Unpredictability, or randomness, of the outcomes of measurements made on an entangled state can be certified provided that the statistics violate a Bell inequality. In the standard Bell scenario where each party performs a single measurement on its share of the system, only a finite amount of randomness, of at most 4 log 2 d bits, can be certified from a pair of entangled particles of dimension d. Our work shows that this fundamental limitation can be overcome using sequences of (nonprojective) measurements on the same system. More precisely, we prove that one can certify any amount of random bits from a pair of qubits in a pure state as the resource, even if it is arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. Introduction.-Bell's theorem [1] has shown that the predictions of quantum mechanics demonstrate non-locality. That is, they cannot be described by a theory in which there are objective properties of a system prior to measurement that satisfy the no-signalling principle (sometimes referred to as "local realism"). Thus, if one requires the no-signalling principle to be satisfied at the operational level then the outcomes of measurements demonstrating non-locality must be unpredictable [1] [2] [3] . This unpredictability, or randomness, is not the result of ignorance about the system preparation but is intrinsic to the theory. Although the connection between quantum non-locality (via Bell's theorem) and the existence of intrinsic randomness is well known [1] [2] [3] [4] it was analyzed in a quantitative way only recently [5, 6] . It was shown how to use nonlocality (probability distributions that violate a Bell inequality) to certify the unpredictability of the outcomes of certain physical processes. This was termed device-independent randomness certification, because the certification only relies on the statistical properties of the outcomes and not on how they were produced. The development of information protocols exploiting this certified form of randomness, such as deviceindependent randomness expansion [5] [6] [7] and amplification protocols [8, 9] , followed. Entanglement is a necessary resource for quantum nonlocality, which in turn is required for randomness certification. It is thus crucial to understand qualitatively and quantitatively how these three fundamental quantities relate to one another. In our work, we focus on asking how much certifiable randomness can be obtained from a single entangled state as a resource. Progress has been made in this direction for entangled states shared between two parties, Alice (A) and Bob (B), in the standard scenario where each party makes a single measurement on his share of the system and then discards it. An argument adapted from Ref. [10] shows that either of the two parties, A or B can certify at most 2log 2 d bits of randomness [11] , where d is the dimension of the local Hilbert space the state lives in, which in turn implies a bound of 4log 2 d bits * Florian.Curchod@icfo.es † Markus.Johansson@icfo.es when the two outputs are combined. This demonstrates a fundamental limitation for device-independent randomness certification in the standard scenario. The main goal of our work is to show that this limitation on the amount of certifiable random bits from one quantum state can be lifted. To do this we will consider the sequential scenario, where sequences of measurements can be applied to each local system. Our main result is to prove that an unbounded amount of random bits can be certified in this scenario. To gain intuition, consider the following set-up where, contrary to the device-independent approach followed here, the functioning of a device can be entirely trusted. The device consists of a quantum state prepared in the Pauli-Z, or σ z eigenstate |0 followed by a measurement in the Pauli-X, or σ x basis {|± = (|0 ± |1 )/ √ 2}. The outcome of this measurement is random and if the device then makes another measurement on the output state, this time in the Pauli-Z basis, it gives yet another random outcome. In this fashion of alternating between the two orthogonal bases, one can potentially obtain an unbounded number of random bits from one qubit. The limitation of this procedure for producing random numbers is that one cannot distinguish this device from a classical one with pre-programmed outcomes-a local model for the outcomes-if one does not fully trust the functioning of the device. Clearly we cannot certify any randomness from a single system (in a device-independent manner) as in the above example, since one needs non-locality for this purpose. But is it possible to build a scheme, that exploits non-locality and makes use of this idea of measuring the state repeatedly, to overcome the bound on the amount of certifiable randomness that one can obtain from a single entangled quantum system? To do so, the main obstacle comes from the fact that the local measurements needed to generate the random outcomes destroy the entanglement present in the state (and nonlocality in the correlations). Thus, one of the challenges is to come up with non-destructive measurements that still produce non-locality but retain some entanglement in the postmeasurement state. In this way, the state can still be used as a resource for subsequent measurements. Bell tests with sequences of measurements have received less attention than the standard ones with a single measurearXiv:1510.03394v2 [quant-ph] 23 Feb 2017 2 ment round in the literature despite the novel features in this scenario [12] , as for example the phenomenon known as hidden nonlocality [13] . In our work we show that they prove useful in the task of randomness certification, which also provides another example [11] where general measurements can overcome limitations of projective ones. More precisely, we describe a scheme where any number m of random bits are certified using a sequence of n > m consecutive measurements on the same system. This work thus shows that the bound of 4log 2 d random bits in the standard scenario can be overcome in the sequential scenario, where it is impossible to establish any bound. The unbounded randomness is certified by a near-maximal violation of a particular Bell inequality for each measurement in the sequence. Moreover, for any finite amount of certified randomness, our protocol has a finite (yet very small) noise robustness. The sequential measurements scenario.-Before presenting our results, let us introduce the scenario we work in. We carry over many of the features from the standard scenario except now we allow party B to make multiple measurements in a sequence on his share of the state. One can visualize this as in Fig. 1 where B is split up into several Bs, each one corresponding to a measurement made on the state and labeled by B i , i ∈ {1, 2, .., n}, where n is the total number of measurements made in the sequence. Each B i makes one measurement and the post-measurement state is sent to B i+1 . We organize the Bobs such that B i is doing his measurement before B j for i < j. Thus in principle B j can receive the information about the inputs and outputs of previous measurements B i for all i < j. To quantify the randomness produced in the setup, we put the above scenario in the setting of non-local guessing games (e.g. Refs. [11, [14] [15] [16] ). Let us consider an additional adversary Eve (E) who is in possession of a quantum system potentially correlated to the one of A and B. The global state is denoted ρ ABE . We assume that at each round of the experiment, E is the one preparing the state ρ ABE and distributes ρ AB = Tr E ρ ABE to A and B. This state will be used to make the measurements in the sequence and the aim of E is to try to guess B's outcomes by using measurements on her share of the state ρ ABE . The parties A and B i s, having no knowledge about the state or the real measurements made on it, see their respective devices as black boxes that receive some classical input x ∈ {0, 1} and y i ∈ {0, 1}, y 1 , y 2 , .., y n ≡ y, respectively, and that generate a classical output a ∈ {±1} and b i ∈ {±1}, (b 1 , b 2 , .., b n ) ≡ b, respectively (see Fig. 1 ). They generate statistics from multiple runs of the experiment to obtain the observed probability distribution P obs with elements p obs (a, b|x, y). This distribution P obs lives inside the set of quantum correlations Q obtained from measurements on quantum states in a sequence as we described. This set is convex and thus can be described in terms of its extreme points, denoted P ext , and any P obs can be written as P obs = From studying the outcome statistics only we can bound E's predictive power by allowing her to have complete knowledge of how P obs is decomposed into extreme points, i.e., she knows the probability distribution q ext over extreme points P ext . This predictive power is quantified via the deviceindependent guessing probability (DIGP) [14] where we fix the particular input string y to guess the outputs b. The DIGP, denoted by G( y 0 , P obs ), is then calculated as the optimal solution to the following optimization problem [15, 16] : subject to: The operational meaning of this quantity is clear: Eve has a complete description of the observed correlations in terms of extreme points. She then guesses the most probable outcome for each extreme point. The standard scenario with a single measurement round can also be represented in this formalism by simply considering that b = b and y (0) = y (0) . To quantify the amount of bits of randomness that is certified, we use the min entropy H( y 0 , P obs ) = − log 2 G( y 0 , P obs ) which returns m bits of randomness if G( y 0 , P obs ) = 2 −m . The amount of bits of randomness quantified in this way is the figure of merit in this work and our goal is to obtain as many bits as possible from a single system. In what follows, problem (2) is relaxed to an optimization where instead of insisting on P obs = ext q ext P ext (2), we only impose that the observed statistics P obs give a particular Bell inequality violation [5] . The optimal solution to this new problem is an upper bound to the optimal solution of (2). Crucially, this relaxation still gives good bounds as shown below. Before presenting our results, it is worth explaining why the causal constraints imposed by the sequential scenario make it stronger than standard Bell tests. At first sight, one could be tempted to group all the measurements in the sequence into a single box receiving an input string y n to output another string 3 b n , as in a standard Bell test. However, in general a sequence of measurements can not be represented as a single measurement. To understand this, note that in the sequential scenario the outcome b i can depend only on variables produced in its past, namely the input choices y 1 , y 2 , ..., y i and the outcomes b 1 , b 2 , ..., b i−1 that were previously obtained. However, in the single measurement scenario, the measurement box receives all inputs and produces all outputs at once. In particular, outcome b i can now be a function of input choices y j>i and outcomes b j>i that are produced in the future. That is, such a big box may violate the physical constraints coming from the sequential arrangement and the assumption that signaling from the future to the past is impossible. These additional causality constraints further limit Eve's predictability with respect to a standard Bell test and are responsible of the unbounded amount of certified randomness. The ingredients.-Alice and Bob share the pure two-qubit state that for all θ ∈]0, π/2[ is entangled. In Ref. [14], a family of Bell inequalities was introduced: . This family of inequalities has the following two useful properties: first, its maximal quantum violation, I max θ = 2 √ 2 1 + β 2 /4, is obtained by measuring the state (3) with measurements: where tan µ = sin(2θ). Second, when maximally violated, the inequality certifies one bit of local randomness on Bob's side for his second measurement choice y 0 = 1: G(y 0 = 1, P max obs ) = 1/2 [14]. These observations are possible because the maximal violation is uniquely achieved by the probability distribution P max obs that arises from the previously-described state and measurements (3) and (5). Therefore, for the maximal violation, P max obs = P ext in (2) and the guessing probability for input choice y 0 = 1 is equal to 1/2. However, in general we may not get correlations that maximally violate our Bell inequality but give a violation that is only close to maximal. In Appendixes A , B, and C we show how to make conclusions about the guessing probability for non-maximal violations. In particular, we show that for any Bell inequality with a unique point of maximal violation, the guessing probability is a continuous function of the value of the inequality close to the maximal violation. This implies in the particular case we are studying that: In Appendix F, we also provide a numerical upper bound on the guessing probability G(y 0 = 1, P obs ) by a concave function of the value of I θ . Bell inequalities (4) are the first main ingredient in our sequential construction below. The second one is the use of general, non-projective measurements. Indeed, if B 1 performs a projective measurement on the shared entangled state, the resulting post-measurement state, now shared between Alice and B 2 , is separable and thus useless for randomness production. Consequently, one needs to consider non-projective measurements to retain some entanglement in the system for the subsequent measurements. For this purpose, let us introduce the following two-outcome quantum measurement (written in the formalism of Kraus operators): corresponding to the two outcomes {±1}. This measurement } can be understood as a generalization of the projective measurement σ x . It varies from being projective (for ξ = 0) to being non-interacting (for ξ = π/4). One can verify that measuring an entangled state (3) for ξ ∈]0, π/4] (non-projective measurement) the postmeasurement state still retains some entanglement, irrespectively of the outcome. Therefore, by tuning the parameter ξ we are able to vary the destruction of the entanglement of the state at the gain of extracting information from it (cf. Ref. [17]): the closer to being a projective measurement, the lower the entanglement in the post-measurement state, but the bigger the violation of the initial Bell inequality. Scheme for unbounded randomness certification.-We now combine the previous observations to demonstrate our main result. First, let us recall that, as shown in [14] , one can obtain one bit of randomness from any pure entangled two qubit state, irrespective of the amount of entanglement in it. Moreover, one can verify that approximately one random bit can be certified if the measurements are close to the ones in Eq. (5) [in the sense thatσ x (ξ) is close to a measurement of σ x for B 1 in Eq. (5)] since I θ is then close to I max θ in Eq. (6). Second, the measurement in Eq. (7) is only close to projective for ξ close to zero and leaves entanglement in the postmeasurement state between Alice and Bob which is thus still useful for randomness certification. By repeated use of these two properties we can certify the production of an unbounded amount of random bits from a single pair of entangled qubits. We now formally describe this process in which Alice makes a single measurement on her share of the state, whereas Bob makes a sequence of n measurements on his. Each B i chooses between measurements of σ z andσ x (ξ i ) for inputs y i = 0 and y i = 1, respectively, with outcomes b i ∈ {±1}. The parameter ξ i is fixed before the beginning of the experiment. The initial entangled state shared between Alice and Bob, before B 1 's measurement, is |ψ  where c = cos(θ b1 (θ 1 , ξ 1 )) and s = sin(θ b1 (θ 1 , ξ 1 )) and the two unitaries, U
Comparison between the device-independent and one-sided device-independent sequential randomness generation between our work and the work in REF .
Unbounded randomness certification using sequences of measurements
Phys. Rev. A 95, 020102 (2017)
Physics