True Quantum Randomness

Abstract

Randomness is a fascinating concept. Since the early days of quantum physics, it became clear that a new form of randomness, with no classical analogue, appears in the quantum regime. Still, it has only been recently that tools to certify and quantify the presence of intrinsic quantum randomness have been introduced. These tools have also been exploited to certify the generation of randomness in an experiment involving two distant atoms. In this contribution, we review the novel approach to quantum randomness and discuss the main differences and advantages when compared to the existing approaches, both in the classical and quantum regime.

Appendices

Appendix A: Quantum Non-Locality and Randomness

In this appendix we explain how to derive the link between randomness and the violation of Bell inequalities. The scenario is the same as in Fig. 2.1: two separate devices generate classical outputs $a$ and $b$given the inputs $x$ and $y$. By testing the devices, it is possible to infer the probability distribution $P\left( {a,b\left| {x,y} \right.} \right)$ describing the input–output relation. Since the devices are assumed to be quantum, this distribution has to be such that there exists a quantum state $\rho$ and measurement operators for each device, $M_a^x$ and $M_b^y$, reproducing it through the standard Born rule,

$$ {P_Q}\left( {a,b\left| {x,y} \right.} \right) = tr\left( {\rho \;M_a^x \otimes M_b^y} \right). $$

(2.1)

The tensor product in this equation follows from the fact that no interaction between the devices is assumed when the measurements take place. All the distributions compatible with condition (2.1) define the set of quantum correlations.

In order to derive a Bell inequality, one considers linear combinations of the input–output probability distributions, specified by a vector of real coefficients $c_{{ab}}^{{xy}}$,

$$ \beta = \sum\limits_{{a,b,x,y}} {c_{{ab}}^{{xy}}\;P\left( {a,b\left| {x,y} \right.} \right)}. $$

(2.2)

For some of these coefficients, this expression (i) is bounded by ${\beta_L}$ for EPR models, which defines the Bell inequality $\beta \leq {\beta_L}$, while (ii) there exist quantum states and measurements leading to a larger value. It is then said that these states and measurements violate the Bell inequality.

The standard measure of randomness in information theory is the min-entropy: for a probability distribution $P(z)$ describing a random variable $Z$ which can take $d$ possible values, the min-entropy is equal to ${H_{{\min }}}(Z) = - {\log_2}\left[ {\mathop{{\max }}\limits_z P(z)} \right]$, measured in bits. If the model is deterministic, this maximum is equal to one and the entropy is zero, while for a perfectly random variable the min-entropy achieves its maximum value ${\log_2}d$. In our case, the randomness of the outcomes generated by the devices for the pair of inputs $x$ and $y$ reads ${H_{{xy}}}\left( {AB} \right) = - {\log_2}{r_{{xy}}}$, where ${r_{{xy}}} = \mathop{{\max }}\limits_{{ab}} P\left( {a,b\left| {x,y} \right.} \right)$.

All these concepts lay the basis for our first result: a lower bound on the min-entropy of the outcomes produced by two quantum devices violating a Bell’s inequality. For a given observed value $\bar{\beta } > {\beta_L}$ of a Bell inequality, we want to solve the following optimization problem: find the quantum realization, that is, the quantum states and measurements that minimizes the min-entropy of the outcomes. At first sight, solving this minimization problem looks extremely hard, as one should look over all possible quantum states and measurements, in any given space of any dimension, compatible with the observed Bell violation. However, we can tackle this problem using the techniques introduced in Navascues et al. (2007). There, a hierarchy of sets is derived which better and better approximates the set of quantum correlations. The important point is that each of these conditions can be mapped into a semi-definite programming instance, for which there exist efficient numerical techniques. Thus, we can relax the previous optimization problem and solve it over the sets in the hierarchy. Since all of them contain the set of quantum correlations, the obtained solution is a lower bound to the minimum of the min-entropy over quantum correlations (in many cases the lower bound is tight). Thus, for any violation of any Bell inequality, denoted by $\bar{\beta }$ as above, we can prove that the randomness of a pair of outcomes satisfies

$$ {H_{{xy}}}\left( {AB} \right) \geq f\left( {\bar{\beta }} \right), $$

(2.3)

where $f$ is a convex function which goes to zero at the point of no violation.

In order to illustrate our results, we plot in Fig. 2.3 the min-entropy for the simplest and best known example of Bell inequality, namely the CHSH inequality. The region below the curve is impossible within quantum physics. As mentioned in the main text, the same bounds can be computed just assuming the validity of the no-signalling principle, and not the entire quantum formalism. The corresponding results are also shown in Fig. 2.3. The derived bounds are worse, as non-signalling correlations are strictly larger than quantum ones.

Appendix B: Device-Independent Quantum Random Number Generator

In this appendix, we exploit (2.3) to construct a novel type of random number generators which are certifiable, private and device-independent. As pointed out by Colbeck in his PhD Thesis (Colbeck 2007), the random character of the generated numbers is guaranteed by the violation of a Bell inequality. As discussed in the main text, we actually propose a randomness expander, a device which expands an initial random seed into a much larger string of random bits.

In order to realize such a randomness expander, we suppose that we have a device, composed of subsystems A and B, that is used $n$ times in succession. The inputs ${x_i},{y_i}$ at each round $i$ are chosen always in the same way and independently of the previous rounds. The amount of randomness needed for this choice can be tuned such that, in the limit of large $n$, it scales as $\sqrt {n}$. Each use of the device produces outputs${a_i}$ and ${b_i}$. We denote by $\vec{x} = \left( {{x_1}, \ldots, {x_n}} \right)$, and similarly $\vec{y}$, $\vec{a}$ and $\vec{b}$ the strings of inputs and outputs. Using the previous bound (2.3), we can show that, with probability $1 - \delta$, where $\delta$ decreases exponentially with $n$, the min-entropy of the final string of outputs satisfies

$$ {H_{{\min }}}\left( {\vec{a},\vec{b}\left| {\vec{x},\vec{y}} \right.} \right) \geq n\;f\left( {\tilde{\beta } - \varepsilon } \right), $$

(2.4)

where $\varepsilon$ is a security parameter which can be taken very small and $\tilde{\beta }$ is an estimation of the Bell parameter $\beta$ derived from the observed symbols. Note that the output randomness scales as $n$, while the initial randomness needed for the tests scaled as $\sqrt {n}$. Beyond the technical details, which are just sketched here, the importance of this bound comes from the fact that it holds even if the devices have internal memories and can adapt their responses to what was produced in the previous rounds. This is a significant advance over the device-independent protocols proposed so far and is the crucial feature that makes our protocol practical.

Finally, note that although the output string may not be uniformly random, we are guaranteed that its entropy is bounded by (2.4). The output string can now be classically processed using a randomness extractor (Nisan and Ta-Shma 1999). An extractor is a well-known technique in information theory that with the help of a small private random seed, maps an initial string of bits whose entropy is bounded by $k$, see (2.4), into $k$ perfect random bits. The use of the extractor, then, concludes the randomness generation (or, more precisely, expansion) process.

Appendix C: Experimental Generation of Private Random Numbers

The experimental realization of our proposal requires the observation of a Bell inequality with the detection loophole closed. This means that almost every event has to be recorded, so that the outputs cannot be deterministically reproduced. This is a strong technological requirement and implies that no photon experiment is possible with current technology, as photon detection is a rather inefficient process. At the moment, the only Bell experiments which are able to close detection loophole consist of trapped atomic particles (Matsukevich et al. 2008, Rowe et al. 2001). Moreover, in our proposal the two devices should also be sufficiently separated so that they do not interact when the measurements are performed. This is needed to assure the tensor product structure in (2.1), which is crucial in the derivation of our results. The only way of guaranteeing this is by considering atoms in two distant traps. At present, the unique setup in the world which satisfies all these requirements is the one in the group of Prof. Monroe, at the University of Maryland. They are able to entangle two atoms in two distant traps and observe a Bell violation with closed detection loophole.

Together with the group of Prof. Monroe, we performed a proof-of-principle demonstration of our proposal. We realize this situation with two ¹⁷¹Yb atomic ions confined in two independent vacuum chambers separated by about 1 m (see Fig. 2.2). First, the atoms are entangled via the coincidence detection of two photons, each one emitted by each ion (Moehring et al. 2007). This process is probabilistic, but when it succeeds, leaves the two ions in a maximally entangled state. The ions are then measured and lead to the violation of the CHSH-Bell inequality. This inequality involves two different measurements per ion. The choice between these two measurements is random and set prior to measurement. Direct interaction between the atoms is negligible and classical microwave and optical fields used to perform measurements on one atom have no influence on the other atom.

To estimate the value of the CHSH inequality $n = 3016$ successive entanglement events were accumulated over the period of about one month. The observed CHSH violation was equal to 2.414 and represents a substantial improvement over previous results (Matsukevich et al. 2008). Using our theoretical formalism, we can prove that the observed CHSH violation implies that at least ${H_{{\min }}}\left( {\vec{a},\vec{b}\left| {\vec{x},\vec{y}} \right.} \right) > 42$ new random bits are generated in the experiment with a 99% confidence level. Thus, we can, for the first time, certify that new randomness is produced in an experiment without a detailed model of the devices.