estimated Bell violation ${J}_{m}-\mu$ be very small. If not, i.e. if ${J}_{m}-\mu$ is smaller than the local bound, no Bell violation can be observed, and thus no randomness can be certified. This is the case for that point of our simulations.