Randomness in quantum mechanics: philosophy, physics and technology

Randomness is a fundamental resource indispensable in numerous scientific and practical applications like Monte-Carlo simulations, taking opinion polls, cryptography etc. In each case one has to generate a 'random sample', or simply a random sequence of digits. Variety of methods to extract such a random sequence from physical phenomena were proposed and, in general successfully, applied in practice. But how do we know that a sequence is 'truly random'? Or, at least. 'Random enough' for all practical purposes? Such problems become particularly acute for cryptography where provably unbreakable security systems are based on the possibility to produce a string of perfectly random, uncorrelated digits used later to encode data. Such a random sequence must be unpredictable for an adversary wanting to break the code, and here we touch a fundamental question concerning the nature of randomness. If all physical processes are uniquely determined by their initial conditions and the only cause of unpredictability is our inability to determine them with an arbitrary precision, or lack of detailed knowledge of actual conditions that can influence their time evolution, the security can be compromised, if an adversary finds finer methods to predict outcomes. On the other hand, if there are processes 'intrinsically' random, i.e. random by their nature and not due to gaps in our knowledge, unconditional secure coding schemes are conceivable.

Two attitudes concerning the nature of randomness in the world mentioned above can be dubbed as epistemic and ontic. Both agree that we observe randomness (indeterminacy, unpredictability) in nature, but differ in identifying the source of the phenomenon. The first claims that the world is basically deterministic, and the only way in which a random behavior demanding probabilistic description appears is due to lack of knowledge of the actual state of the observed system or details of its interaction with the rest of the universe. In contrast, according to the second, the world is nondeterministic, randomness is its intrinsic property, independent of our knowledge and resistant to attempts aiming at circumventing its consequences by improving precision of our observations. In other words, 'intrinsic' means that this kind of randomness cannot be understood in terms of a deterministic 'hidden variable' model. The debate on both epistemic and ontic nature of randomness can be traced back to the pre-Socratic beginnings of the European philosophy. For early atomists, Leucippus⁴ and Democritus⁵, the world was perfectly deterministic. Any occurrence of chance is a consequence of our limited abilities⁶. One century later Epicurus took the opposite side. To accommodate an objective chance the deterministic motion of atoms must be interrupted, without a cause, by 'swerves'. Such an indeterminacy propagates then to macroscopic world. The main motivation was to explain, or at least to give room for human free will, hardly imaginable in a perfectly deterministic world⁷. It should be clear, however, that purely random nature of human actions is as far from free will, as the latter from a completely deterministic process. A common feature of both extreme cases of pure randomness and strict determinism is lack of any possibility to control or influence the course of events. Such a possibility is definitely indispensable component of the free will. The ontological status of randomness is thus here irrelevant and the discussion whether 'truly random theories', (as supposedly should quantum mechanics be), can 'explain the phenomenon of the free will' is pointless. It does not mean that free will and intrinsic randomness problems are not intertwined. On one side, as we explain later, the assumption that we may perform experiments in which we can freely choose what we measure, is an important ingredient in arguing that violating of Bell-like inequalities implies 'intrinsic randomness' of quantum mechanics. On the second side, as strict determinism in fact precludes the free will, the intrinsic randomness seems to be a necessary condition for its existence. But, we need more to produce a condition that is sufficient. An interesting recent discussion of connections between free will and quantum mechanics may be found in Part I of Suarez and Adams (2013). In Gisin (2013) and Brassard and Robichaud (2013) the many-world interpretation of quantum mechanics, which is sometimes treated as a cure against odds of orthodox quantum mechanics, is either dismissed as a theory that can accommodate free will (Gisin 2013) or, taken seriously in Brassard and Robichaud (2013), as admitting the possibility that free will might be a mere illusion. In any case it is clear that one needs much more than any kind of randomness to understand how free will appears. In Suarez (2013) the most radical attitude to the problem (apparently present also in Gisin (2013)) is that 'not all that matters for physical phenomena is contained in space-time'.

A seemingly clear distinction between two possible sources of randomness outlined in the previous section becomes less obvious if we try to make the notion of determinism more precise. Historically, its definition usually had a strong epistemic flavor. Probably the most famous characterization of determinism is that of Laplace (1814): ' Une intelligence qui, pour un instant donné, connaîtrait toutes les forces dont la nature est animée, et la situation respective des êtres qui la composent, si d'ailleurs elle était assez vaste pour soumettre ces données à l'analyse, embrasserait dans la même formule les mouvemens des plus grands corps de l'univers et ceux du plus léger atome : rien ne serait incertain pour elle, et l'avenir comme le passé, serait présent á ses yeux⁸'. Two hundred years later Karl Raimund Popper writes 'We can ... define 'scientific' determinism as follows: the doctrine of 'scientific' determinism is the doctrine that the state of any closed physical system at any given future instant of time can be predicted, even from within the system, with any specified degree of precision, by deducing the prediction from theories, in conjunction with initial conditions whose required degree of precision can always be calculated (in accordance with the principle of accountability) if the prediction task is given' (Popper 1982). By contraposition thus, unpredictability implies indeterminism. If we now equate indeterminism with existence of randomness, we see that a sufficient condition for the latter is the unpredictability. But, unpredictable can be equally well events about which we do not have enough information, and those that are 'random by themselves'. Consequently, as it should have been obvious, Laplacean-like descriptions of determinism are of no help when we look for sources of randomness.

Let us thus simply say that a course of events is deterministic if there is only one future way for it to develop. Usually we may also assume that its past history is also unique. In such cases the only kind of randomness is the epistemic one.

As an exemplary theory describing such situations one usually invokes classical mechanics. Arnol'd in his treatise on ordinary differential equations, after adopting the very definition of determinism advocated above⁹, writes: 'thus for example, classical mechanics considers the motion of systems whose past and future are uniquely determined by the initial positions and velocities of all points of the system'¹⁰. The same can be found in his treatise on classical mechanics¹¹. He gives also a kind of justification, 'It is hard to doubt this fact, since we learn it very early'¹². But, what he really means is that a mechanical system are uniquely determined by positions and momenta of its constituents, 'one can imagine a world, in which to determine the future of a system one must also know the acceleration at the initial moment, but experience shows us that our world is not like this'¹³. It is clearly exposed in another classical mechanics textbook, Landau and Lifschitz's Mechanics: 'if all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically, this means that, if all the co-ordinates q and velocities $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined'¹⁴. Apparently, also here the 'experience' concerns only the observation that positions and velocities, and not higher time-derivatives of them, are sufficient to determine the future.

In such a theory there are no random processes. Everything is in fact completely determined and can be predicted with desired accuracy once we improve our measuring and computing devices. Statistical physics, which is based on classical mechanics, is a perfect example of indeterministic theory where measurable quantities like pressure or temperature are determined by mean values of microscopical 'hidden variables', for example positions and momenta of gas particles. These hidden variables, however, are completely determined at each instant of time by the laws of classical mechanics, and with an appropriate effort can be, in principle, measured and determined. What makes the theory 'indeterministic' is a practical impossibility to follow trajectories of individual particles because of their number and/or the sensitiveness to changes of initial conditions. In fact such a sensitiveness was pointed as a source of chance by Poincaré¹⁵ and Smoluchowski¹⁶ soon after modern statistical physics was born, but it is hard to argue that this gives to the chance an ontological status. It is, however, worth mentioning that Poincaré was aware that randomness might have not only epistemic character. In the above cited Introduction to his Calcul des probabilités he states 'Il faut donc bien que le hasard soit autre chose que le nom que nous donnons à notre ignorance'¹⁷, ('So it must be well that chance is something other than the name we give our ignorance'¹⁸).

Still, the very existence of deterministic chaos implies that classical mechanics is unpredictable in general in any practical sense. The technical question how important this unpredictability can be is, actually, the subject of intensive studies in the last decades (for recent monographs see Vallejo and Sanjuan (2017) and Rajasekar and Sanjuan (2016)).

It is commonly believed (and consistent with the above cited descriptions of determinism in mechanical systems) that on the mathematical level the deterministic character of classical mechanics takes form of Newton's second law

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{N2} m\frac{{\rm d}^2{\bf x}(t)}{{\rm d}t^2}={\bf F}({\bf x(t)},t), \nonumber \end{align} \tag{ 1 }$$

where the second derivatives of the positions, ${\bf x}(t)$, are given in terms of some (known) forces ${\bf F}({\bf x(t)}, t)$. But, to be able to determine uniquely the fate of the system we need something more than merely initial positions ${\bf x}(0)$ and velocities ${\rm d}{\bf x}(t)/{\rm d}t\vert _{t=0}$. To guarantee uniqueness of the solutions of the Newton's equations (1), we need some additional assumptions about the forces ${\bf F}({\bf x(t)}, t)$. According to the Picard theorem¹⁹ Coddington and Levinson (1955), an additional technical condition that is sufficient for the uniqueness is the Lipschitz condition, limiting the variability of the forces with respect to the positions. Breaking it opens possibilities to have initial positions and velocities that do not determine uniquely the future trajectory. A world, in which there are systems governed by equations admitting non-unique solutions is not deterministic according to our definition. We can either defend determinism in classical mechanics by showing that such pathologies never occur in our world, or agree that classical mechanics admits, at least in some cases, a nondeterministic evolution. Each choice is hard to defend. In fact it is relatively easy to construct examples of more or less realistic mechanical systems for which the uniqueness is not guaranteed. Norton (2007) (see also Norton (2008)) provided a model of a point particle sliding on a curved surface under the gravitational force, for which the Newton equation reduces to $\frac{{\rm d}^2r}{{\rm d}t^2}=\sqrt{r}$. For the initial conditions $r(0)=0, \frac{{\rm d}r}{{\rm d}t}\vert _{t=0}=0$, the equation has not only an obvious solution $r(t)=0$, but, in addition, a one parameter family given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r(t)=\left\{\begin{array}{@{}cl@{}} 0, & {\rm for\ } t\leqslant T \nonumber \\ \frac{1}{144}(t-T)^4, & {\rm for\ } t\geqslant T \end{array} \right. \nonumber \end{align} \tag{ 2 }$$

where T is an arbitrary parameter. For a given T the solution describes the particle staying at rest at $r=0$ until T and starting to accelerate at T. Since T is arbitrary we can not predict when the change from the state of rest to the one with a non-zero velocity takes place.

The example triggered discussions (Korolev 2007a, 2007b, Kosyakov 2008, Malament 2008, Roberts 2009, Wilson 2009, Zinkernagel 2010, Fletcher 2012, Laraudogoitia 2013), raising several problems, in particular its physical relevance in connection with simplifications and idealizations made to construct it. However, they do not seem to be different from ones commonly adopted in descriptions of similar mechanical situations, where the answers given by classical mechanics are treated as perfectly adequate. At this point classical mechanics is not a complete theory of the part of the physical reality it aspires to describe. We are confronted with a necessity to supplement it by additional laws dealing with situations where the Newton's equation do not posses unique solutions.

The explicit assumption of incompleteness of classical mechanics has its history, astonishingly longer than one would expect. Possible consequences of non-uniqueness of solutions attracted attention of Boussinesq who in his Mémoire for the French Academy of Moral and Political Sciences writes: '...les phénomènes de mouvement doivent se diviser en deux grandes classes. La première comprendra ceux où les lois mécaniques exprimées par les équations différentielles détermineront à elles seules la suite des états par lesquels passera le système, et où, par conséquent, les forces physico-chimiques ne laisseront aucun rôle disponible à des causes d'une autre nature. Dans la seconde classe se rangeront, au contraire, les mouvements dont les équations admettront des intégrales singulières, et dans lesquels il faudra qu'une cause distincte des forces physico-chimiques intervienne, de temps en temps ou d'une manière continue, sans d'ailleurs apporter aucune part d'action mécanique, mais simplement pour diriger le système a chaque bifurcation d'intégrales qui se présentera'²⁰.

Boussinesq does not introduce any probabilistic ingredient to the reasoning, but definitely, there is a room to go from mere indeterminism to the awaited 'intrinsic randomness'. To this end, however, we need to postulate an additional law supplementing classical mechanics by attributing probabilities to different solutions of non-Lipschitzian equations²¹. It is hard to see how to discover (or even look for) such a law, and how to check its validity. What we find interesting is an explicit introduction to the theory a second kind of motion. It is strikingly similar to what we encounter in quantum mechanics, where to explain all observed phenomena one has to introduce two kinds of kinematics of a perfectly deterministic Schrödinger evolution and indeterministic state reductions during measurements. Similarity consist in the fact, that deterministic (Schrödinger, Newton) equations are not sufficient to describe the full evolution: they have to be completed, for instance by probabilistic description of the results of measurements in quantum mechanics, or by probabilistic choice of non-unique solutions in the Norton's example²².

It is interesting to note the ideas of Boussinesq have been in fact a subject of intensive discussion in the recent years in philosophy of science within the, so called, 'second Boussinesq debate'. The first Boussinesq debate took place in France between 1874–1880. As stated by Michael Mueller (2015): 'in 1877, a young mathematician named Joseph Boussinesq presented a mémoire to the Académie des Sciences, which demonstrated that some differential equations may have more than one solution. Boussinesq linked this fact to indeterminism and to a possible solution to the free will versus determinism debate'. The more recent debate discovered, in fact, that some hints for the Boussinesq ideas can be also found in the works of James Clerk Maxwell (Isley 2017). The views of Maxwell, important in this debate and not known very much by physicists, show that he was very much influenced by the work of Joseph Boussinesq and Adhémar Jean Claude Barré de Saint-Venant (Michael Mueller 2015). What is also quite unknown by many scientists is that Maxwell learn the statistical ideas from Adolphe Quetelet, a Belgian mathematician, considered to be one of the founders of statistics. Excellent account on the concepts of determinism versus indeterminism, on the notion of uncertainty, also associated to the idea of randomness, as well as on different meanings that randomness has for different audiences may be found in the set of blogs of Sanjuán (2009c), Sanjuán (2009b), Sanjuán (2009a) and in the outstanding book (Dahan-Dalmedico et al 1992). These references cover also a lot of details of the the first and recent Boussinesq debate. A Polish text by Koleżyński (2007) discusses related quotations from Boussinesq, Maxwell and Poincaré in a philosophical context of the determinism.

Of course, to great extend Boussinesq debate was stimulated by the attempts toward understanding of nonlinear dynamics and hydrodynamics in general, and the phenomenon of turbulence in particular. A nice review of of various approaches and ideas until 1970s is presented in the lecture by Farge (1991). The contemporary approach to turbulence is very much related to the Boussinesq suggestions and the use of non-Lipschitzian, i.e. nondeterministic hydrodynamics, has been develop in the recent years by Falkovich, Gawedzki, Vargassola and others (for outstanding reviews see Falkovich et al (2001) and Gawedzki (2001)). The history of these works is nicely described in the presentation (Bernard et al 1998), while the most important particular articles include the series of papers by Bernard et al (1998), Gawedzki and Vergassola (2000), Vanden Eijnden and Vanden Eijnden (2000), Weinan and Vanden Eijnden (2001), Le Jan and Raimond (1998, 2002) and Le Jan and Raimond (2004).

The chances of proving the existence of 'intrinsic randomness' in the world seem to be much higher, when we switch to quantum mechanics. The Born's interpretation of the wave function implies that we can count only on a probabilistic description of reality, therefore quantum mechanics is inherently probabilistic.

Obviously, one should ask what is the source of randomness in quantum physics. As pointed out by one of the referees: 'in my view all the sources of randomness originate because of interaction of the system (and/or the measurement apparatus) with an environment. The randomness that affects pure states due to measurement is, in my view, due to the interaction of the measurement apparatus with an environment. The randomness that affects open systems (those that directly interact with an environment) is again due to environmental effects'. This point of view is, as considered by many physicists, of course, parallel to the contemporary theory of quantum measurements, and collapse of the wave function (Wheeler and Zurek 1983, Zurek 2003, 2009).

Still, the end result of such approach to randomness and quantum measurements is that the Born's rule and the traditional Copenhagen interpretation is not far from being rigorously correct. At the same time, quantum mechanics viewed from the device independent point of view, i.e. by regarding only probabilities of outcomes of individual or correlated measurements, incorporates randomness, which cannot be reduced to our lack of knowledge or imperfectness of our measurements (this will be discussed with more details below). In this sense for the purpose of the present discussion, the detailed form of the major source of the randomness is not relevant, as long as this randomness leads to contextual results of measurements, or nonlocal correlations.

Let us repeat, both the pure Born's rule and the advanced theory of quantum measurement imply that the measurement outcomes (or expectation value of an observable) may have some randomness. However, a priori there are no obvious reasons for leaving the Democritean ground and switch to the Epicurean view. It might be so that quantum mechanics, just as statistical physics, is an incomplete theory admitting deterministic hidden variables, values of which were beyond our control. To be precise, one may ask how 'intrinsic' this randomness is and if it can be considered as an epistemic one. To illustrate it further, we consider two different examples in the following.

2.3.1. Contextuality and randomness.

2.3.2. Nonlocality and randomness.

It is important to extend the situation beyond the one mentioned above to multi-party systems and local measurements. For example, consider multi-particle system with each particle placed in a separated region. Now, instead of observing the system as a whole, one may get interested to observe only a part of it, i.e. perform local measurements. Given two important facts that QM allows superposition and no quantum system can be absolutely isolated, spatially separated quantum systems can be non-trivially correlated, beyond classical correlations allowed by classical mechanics. In such situation, the information contained in the whole system is certainly larger than that of sum of individual local systems. The information residing in the nonlocal correlations cannot be accessed by observing locally individual particles of the systems. It means that local descriptions cannot completely describe the total state of the system. Therefore, outcomes due to any local observation are bound to incorporate a randomness in the presence of nonlocal correlation, as long as we do not have access to the global system or ignore the nonlocal correlation. In technical terms, the randomness appearing in the local measurement outcomes cannot be understood in terms of deterministic local hidden variable model and a 'true' local indeterminacy is present²⁵. Moreover, randomness on the local level appears even if the global state of the system is pure and completely known—the necessary condition for this is just entanglement of the pure state in question. That is typically referred as 'intrinsic' randomness in the literature, and that is the point of view we adopt in this report.

Before we move further in discussing quantum randomness in the presence of quantum correlation, let us make a short detour through the history of foundation of quantum mechanics. The possibility of nonlocal correlation, also known as quantum entanglement, was initially put forward with the question if quantum mechanics respects local realism, by Einstein et al (1935). According to EPR, two main properties any reasonable physical theory should satisfy are realism and locality. The first one states that if a measurement outcome of a physical quantity, pertaining to some system, is predicted with unit probability, then there must exits 'an element of reality' correspond to this physical quantity having a value equal to the predicted one, at the moment of measurement. In other words, the values of observables, revealed in measurements, are intrinsic properties of the measured system. The second one, locality, demands that elements of reality pertaining to one system cannot be affected by measurement performed on another sufficiently far system. Based on these two essential ingredients, EPR studied the measurement correlations between two entangled particles and concluded that the wave function describing the quantum state 'does not provide a complete description of physical reality'. Thereby they argued that quantum mechanics is an incomplete but effective theory and conjectured that a complete theory describing the physical reality is possible.

In these discussions, one needs to clearly understand what locality and realism mean. In fact, they could be replaced with no-signaling and determinism, respectively. The no-signaling principle states that infinitely fast communication is impossible. The relativistic limitation of the speed, by the velocity of light, is just a special case of no-signaling principle. If two observers are separated and perform space-like separated measurements (as depicted in figure 1), then the principle ascertains that the statistics seen by one observer, when measuring her particle, is completely independent of the measurement choice made on the space-like separated other particle. Clearly, if it were not the case, one observer could, by changing her measurement choice, make a noticeable change on the other and thereby instantaneously communicate with an infinite speed.

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Determinism, the other important ingredient, implies that correlations observed in an experiment can be decomposed as mixtures of deterministic ones i.e. occurring in situations where all measurements have deterministic outcomes. A deterministic theory accepts the fact that the apparent random outcomes in an experiment, like in coin tossing, are only consequences of ignorance of the actual state of the system. Therefore, each run of the experiment does have an a priori definite result, but we have only an access to averages.

In 1964, Bell showed that all theories that satisfy locality and realism (in the sense of EPR) are incompatible with quantum mechanics (Bell 1964, 1966). In a simple experiment, mimicking Bell's scenario, two correlated quantum particles are sent to two spatially separated measuring devices (see figure 1), and each device can perform two different measurements with two possible outcomes. The measurement processes are space-like separated and no communication is possible when these are performed. With this configuration a local-realistic model gives bounds on the correlation between the outcomes observed in the two measurement devices, called Bell inequalities (Bell 1964). In other words, impossibility of instantaneous communication (no-signaling) between spatially separated systems together with full local determinism imply that all correlations between measurement results must obey the Bell inequalities.

Strikingly, these inequalities are violated with correlated (entangled) quantum particles, and therefore have no explanations in terms of deterministic local hidden variables. In fact, the correlations predicted by the no-signaling and determinism are exactly the same as predicted by EPR model, and they are equivalent. The experimental violations of the Bell inequalities in 1972 (Freedman and Clauser 1972), in 1981 (Aspect et al 1981) and in 1982 (Aspect et al 1982), along with the recent loophole-free Bell-inequality violations (Hensen et al 2015, Giustina et al 2015) confirm that any local-realistic theory is unable to predict the correlations observed in quantum mechanics. It immediately implies that either no-signaling or local determinism has to be abandoned. For the most physicists, it is favorable to dump local determinism and save no-signaling. Assuming that the nature respects no-signaling principle, any violation of Bell inequality implies thus that the outcomes could not be predetermined in advance.

Thus, once the no-signaling principle is accepted to be true, the experimental outcomes, due to local measurements, cannot be deterministic and therefore are random. Of course, a valid alternative is to abandon the no-signaling principle, allow for non-local hidden variables, but save the determinism, as for instance is done in Bohm's theory (Bohm 1951, 1952). In any case, some kind of non-locality is needed to explain Bell correlations. One can, also, abandon both no-signaling and local determinism: such sacrifice is, however, hard to be accepted by majority of physicists, and scientists in general.

Another crucial assumption is considered for Bell experiments, that is the measurements performed with the local measurement devices have to be chosen 'freely'. In other words, the measurement choices cannot, in principle, be predicted in advance. If the free-choice condition is relaxed and the chosen measurements could be predicted in advance, then it is easy to construct a no-signaling, but deterministic theory that leads to Bell violations. It has been shown in Hall (2010) and Koh et al (2012) that one does not have to give up measurement independence completely to violate Bell inequalities. Even, relaxing free-choice condition to a certain degree, the Bell inequities could be maximally violated using no-signaling and deterministic model (Hall 2010). However, in the Bell-like experiment scenarios where the local observers are separated, it is very natural to assume that the choices of the experiments are completely free (this is often referred to free-will assumption). Therefore, the Bell-inequality violation in the quantum regime, with the no-signaling principle, implies that local measurement outcomes are 'intrinsically' random.

The lesson that we should learn from the above discussion is that the question raised by Einstein, Rosen, Podolsky found its operational meaning in Bell's theorem that showed incompatibility of hidden-variable theories with quantum mechanics (Bell 1964, 1966). Experiment could now decide about existence or non-existence of nonlocal correlations. Exhibiting non-local correlations in an experiment gave, under the assumption of no-signaling, a proof of a non-deterministic nature of quantum mechanical reality, and allowed certifying the existence of truly random processes. These experiments require, however, random adjustments of measuring devices (Bell 1964). There must exist a truly random process controlling their choice. This, ironically, closes an unavoidable circulus vitiosus. We can check the indeterministic character of the physical reality only assuming that it is, in fact, indeterministic.

According to standard textbook approach, quantum mechanics (QM) is an inherently probabilistic theory (see Messiah (2014), Cohen-Tannoudji et al (1991) and Wheeler and Zurek (1983)—the prediction of QM concerning results of measurements are typically probabilistic. Only in very rare instances measurements give deterministic outcomes—this happens when the systems is in an eigenstate of an observable to be measured. Note, that in general, even if we have full information about the quantum mechanical state of the system, the outcome of the measurements is in principle random. The paradigmatic example is provided a d-state system (a qudit), whose space of states is spanned by the states $\vert 1\rangle$, $\vert 2\rangle$,..., $\vert d\rangle$. Suppose that we know the system is in the superposition state $\vert \phi\rangle=\sum_{j=1}^d\alpha_j\vert\,j\rangle$, where $\alpha_j$ are complex probability amplitudes and $\sum_{j=1}^d\vert \alpha_j\vert ^2 =1$, and we ask whether it is in a state $\vert i\rangle$. To find out, we measure an observable $\hat P=\vert i\rangle \langle i\vert$ that projects on the state $\vert i\rangle$. The result of such measurement will be one (yes, the system is in the state $\vert i\rangle$) with probability $\vert \alpha_i\vert ^2$ and zero with probability $1-\sum_{j \ne i}^d\vert \alpha_j\vert ^2$.

We do not want to enter here deeply into the subject of the foundations of QM, but we want to remind the readers the 'standard' approach to QM.

3.1.1. Postulates of QM.

3.1.2. Measurement theory.

Evidently, the inherent randomness of QM is associated with the measurement processes (Fourth and Fifth Postulates). The quantum measurement theory has been a subject of intensive studies and long debate, see e.g. Wheeler and Zurek (1983). In particular the question of the physical meaning of the wave function collapse has been partially solved only in the last 30 years by analyzing interactions of the measured system with the environment (reservoir), describing the measuring apparatus (see seminal works of Zurek (2003, 2009))

In the abstract formulation in the early days of QM, one has considered von Neumann measurements (Neumann 1955), defined in the following way. Let the observable $\hat Q$ has (possibly degenerated) eigenvalues q_n and let $\hat E_n$ denote projectors on the corresponding invariant subspaces (one dimensional for non-degenerate eigenvalues, k-dimensional for k-fold degenerated eigenvalues). Since the invariant subspace are orthogonal, we have $\hat E_n\hat E_m=\delta_{nm}\hat E_n$, where $\delta_{mn}$ is the Kronecker delta. If $\hat P_\psi$ denotes the projector, which describes a state of a system, the measurement outcome corresponds to the eigenvalue q_n of the observable will appear with probability $p_ n={\rm Tr}(\hat P_\psi\hat E_n)$, where ${\rm Tr}(.)$ denotes the matrix trace operation. Also, after the measurement, the systems is found in the state $\hat E_n\hat P_\psi\hat E_n/p_n$ with probability p_n.

In the contemporary quantum measurement theory the measurements are generalized beyond the von Neumann projective ones. To define the, so called, positive-operator valued measures (POVM), one still considers von Neumann measurements, but on a system plus an additional ancilla system (Peres 1995). POVMs are defined by a set of Hermitian positive semidefinite operators $\{F_i\}$ on a Hilbert space $\mathcal{H}$ that sum to the identity operator,

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sum_{i=1}^K F_i = \mathbb{I}_H.\nonumber \end{align*}$$

This is a generalization of the decomposition of a (finite-dimensional) Hilbert space by a set of orthogonal projectors, $\{E_i\}$, defined for an orthogonal basis $\{\left\vert \phi_{i}\right\rangle\}$ by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle E_i=\left\vert \phi_{i}\right\rangle \left\langle\phi_{i}\right\vert, \nonumber \end{align*}$$

hence,

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sum_{i=1}^N E_i = \mathbb{I}_H, \quad E_i E_j = \delta_{i j} E_i.\nonumber \end{align*}$$

An important difference is that the elements of POVM are not necessarily orthogonal, with the consequence that the number K of elements in the POVM can be larger than the dimension N of the Hilbert space they act on.

The post-measurement state depends on the way the system plus ancilla are measured. For instance, consider the case where the ancilla is initially a pure state $\vert 0\rangle_B$. We entangle the ancilla with the system, taking

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \vert \psi\rangle_A \vert 0\rangle_B \rightarrow \sum_i M_i \vert \psi\rangle_A \vert i\rangle_B,\nonumber \end{align*}$$

and perform a projective measurement on the ancilla in the $\{\vert i\rangle_B\}$ basis. The operators of the resulting POVM are given by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle F_i = M_i ^\dagger M_i .\nonumber \end{align*}$$

Since the M_i are not required to be positive, there are an infinite number of solutions to this equation. This means that there are infinitely many different experimental apparatus giving the same probabilities for the outcomes. Since the post-measurement state of the system (expressed now as a density matrix)

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \rho_i = {M_i \rho M_i^\dagger \over {\rm tr}(M_i \rho M_i^\dagger)}\nonumber \end{align*}$$

depends on the M_i, in general it cannot be inferred from the POVM alone.

If we accept quantum mechanics and its inherent randomness, then it is possible in principle to implement measurements of an observable on copies of a state that is not an eigenstate of this observable, to generate a set of perfect random numbers. Early experiments and commercial devices attempted to mimic a perfect coin with probability 1/2 of getting head and tail. To this aim quantum two-level systems were used, for instance single photons of two orthogonal circular polarizations. If such photons are transmitted through a linear polarizer of arbitrary direction then they pass (do not pass) with probability 1/2. In practice, the generated numbers are never perfect, and randomness extraction is required to generate good random output. The challenges of sufficiently approximating the ideal two-level scenario, and the complexity of detectors for single quantum systems, have motivated the development of other randomness generation strategies. In particular, continuous-variable techniques are now several orders of magnitude faster, and allow for randomness extraction based on known predictability bounds. See section 4.

It is worth mentioning that the Heisenberg uncertainty relation (Heisenberg 1927) and its generalized version, i.e. the Robertson–Scrödinger relation (Robertson 1929, Schrödinger 1930, Wheeler and Zurek 1983), often mentioned in the context of quantum measurements, signify how precisely two non-commuting observables can be measured on a quantum state. Quantitatively, for a given state ρ and observables X and Y, it gives a lower bound on the uncertainty when they are measures simultaneously, as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \delta X^2 \delta Y^2 \geqslant \frac{1}{4} \vert {\rm{Tr}} \rho [X,Y]\vert ^2, \label{eq:RSUR} \nonumber \end{align} \tag{ 3 }$$

where $\delta X^2={\rm{Tr}}\rho X^2-({\rm{Tr}} \rho X){\hspace{0pt}}^2$ is the variance and $[X, Y]=XY-YX$ is the commutator. A non-vanishing $\delta X$ represents a randomness in the measurement process and that may arise from non-commutativity (misalignment in the eigenbases) between state and observable, or even it may appear due to classical uncertainty present in the state (i.e. not due to quantum superposition). In fact equation (3) does allow to have either $\delta X$ or $\delta Y$ vanishing, but not simultaneously for a given state ρ and $[X, Y]\neq 0$. However, when $\delta X$ vanishes, it is nontrivial to infer on $\delta Y$ and vice versa. To overcome this drawback, the uncertainty relation is extended to sum-uncertainty relations, both in terms of variance (Maccone and Pati 2014) and entropic quantities (Beckner 1975, Białynicki-Birula and Mycielski 1975, Deutsch 1983, Maassen and Uffink 1988). We refer to Coles et al (2015) for an excellent review on this subject. The entropic uncertainty relation was also considered in the presence of quantum memory (Berta et al 2010). It has been shown that, in the presence of quantum memory, any two observables can simultaneously be measured with arbitrary precision. Therefore the randomness appearing in the measurements can be compensated by the side information stored in the quantum memory. As we mentioned in the previous section, Heisenberg uncertainty relations are closely related to the contextuality of quantum mechanics at the level of single systems. Non-commuting observables are indeed responsible for the fact that there does not exist non-contextual hidden variable theories that can explain all results of quantum mechanical measurements on a given system.

The inherent randomness considered in this work is steaming out the Born's rule in quantum mechanics, irrespective of the fact if there is more than one observable being simultaneously measured or not. Furthermore the existence of nonlocal correlations (and quantum correlations) in the quantum domain give rise to possibility of, in a sense, a new form of randomness. In the following we consider such randomness and its connection to nonlocal correlations. Before we do so, we shall discuss nonlocal correlations in more detail.

3.2.1. Two-party nonlocality.

Let us now turn to nonlocality, i.e. property of correlations that violate Bell inequalities (Bell 1964, Brunner et al 2014). As we will see below, nonlocality is intimately connected to the intrinsic quantum randomness. In the traditional scenario a Bell nonlocality test relies on two spatially separated observers, say Alice and Bob, who perform space-like measurements on a bipartite system possibly produced by a common source. For a schematic see figure 1. Suppose Alice's measurement choices are $x\in \mathcal{X}=\{1, \ldots, M_A\}$ and Bob's choices $y\in \mathcal{Y}=\{1, \ldots, M_B\}$ and the corresponding outcomes $a\in \mathcal{A}=\{1, \ldots, m_A\}$ and $b\in \mathcal{B}=\{1, \ldots, m_B\}$ respectively. After repeating many times, Alice and Bob communicate their measurement settings and outcomes to each other and estimate the joint probability $p(a, b\vert x, y)=p(A=a, B=b\vert X=x, Y=y)$ where X, Y are the random variables that govern the inputs and A, B are the random variables that govern the outputs. The outcomes are considered to be correlated, for some $x, y, a, b$, if

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p(a,b\vert x,y)\neq p(a\vert x)\,p(b\vert y). \nonumber \end{align} \tag{ 4 }$$

Observing such correlations is not surprising as there are many classical sources and natural processes that lead to correlated statistics. These can be modeled with the help of another random variable Λ with the outcomes λ, which has a causal influence on both the measurement outcomes and is inaccessible to the observers or ignored.

In a local hidden-variable model, considering all possible causes Λ, the joint probability can then be expressed as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p(a,b\vert x,y,\lambda) = p(\lambda)\,p(a\vert x, \lambda)\,p(b\vert y,\lambda). \nonumber \end{align} \tag{ 5 }$$

One, thereby, could explain any observed correlation in accordance with the fact that Alice's outcomes solely depends on her local measurement settings x, on the common cause λ, and are independent of Bob's measurement settings. Similarly Bob's outcomes are independent of Alice's choices. This assumption—the no-signaling condition—is crucial—it is required by the theory of relativity, where nonlocal causal influence between space-like separated events is forbidden. Therefore, any joint probability, under the local hidden-variable model, becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p(a,b\vert x,y) = \int_{\Lambda} {\rm d}\lambda p(\lambda)\,p(a\vert x, \lambda)\,p(b\vert y,\lambda), \label{eq:lhv} \nonumber \end{align} \tag{ 6 }$$

with the implicit assumption that the measurement settings x and y could be chosen independently of λ, i.e. $p(\lambda\vert x, y)=p(\lambda)$. Note that so far we have not assumed anything about the nature of the local measurements, whether they are deterministic or not. In a deterministic local hidden-variable model, Alice's outcomes are completely determined by the choice x and the λ. In other words, for an outcome a, given input x and hidden cause λ, the probability $p(a\vert x, \lambda)$ is either 1 or 0 and so as for Bob's outcomes. Importantly, the deterministic local hidden-variable model has been shown to be fully equivalent to the local hidden-variable model (Fine 1982). Consequently, the observed correlations that admit a join probability distribution as in (6), can have an explanation based on a deterministic local hidden-variable model.

In 1964, Bell showed that any local hidden-variable model is bound to respect a set of linear inequalities, which are commonly know as Bell inequalities. In terms of joint probabilities they can be expressed as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum_{a,b,x,y} \alpha^{xy}_{ab} \ p(a,b\vert x,y) \leqslant \mathcal{S}_L, \label{eq:bi} \nonumber \end{align} \tag{ 7 }$$

where $\alpha^{xy}_{ab}$ are some coefficients and $\mathcal{S}_L$ is the classical bound. Any violation of Bell inequalities (7) implies a presence of correlations that cannot be explained by a local hidden-variable model, and therefore have a nonlocal character. Remarkably, there indeed exists correlations violating Bell inequalities that could be observed with certain choices of local measurements on a quantum system, and hence do not admit a deterministic local hidden-variable model.

To understand it better, let us consider an example of the most studied two-party Bell inequalities, also known as Clauser–Horne–Shimony–Holt (CHSH) inequalities, introduced in Clauser et al (1969). Assume the simplest scenario (as in figure 1) in which Alice and Bob both choose one of two local measurements $x, y \in \{0, 1\}$ and obtain one of two measurement outcomes $a, b \in \{-1, 1\}$. Let the expectation values of the local measurements are $\langle a_x b_y \rangle = \sum_{a, b} a\cdot b \cdot p(a, b\vert x, y)$, then the CHSH-inequality reads:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I_{\rm CHSH}=\langle a_0 b_0 \rangle + \langle a_0 b_1 \rangle + \langle a_1 b_0 \rangle - \langle a_1 b_1 \rangle \leqslant 2. \label{eq:CHSHprob} \nonumber \end{align} \tag{ 8 }$$

One can maximize I_CHSH using local deterministic strategy and to do so one needs to achieve the highest possible values of $\langle a_0 b_0 \rangle, \ \langle a_0 b_1 \rangle, \ \langle a_1 b_0 \rangle$ and the lowest possible value of $\langle a_1 b_1 \rangle$. By choosing $p(1, 1\vert 0, 0)=p(1, 1\vert 0, 1)=p(1, 1\vert 1, 0)=1$, the first three expectation values can be maximized. However, in such situation the $p(1, 1\vert 1, 1)=1$ and I_CHSH could be saturated to 2. Thus the inequality is respected. However, it can be violated in a quantum setting. For example, considering a quantum state $\vert \Psi^+\rangle=\frac{1}{\sqrt{2}}(\vert 00\rangle + \vert 11\rangle)$ and measurement choices $A_0=\sigma_z$, $A_1=\sigma_x$, $B_0=\frac{1}{\sqrt{2}}(\sigma_z+\sigma_x)$, $B_1=\frac{1}{\sqrt{2}}(\sigma_z-\sigma_x)$ one could check that for the quantum expectation values $\newcommand{\ot}[0]{\otimes} \langle a_\alpha b_\beta \rangle = \langle \Psi^+ \vert A_\alpha \otimes B_\beta \vert \Psi^+ \rangle$ we get $I_{\rm CHSH}=2\sqrt{2}$. Here $\sigma_z$ and $\sigma_x$ are the Pauli spin matrices and $\vert 0\rangle$ and $\vert 1\rangle$ are two eigenvectors of $\sigma_z$. Therefore the joint probability distribution $p(a, b\vert x, y)$ cannot be explained in terms of local deterministic model.

3.2.2. Multi-party nonlocality and device independent approach.

Bell-type inequalities can be also constructed in multi-party scenario. Their violation signifies nonlocal correlations distributed over many parties. A detailed account may be found in Brunner et al (2014).

Here we introduce the concept of nonlocality using the contemporary language of device independent approach (DIA) (Brunner et al 2014). Recent successful hacking attacks on quantum cryptographic devices (Lydersen et al 2010) triggered this novel approach to quantum information theory in which protocols are defined independently of the inner working of the devices used in the implementation. That leads to avalanches of works in the field of device independent quantum information processing and technology (Brunner 2014, Pironio et al 2015).

The idea of DIA is schematically given in figure 2. We consider here the following scenario, usually referred to as the $(n, m, d)$ scenario. Suppose n spatially separated parties $A_1, \ldots, A_n$. Each of them possesses a black box with m measurement choices (or observables) and d measurement outcomes. Now, in each round of the experiment every party is allowed to perform one choice of measurement and acquires one outcome. The accessible information, after the measurements, is contained in a set of $(md){\hspace{0pt}}^n$ conditional probabilities $p(a_1, \ldots, a_n\vert x_1, \ldots, x_n)$ of obtaining outputs $a_1, a_2, \ldots, a_n$, provided observables $x_1, x_2, \ldots, x_n$ were measured. The set of all such probability distributions forms a convex set; in fact, it is a polytope in the probability manifold. From the physical point of view (causality, special relativity) the probabilities must fulfill the no-signaling condition, i.e. the choice of measurement by the k-th party, cannot be instantaneously signalled to the others. Mathematically it means that for any $k=1, \ldots, n$, the following condition

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:no-sig} &\sum_{a_k}p(a_1, \ldots, a_k, \ldots, a_n\vert x_1,\ldots, x_k,\ldots, x_n)\nonumber \\ & = p(a_1,\ldots, a_{k-1}, a_{k+1}\ldots, a_n\vert x_1, \ldots, x_{k-1},x_{k+1}\ldots, x_n), \nonumber \end{align} \tag{ 9 }$$

is fulfilled.

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The local correlations are defined via the concept of a local hidden variable λ with the associated probability $q_{\lambda}$. The correlations that the parties are able to establish in such case are of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p(a_1, & \ldots, a_n\vert x_1, \ldots, x_n) \nonumber \\ & = \sum_{\lambda} q_\lambda D(a_1\vert x_1,\lambda) \ldots D(a_n\vert x_n,\lambda), \nonumber \end{align} \tag{ 10 }$$

where $D(a_k\vert x_k, \lambda)$ are deterministic probabilities, i.e. for any λ, $D(a_k\vert x_k, \lambda)$ equals one for some outcome, and zero for all others. What is important in this expression is that measurements of different parties are independent, so that the probability is a product of terms corresponding to different parties. In this n-party scenario the local hidden variable model bounds the joint probabilities to follow the Bell inequalities, given as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum_{a_1, \ldots, a_n,x_1, \ldots, x_n} \alpha_{a_1, \ldots, a_n}^{x_1, \ldots, x_n} \ p(a_1,\ldots, a_n\vert x_1, & \ldots, x_n) \nonumber \\ &\leqslant \mathcal{S}_L^n, \nonumber \end{align} \tag{ 11 }$$

where $\alpha_{a_1, \ldots, a_n}^{x_1, \ldots, x_n}$ are some coefficients and $\mathcal{S}_L^n$ is the classical bound.

The probabilities that follow local (classical) correlations form a convex set that is also a polytope, denoted $\mathbb {P}$ (see figure 3). Its extremal points (or vertices) are given by $\prod_{i=1}^n D(a_i\vert x_i, \lambda)$ with fixed λ. The Bell theorem states that the quantum-mechanical probabilities, which also form a convex set $\mathcal{Q}$, may stick out of the classical polytope (Bell 1964, Fine 1982). The quantum probabilities are given by the trace formula for the set of local measurements

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ot}[0]{\otimes} \displaystyle p(a_1, \ldots, a_n \vert x_1, \ldots, x_n)={\rm Tr}(\rho \otimes_{i=1}^n M_{a_i}^{x_i}), \nonumber \end{align} \tag{ 12 }$$

where ρ is some n-partite state and $M_{a_i}^{x_i}$ denote the measurement operators (POVMs) for any choice of the measurement x_i and party i. As we do not impose any constraint on the local dimension, we can always choose the measurements to be projective, i.e. the measurement operators additionally satisfy $M_{a^{\prime}_i}^{x_i}M_{a_i}^{x_i}=\delta_{a^{\prime}_i, a_i} M_{a_i}^{x_i}$.

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This approach towards the Bell inequalities is explained in figure 3. Any hyperplane in the space of probabilities that separates the classical polytope from the rest determines a Bell inequality: everything that is above the upper horizontal dashed line is obviously nonlocal. But the most useful are the tight Bell inequalities corresponding to the facets of the classical polytope, i.e. its walls of maximal dimensions (lower horizontal dashed line).

In general $(n, m, d)$ scenarios, the complexity of characterizing the corresponding classical polytope is enormous. It is fairly easy to see that, even for $(n, 2, 2)$, the number of its vertices (extremal points) is equal to $2^{2n}$, hence it grows exponentially with n. Nevertheless, a considerable effort has been made in recent time to characterize multi-party nonlocality (Brunner et al 2014, Tura et al 2014a, 2014b, Liang et al 2015, Tura et al 2015, Rosicka et al 2016).

Among the many other device independent applications, the nonlocality appears to be a valuable resource in random number generation, certification, expansion and amplification, which we outline in the following sections. In fact, it has been shown that Bell nonlocal correlation is a genuine resource, in the framework of a resource theory, where the allowed operations are restricted to device independent local operations (Gallego et al 2012, Vicente 2014).

Before turning to the quantum protocols involving randomness, we discuss in this section randomness from the information theory standpoint. It is worth mentioning the role of randomness in various applications, beyond its fundamental implications. In fact randomness is a resource in many different areas—for a good overview see Motwani and Raghavan (1995) and Menezes et al (1996). Random numbers play important role in cryptographic applications, in numerical simulations of complex physical, chemical, economical, social and biological systems, not to mention gambling. That is why, much efforts were put forward to (1) develop good, reliable sources of random numbers, and (2) to design reliable certification tests for a given source of random numbers.

In general, there exists three types of random number generators (RNG). They are 'true' RNGs, pseudo-RNGs and the quantum-RNGs. The true RNGs are based on some physical processes that are hard to predict, like noise in electrical circuits, thermal or atmospheric noises, radioactive decays etc. The pseudo-RNGs rely on the output of a deterministic function with a shorter random seed possibly generated by a true RNG. Finally, quantum RNGs use genuine quantum features to generate random bits.

We consider here a finite sample space and denote it by the set Ω. The notions of ideal and weak random strings describe distributions over Ω with certain properties. When a distribution is uniform over Ω, we say that it has ideal randomness. A uniform distribution over n-bit strings is denoted by U_n. The uniform distributions are very natural to work with. However, when we are working with physical systems, the processes or measurements are usually biased. Then the bit strings resulting from such sources are not uniform. A string with nonuniform distribution, due to some bias (could be unknown), is referred to have weak randomness and the sources of such strings are termed as weak sources.

Consider the random variables denoted by the letters $(X, Y, \ldots)$. Their values will be denoted by $(x, y, \ldots)$. The probability of a random variable X with a value x is denoted as $p(X=x)$ and when the random variable in question is clear we use the shorthand notation $p(x)$. Here we briefly introduce the operational approach to define randomness of a random variable. In general, the degree of randomness or bias of a source is unknown and it is insufficient to define a weak source by a random variable X with a probability distribution $P(X)$. Instead one needs to model the weak randomness by a random variable with an unknown probability distribution. In another words, one need to characterize a set of probability distributions with desired properties. If we suppose that the probability distribution $P(X)$ of the variable X comes from a set Ω, then the degree of randomness is determined by the properties of the set, or more specifically, by the least random probability distribution(s) in the set. The types of weak randomness differ with the types of distribution $P(X)$ on Ω and the set Ω itself—they are determined by the allowed distributions motivated by a physical source. There are many ways to classify the weak random sources, and an interested reader may go through (Pivoluska and Plesch 2014). Here we shall consider two types of weak random sources, Santha–Vazirani (SV) and min-entropy (ME) sources, which will be sufficient for our later discussions.

A Santha–Vazirani (SV) source (Santha and Vazirani 1986) is defined as a sequence of binary random variables $(X_1, X_2, \ldots, X_n)$, such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:SVsource} \frac{1}{2} - \epsilon \leqslant & p(x_i=1\vert x_1,\ldots,x_{i-1}) \leqslant \frac{1}{2} + \epsilon, \nonumber \\ & \forall i \in \mathbb{N}, \forall x_1,\ldots,x_{i-1} \in \{0,1\} \nonumber, \nonumber \end{align} \tag{ 13 }$$

where the conditional probability $p(x_i=1\vert x_1, \ldots, x_{i-1})$ is the probability of the value $x_i=1$ conditioned on the values $x_1, \ldots, x_{i-1}$. The $\newcommand{\e}{{\rm e}} 0 \leqslant \epsilon \leqslant \frac{1}{2}$ represents bias of the source. For fixed and n, the SV-source represents a set of probability distributions over n-bit strings. If a random string satisfies (13), then we say that it is -free. For $\newcommand{\e}{{\rm e}} \epsilon=0$ the string is perfectly random—uniformly distributed sequence of bits U_n. For $\newcommand{\e}{{\rm e}} \epsilon=\frac{1}{2}$, nothing can be inferred about the string and it can be even deterministic. Note that in SV sources the bias can not only change for each bit X_i, but it also can depend on the previously generated bits. It requires that each produced bit must have some amount of randomness, when $\newcommand{\e}{{\rm e}} \epsilon \neq \frac{1}{2}$, and even be conditioned on the previous one.

In order the generalize it further one considers block source (Chor and Goldreich 1988), where the randomness is not guaranteed in every single bit, but rather for a block of n-bits. Here, in general, the randomness is quantified by the min-entropy, which is defined as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{\infty}(Y)={\rm{min}}_{y} [-{\rm{log}}_2(\,p(Y=y))], \nonumber \end{align} \tag{ 14 }$$

for an n-bit random variable Y. For a block source, the randomness is guaranteed by the most probable n-bit string appearing in the outcome of the variable—simply by guessing the most probable element—provided that the probability is less than one. A block $(n, k)$ source can now be modeled, for n-bit random variables $(X_1, X_2, ..., X_n)$, such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{\infty}&(X_i\vert X_{i-1}=x_{i-1},...,X_1=x_1)\geqslant k, \nonumber \\ & \forall i \in \mathbb{N}, \forall x_1,...,x_{i-1} \in \{0,1\}^n.\nonumber \end{align} \tag{ 15 }$$

These block sources are generalizations of SV-sources; the latter are recovered with $n=1$ and $\newcommand{\e}{{\rm e}} \epsilon = 2^{-H_{\infty}(X)}-\frac{1}{2}$. The block sources can be further generalized to sources of randomness of finite output size, where no internal structure is given, e.g. guaranteed randomness in every bit (SV-sources) or every block of certain size (block sources). The randomness is only guaranteed by its overall min-entropy. Such sources are termed as the min-entropy sources (Chor and Goldreich 1988) and are defined, for an n-bit random variable X, such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{\infty}(X)\geqslant k. \nonumber \end{align} \tag{ 16 }$$

Therefore, a min-entropy source represents a set of probability distributions where the randomness is upper-bounded by the probability of the most probable element, measured by min-entropy.

Let us now briefly outline the randomness extraction (RE), as it is one of the most common operations that is applied in the post-processing of weak random sources. The randomness extractors are the algorithms that produce nearly perfect (ideal) randomness useful for potential applications. The aim of RE is to convert randomness of a string from a weak source into a possibly shorter string of bits that is close to a perfectly random one. The closeness is defined as follows. The random variables X and Y over a same domain Ω are ε-close, if:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta(X,Y)=\frac{1}{2}\sum_{x \in \Omega}\vert\,p(X=x)-p(Y=x)\vert \leqslant \varepsilon. \nonumber \end{align} \tag{ 17 }$$

With respect to RE, the weak sources can be divided into two classes—extractable sources and non-extractable sources. Only from extractable sources a perfectly random string can be extracted by a deterministic procedure. Though there exist many non-trivial extractable sources (see for example Kamp et al (2011)), most of the natural sources, defined by entropies, are non-extractable and in such cases non-deterministic (stochastic) randomness extractors are necessary.

Deterministic extraction fails for the random strings from SV-sources, but it is possible to post-process them with a help of an additional random string. As shown in Vazirani (1987), for any and two mutually independent -free strings from SV-sources, it is possible to efficiently extract a single almost perfect bit ($\newcommand{\e}{{\rm e}} \epsilon^\prime \rightarrow 0$). For two n-bit independent strings $X=(X_1, ..., X_n)$ and $Y=(Y_1, ...Y_n)$, the post-processing function, Ex, has been defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Ex(X,Y)=(X_1\cdot Y_1)\oplus(X_2 \cdot Y_2)\oplus \cdots \oplus (X_n \cdot Y_n), \nonumber \end{align} \tag{ 18 }$$

where $\oplus$ denotes the sum modulo 2. The function Ex is the inner product between the n-bit strings X and Y modulo 2. Randomness extraction of SV-sources are sometime referred as randomness amplification as two -free strings from SV-sources are converted to one-bit string of $\newcommand{\e}{{\rm e}} \epsilon^\prime$-free and with $\newcommand{\e}{{\rm e}} \epsilon^\prime < \epsilon$.

Deterministic extraction is also impossible for the min-entropy sources. Nevertheless, an extraction might be possible with the help of seeded extractor in which an extra resource of uniformly distributed string, called the seed, is exploited. A function, $Ex: \ \{0, 1\}^n \ \times \ \{0, 1\}^r \mapsto \{0, 1\}^m$ is seeded $(k, \varepsilon)$-extractor, for every string from block $(n, k)$-source of random variable X, if

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta (Ex(X,U_r),U_m) \leqslant \varepsilon. \nonumber \end{align} \tag{ 19 }$$

Here U_r (U_m) is the uniformly distributed r-bit (m-bit) string. In fact, for a variable X, min-entropy gives the upper bound on the number of extractable perfectly random bits (Shaltiel 2002). Randomness extraction is well developed area of research in classical information theory. There are many randomness extraction techniques using multiple strings (Nisan and Ta-Shma 1999, Shaltiel 2002, Dodis et al 2004, Raz 2005, Gabizon and Shaltiel 2008, Barak et al 2010), such as universal hashing extractor, Hadamard extractor, DEOR extractor, BPP extractor etc, useful for different post-processing.

Here we link the new form of randomness, i.e the presence of nonlocality (in terms of Bell violation) in the quantum regime, to random number generation and certification. To do so, we outline how nonlocal correlations can be used to generate new types of random numbers, what has been experimentally demonstrated in Pironio et al (2010). Consider the Bell-experiment scenario (figure 1), as explained before. Two separated observers perform different measurements, labeled as x and y, on two quantum particles in their possession and get measurement outcomes a and b, respectively. With many repetitions they can estimate the joint probability, $p(a, b\vert x, y)$, for the outcomes a and b with the measurement choices x and y. With the joint probabilities the observers could check if the Bell inequalities are respected. If a violation is observed the outcomes are guaranteed to be random. The generation of these random numbers is independent of working principles of the measurement devices. Hence, this is a device independent random number generation. In fact, there is a quantitative relation between the amount of Bell-inequality violation and the observed randomness. Therefore, these random numbers could be (a) certifiable, (b) private, and (c) device independent (Colbeck 2007, Pironio et al 2010). The resulting string of random bits, obtained by N uses of measurement device, would be made up of N pair of outcomes, $(a_1, b_1, ..., a_N, b_N)$, and their randomness could be guaranteed by the violation of Bell inequalities.

There is however an important point to be noted. A priori, the observers do not know whether the measurement devices violate Bell inequalities or not. To confirm they need to execute statistical tests, but such tests cannot be carried out in predetermined way. Of course, if the measurement settings are known in advance, then an external agent could prepare the devices that are completely deterministic and the Bell-inequality violations could be achieved even in the absence of nonlocal correlations. Apparently there is a contradiction between the aim of making random number generator and the requirement of random choices to test the nonlocal nature of the devices. However, it is natural to assume that the observers can make free choices when they are separated.

Initially it was speculated that the more particles are nonlocally correlated (in the sense of Bell violation), the stronger would be the observed randomness. However, this intuition is not entirely correct, as shown in Acín et al (2012)—a maximum production of random bits could be achieved with a non-maximal CHSH violation. To establish a quantitative relation between the nonlocal correlation and generated randomness, let us assume that the devices follow quantum mechanics. There exist thus a quantum state ρ and measurement operators (POVMs) of each device $M^x_a$ and $M^y_b$ such that the joint probability distribution $P(a, b\vert x, y)$ could be expressed, through Born rule, as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ot}[0]{\otimes} \displaystyle P_Q(a,b\vert x,y)={\rm{Tr}}(\rho M^x_a \otimes M^y_b), \nonumber \end{align} \tag{ 20 }$$

where the tensor product reflects the fact that the measurements are local, i.e. there are no interactions between the devices, while the measurement takes place. The set of quantum correlations consists of all such probability distributions. Consider a linear combinations of them,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum_{x,y,a,b} \alpha^{xy}_{ab} P_Q(a,b\vert x,y) = \mathcal{S}, \nonumber \end{align} \tag{ 21 }$$

specified by real coefficients $\alpha^{xy}_{ab}$. For local hidden-variable model, with certain coefficients $\alpha^{xy}_{ab}$, the Bell inequalities can be then expressed as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{S} \leqslant \mathcal{S}_L. \label{eq:BI} \nonumber \end{align} \tag{ 22 }$$

This bound could be violated ($\mathcal{S} > \mathcal{S}_L$) for some quantum states and measurements indicating that the state contains nonlocal correlation.

Let us consider the the measure of randomness quantified by min-entropy. For a d-dimensional probability distribution $P(X)$, describing a random variable X, the min-entropy is defined as $H_{\infty}(X)=-{\rm{log}}_2\left[ {\rm{max}}_x p(x) \right]$. Clearly, for the perfectly deterministic distribution this maximum equals one and the min-entropy is zero. On the other hand, for a perfectly random (uniform) distribution, the entropy acquires the maximum value, ${\rm{log}}_2 d$. In the Bell scenario, the randomness in the outcomes, generated by the pair of measurements x and y, reads $H_{\infty}(A, B\vert x, y)=-{\rm{log}}_2 c_{xy}$, where $c_{xy}={\rm{max}}_{ab}P_Q(a, b\vert x, y)$. For a given observed value $\mathcal{S} > \mathcal{S}_L$, violating Bell inequality, one could find a quantum realization, i.e. the quantum states and set of measurements, that minimizes the min-entropy of the outcomes $H_{\infty}(A, B\vert x, y)$ (Navascués et al 2008). Thus, for any violation of Bell inequalities, the randomness of a pair of outcomes satisfies

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{\infty}(A,B\vert x,y) \geqslant f(\mathcal{S}), \label{eq:RandBound1} \nonumber \end{align} \tag{ 23 }$$

where f is a convex function and vanishes for the case of no Bell-inequality violation, $\mathcal{S} \leqslant \mathcal{S}_L$. Hence, the (23) quantitatively states that a violation of Bell inequalities guarantees some amount randomness. Intuitively, if the joint probabilities admit (22), then for each setting $x, \ y$ and a hidden cause λ, the outcomes a and b can be deterministically assigned. However, the violation of (22) rules out such possibility. As a consequence, the observed correlation cannot be understood with a deterministic model and the outcomes are fundamentally undetermined at the local level.

Although there are many different approaches to generate random numbers (Marsaglia 2008, Bassham et al 2010), the certification of randomness is highly non-trivial. However, this problem could be solved, in one stroke, if the random sequence shows a Bell violation, as it certifies a new form of 'true' randomness that has no deterministic classical analogue.

Nonlocal correlations can be also used for the construction of novel types of randomness expansion protocols. In these protocols, a user expands an initial random string, known as seed, into a larger string of random bits. Here, we focus on protocols that achieve this expansion by using randomness certified by a Bell inequality violation. Since the first proposals in Colbeck (2007) and Pironio et al (2010), there have been several different works studying Bell-based randomness expansion protocols, see for instance (Colbeck 2007, Pironio et al 2010, Colbeck and Kent 2011, Vazirani and Vidick 2012, Coudron and Yuen 2013, Miller and Shi 2016, Chung et al 2014, Miller and Shi 2014, Arnon-Friedman et al 2016). It is not the scope of this section to review the contributions of all these works, which in any case should be interpreted as a representative but non-exhaustive list. However, most of them consider protocols that have the structure described in what follows. Note that the description aims at providing the main intuitive steps in a general randomness expansion protocols and technicalities are deliberately omitted (for details see the references above).

The general scenario consists of a user who is able to run a Bell test. He thus has access to $n\geqslant 2$ devices where he can implement m local measurements of d outputs. For simplicity, we restrict the description in what follows to protocols involving only two devices, which are also more practical form an implementation point of view. The initial seed is used to randomly choose the local measurement settings in the Bell experiment. The choice of settings does not need to be uniformly random. In fact, in many situations, there is a combination of settings in the Bell test that produces more randomness than the rest. It is then convenient to bias the choice of measurement towards these settings so that (i) the amount of random bits consumed from the seed, denoted by N_s, is minimize and (ii) the amount of randomness produced during the Bell test is maximized.

The choice of settings is then used to perform the Bell test. After N repetitions of the Bell test, the user acquires enough statistics to have a proper estimation of the non-locality of the generated data. If not enough confidence about a Bell violation is obtained in this process, the protocol is aborted or more data are generated. From the observed Bell violation, it is possible to bound the amount of randomness in the generated bits. This is often done by means of the so-called min-entropy, $H_{\infty}$. In general, for a random variable X, the min-entropy is expressed in bits and is equal to $H_{\infty}=-\log_2\max_x P(X=x)$. The observed Bell violation is used to establish a lower bound on the min-entropy of the generated measurement outputs. Usually, after this process, the user concludes that with high confidence the $N_G\leqslant N$ generated bits have an entropy at least equal to $R\leqslant N_g$, that is $H_{\infty}\geqslant R$.

This type of bounds is useful to run the last step in the protocol: the final randomness distillation using a randomness extractor (Nisan and Ta-Shma 1999, Trevisan 2001). This consists of classical post-processing of the measurement outcomes, in general assisted by some extra N_e random bits from the seed, which map the N_g bits with entropy at least R to R bits with the same entropy, that is, R random bits. Putting all the things together, the final expansion of the protocols is given by the ration $R/(N_s+N_e)$.

Every protocol comes with a security proof, which guarantees that the final list of R bits is unpredictable to any possible observer, or eavesdropper, who could share correlated quantum information with the devices used in the Bell test. Security proofs are also considered in the case of eavesdroppers who can go beyond the quantum formalism, yet without violating the no-signaling principle. All the works mentioned above represent important advances in the design of randomness expansion protocols. At the moment, it is for instance known that (i) any Bell violation is enough to run a randomness expansion protocol (Miller and Shi 2016) and (ii) in the previous simplest configuration presented here, there exist protocol attaining an exponential randomness expansion (Vazirani and Vidick 2012). More complicated variants, where devices are nested, even attain an unbounded expansion (Coudron and Yuen 2013, Chung et al 2014). Before concluding, it is worth mentioning that another interesting scenario consists of the case in which a trusted source of public randomness is available. Even if public, this trusted randomness can safely be used to run the Bell test and does not need to be taken into account in the final rate.

Here we discuss the usefulness of nonlocal correlation for randomness amplification, a task related but in a way complementary to randomness expansion. While in randomness expansion one assumes the existence of an initial list of perfect random bits and the goal is to generate a longer list, in randomness amplification the user has access to a list of imperfect randomness and the goal is to generate a source of higher, ideally arbitrarily good, quality. As above, the goal is to solve this information task by exploiting Bell violating correlations.

Randomness amplification based on non-locality was introduced in Colbeck and Renner (2012). There, the initial source of imperfect randomness consisted of a SV source. Recall that the amplification of SV sources is impossible by classical means. A protocol was constructed based on the two-party chained Bell inequalities that was able to map an SV source with parameter $\newcommand{\e}{{\rm e}} \epsilon < 0.058$ into a new source with arbitrarily close to zero. This result is only valid in an asymptotic regime in which the user implements the chained Bell inequality with an infinite number of measurements. Soon after, a more complicated protocol attained full randomness amplification (Gallego et al 2013), that is, it was able to map SV sources of arbitrarily weak randomness, $\newcommand{\e}{{\rm e}} \epsilon<1/2$, to arbitrarily good sources of randomness, $\newcommand{\e}{{\rm e}} \epsilon\rightarrow 0$. The final result was again asymptotic, in the sense that to attain full randomness amplification the user now requires an infinite number of devices. Randomness amplification protocols have been studied by several other works, see for instance (Coudron and Yuen 2013, Bouda et al 2014, Chung et al 2014, Grudka et al 2014, Mironowicz et al 2015, Wojewódka et al 2016, Ramanathan et al 2016, Brandão et al 2016). As above, the scope of this section is not to provide a complete description of all the works studying the problem of randomness amplification, but rather to provide a general framework that encompasses most of them. In fact, randomness amplification protocols (see e.g. figure 4) have a structure similar to randomness expansion ones.

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The starting point of a protocol consists of a source of imperfect randomness. This is often modelled by an SV source, although some works consider a weaker source of randomness, known as min-entropy source, in which the user only knows a lower bound on the min-entropy of the symbols generated by the source (Bouda et al 2014, Chung et al 2014). The bits of imperfect randomness generated by the user are used to perform N repetitions of the Bell test. If the observed Bell violation is large enough, with enough statistical confidence, bits defining the final source are constructed from the measurement outputs, possibly assisted by new random bits from the imperfect source. Note that contrarily to the previous case, the extraction process cannot be assisted with a seed of perfect random numbers, as this seed could be trivially be used to produce the final source. As in the case of expansion protocols, any protocol should be accompanied by a security proof showing that the final bits are unpredictable to any observer sharing a system correlated with the devices in the user's hands.