Device-independent certification of desirable properties with a confidence interval

Abstract

In the development of quantum technologies, a reliable means for characterizing quantum devices, be it a measurement device, a state-preparation device, or a transformation device, is crucial. However, the conventional approach based on, for example, quantum state tomography or process tomography relies on assumptions that are often not necessarily justifiable in a realistic experimental setting. Although the device-independent (DI) approach to this problem bypasses the shortcomings above by making only minimal, justifiable assumptions, most of the theoretical proposals to date only work in the idealized setting where independent and identically distributed (i.i.d.) trials are assumed. Here, we provide a versatile solution for rigorous device-independent certification that does not rely on the i.i.d. assumption. Specifically, we describe how the prediction-based ratio (PBR) protocol and martingale-based protocol developed for hypothesis testing can be applied in the present context to achieve a device-independent certification of desirable properties with confidence interval (CI). To illustrate the versatility of these methods, we demonstrate how we can use them to certify—with finite data—the underlying negativity, Hilbert space dimension, entanglement depth, and fidelity to some target pure state. In particular, we provide examples showing how the amount of certifiable negativity and fidelity scales with the number of trials and how many experimental trials one needs to certify a qutrit state space or the presence of genuine tripartite entanglement. Overall, we have found that the PBR protocol and the martingale-based protocol often offer similar performance, even though the latter does have to presuppose any witness (Bell-like inequality). In contrast, our findings also show that the performance of the martingale-based protocol may be severely affected by one’s choice of Bell-like inequality. Intriguingly, a Bell function useful for self-testing does not necessarily give the optimal confidence-gain rate for certifying the fidelity to the corresponding target state.

1 Introduction

The proper analysis of quantum experiments is an indispensable part in the development of quantum technologies. However, it is not trivial to reliably characterize a quantum setup, which may include, e.g., measurement and state-preparation devices. Moreover, imperfections in the experimental setup can easily result in a mismatch [1–3] between the characterization tools developed for an idealized situation and an actual experimental situation. However, we can circumvent this problem by the so-called “device-independent approach” [4, 5]. In quantum information, the term “device-independent” (DI) was first coined [6] in the task of quantum key distribution [7–9], even though the idea was already perceived independently but implicitly in [10, 11].

In a nutshell, the DI approach is a framework for analyzing physical systems without relying on any assumption about the degrees of freedom measured. Its basis is Bell nonlocality [5, 12], which shows that no local-hidden-variable (LHV) theory can reproduce all quantum predictions, even though no further assumption is made about the details of such a theory. For example, it is known that the violation of Bell inequalities [12] obtained by locally measuring a shared state implies [13] shared entanglement [14], which is a powerful resource in many quantum information processing tasks. More generally, many other desirable properties of the underlying state [15–23], measurements [22–28], and channel [21, 29, 30] may be derived directly from the observation of a Bell inequality-violating correlation between measurement outcomes. Recently, the DI approach has also been incorporated into the security analysis of quantum secure direct communication; see, for example, [31] and references therein.

However, due to statistical fluctuations, even when the experimental trials are independent and identically distributed (i.i.d.), relative frequencies of the measurement outcomes obtained from a Bell experiment do not faithfully represent the underlying distribution. In particular, such raw distributions estimated from the experimental results typically [32–34] lead to a violation of the nonsignaling conditions [35, 36], which is a prerequisite for the analysis shown in [16–30]. In other words, statistical fluctuations render many theoretical tools developed for such a purpose inapplicable. To address this issue, some ad hoc methods [32–34] have been proposed to regularize the relative frequencies obtained to ensure that the resulting distribution satisfy the nonsignaling conditions. In [37], a more in-depth discussion was provided, and two better-motivated regularization methods were proposed.

Although these more recent attempts do provide a point estimator that fits within the framework of the usual DI analysis, they are still problematic in two aspects. First, they do not provide any confidence region associated with the estimate. However, any real experiment necessarily involves only a finite number of experimental trials. Therefore a useful analysis should provide not only an estimate but also an indication of the reliability of such an estimate. In many of the Bell experiments reported [38–41], this is achieved by reporting the standard deviations of Bell violations. However, for finite, especially relatively small numbers of trials, the central limit theorem is not warranted, so the usual interpretation of standard deviations may become dubious. Second, these usual approaches and those that provide a DI point estimator [32–34, 37] implicitly assume that the experimental trials are i.i.d. and hence free of the memory effect [42, 43] (see more discussions in [5, 44–46]). Again, in a realistic experimental setting, the i.i.d. assumption may be difficult to justify.

For the tasks of DI randomness expansion [47, 48] and DI quantum key distribution [49, 50], specific tools [51–59] have been developed to overcome the abovementioned problems. Here, we are interested in providing a general solution to other device-independent certification tasks¹ that 1) can provide a confidence region and two) does not a priori require the i.i.d. assumption. Our approach is inspired by the prediction-based ratio (PBR) protocol developed in [60] and the martingale-based method proposed by Gill [43, 61] for performing a hypothesis testing against the assumption of Bell locality. Following [62], we further adapt these earlier methods and illustrate how they can be used for the device-independent certification of various properties of interest, including the underlying amount of entanglement and its fidelity with respect to some target quantum state.

To this end, we structure the rest of this paper as follows. In Section 2.1, we explain the basic concepts relevant to the understanding of DI certification in the ideal setting. After that, we introduce in Section 2.2 our adapted statistical tools for performing a rigorous device-independent certification. Results obtained from these tools are then presented in Section 3.1. Finally, we provide some concluding remarks and future directions in Section 4.

2 Materials and methods

2.1 Preliminaries

2.1.1 Correlations and Bell inequalities

The starting point of the DI approach is a Bell test. To this end, a bipartite Bell scenario was considered, where two observers, Alice and Bob, can choose, respectively, their measurements labeled by and register outcomes .² In the i.i.d. setting, one can estimate the underlying correlation between measurement outcomes, i.e., , from the registered empirical frequencies. Interestingly, as Bell first showed in [12], highly nontrivial conclusions can be drawn by inspecting alone.

For example, correlations that can be produced in an LHV theory have to satisfy a Bell inequality:where the Bell coefficients, , and is the so-called local (upper) bound. Here, we use to signify that the inequality holds under the assumption that is compatible with the LHV theory. Explicitly, the nature of such a theory demands that is factorizable in the form [5, 12]:where for all , , and are local response functions.

In an actual Bell test, the measurement settings ought to be chosen randomly according to some predetermined distributions . To manifest this fact, one may write Equation 1 using the unconditional joint distribution such that . In turn, we can then write a Bell inequality as a bound on the expectation value of a Bell function, defined in terms of and , i.e.,where is the quadruple of random variables for the measurement outcomes and settings . As an example, the famous Clauser–Horne–Shimony–Holt (CHSH) Bell inequality [63] may be specified viaor equivalently, in terms of the correlator , aswhere .

In contrast, quantum theory allows correlations that cannot be cast in the form of Equation 2. In fact, in a bipartite Bell test, general quantum correlations read aswhere and are, respectively, the local positive-operator-valued measure (POVM) describing Alice and Bob’s local measurements. For the benefits of subsequent discussions, it is also worth noting that both LHV and quantum correlations satisfy the nonsignaling conditions [35, 36]:

For the CHSH Bell function, cf. Equation 4, quantum theory dictates the upper bound aswhich can be seen as a Bell-like inequality. Other Bell and Bell-like inequalities relevant to this work will be presented in the corresponding sections below.

2.1.2 Examples of properties to be certified

2.1.2.1 Negativity and dimension

As mentioned above, with local measurements on a quantum system, a Bell inequality-violating correlation necessarily originates [13] from an entangled state . Interestingly, the entanglement of the underlying can also be lower bounded [17, 18, 20, 23] directly from the observed correlation . In this work, we focus on negativity [64], but it is worth noting that DI entanglement quantification can also be achieved, e.g., for the linear entropy of entanglement [18], generalized robustness of entanglement [23], and one-shot distillable entanglement [20].

For a bipartite density operator , let be its partial transposition [65] with respect to subsystem . Then, the negativity for a bipartite density operator is defined as [64] , i.e., the sum of the absolute value of all negative eigenvalues of . Using a variational characterization of negativity provided in [64], it was shown in [17] that is lower bounded by the optimum value of the following semidefinite program (SDP):where is the moment matrix that can be obtained by applying a particular local map on (see [17] for details), is the output Hilbert space of the local map on , and represents the trace of the underlying operator . It is worth noting that for every integer , the constraints of Equation 9c provide a superset characterization of the quantum set of correlations, analogous to those considered in [66–68]. Indeed, all entries from appear somewhere in the moment matrix ; see [17].

As an explicit example, note that an observed violation of the CHSH Bell inequality of Equation 5 gives the following nontrivial negativity lower bound of the underlying state :

In addition, it is worth noting that if acts on with , then the maximal possible negativity is upper bounded by . Consequently, the observation of a large enough negativity also provides a nontrivial lower bound on the local Hilbert space dimension of the underlying system. More precisely, if the lower bound on obtained from Equations 9a, 9b and 9c exceeds , one immediately deduces that must act on a local Hilbert space of dimension , thereby giving a dimension witness [15].

From Equations 5, 8 and 10, nonetheless, we see that a violation of the CHSH Bell inequality can never witness a local Hilbert space dimension . Instead, witnessing a local Hilbert space beyond qubits can be achieved by observing a reasonably strong violation of the three-outcome Collins–Gisin–Linden–Massar–Popescu (CGLMP) Bell inequality [69] (see also [70]), defined bywhere if and vanishes otherwise. Denoting the corresponding expectation value by , the results from [17, 71, 72] suggest a negativity lower bound that increases linearly with from whenever .

2.1.2.2 Entanglement depth

In a many-body system, entanglement can occur in various forms or structures [73]. In particular, an -partite quantum state that is not fully separable is not necessarily genuinely -partite entangled either. To witness the latter, one could rely on the demonstration of so-called genuine multipartite nonlocality [74]. However, as remarked in [16], it is possible to witness genuine multipartite entanglement without relying on this strong form of multipartite nonlocality. In fact, using the SDP introduced in [17], one can even systematically construct DI witnesses of this kind, starting from a given multipartite Bell function, say . Later, it was further shown in [19] (see also [75]) that the extent to which a multipartite Bell inequality is violated can be used to witness (lower-bound) the underlying entanglement depth [76, 77], i.e., the extent to which a many-body entanglement is needed to prepare the given multipartite state.

For illustration, consider the expectation value of the Mermin Bell function [78] with :where is the tripartite correlator, the restricted sum is over all combinations of such that , for the same combinations of , and the Bell coefficients is

Then, it is known [19] that the following Bell-like inequalities hold, respectively, for fully separable states, 2-producible [76] tripartite quantum states (i.e., quantum states that can be generated using only two-body entanglement), and general tripartite quantum states:

2.1.2.3 State fidelity

The strongest form of device certification one can hope for within a DI paradigm is called self-testing [79], which was first proposed in [10]. The key observation behind this feat is that the quantum strategy compatible with certain extremal quantum correlations is essentially unique. Hence, with the observation of in a Bell test, we can conclude unambiguously that some degree of freedom (DOF) of the measured system must match a specific target state . Often, one can also self-test the underlying measurements alongside the state (see, however, [80, 81] for some examples of exceptions).

For instance, it is long known [82–85] that the maximal CHSH Bell-inequality violation of can only be obtained (up to local isometry) by measuring the following observables on a shared maximally entangled state (MES):

where the respective POVM elements (with ) are

Moreover, to obtain the maximal CHSH Bell-inequality violation for a partially entangled two-qubit state,it suffices [72] to consider of Equation 15b but generalize to [86]:thereby giving

Interestingly, the resulting correlation also self-tests [86, 87] the corresponding quantum strategy of Equation 17a, b and maximally violate the family of tilted CHSH Bell inequalities for :giving . Note that in Equation 19, thanks to the nonsignaling [35, 36] property of , the expression for is, in fact, independent of whether or 1.

In practice, however, due to various imperfections, one can, at best, attain a correlation close to the ideal correlation . In other words, in a realistic experimental setting, one can only hope to lower bound the similarity of the measured state with respect to the target state via a fidelity measure. To this end, a powerful numerical technique known as the SWAP method has been introduced in [88] (see also [86]) for exactly this purpose. More precisely, for any observed quantum correlation , the method allows one to lower bound the fidelity:with the help of an SDP outer approximation of the quantum set (e.g., due to [17, 66, 67]). Here, is the “swapped” state:which is extracted from the underlying quantum state via some local extraction map , which is a function of the actual POVM elements. Consequently, is a function of the entries of the moment matrix , as discussed below Equations 9a, b and c. For the details of the method, we refer the readers to [86].

2.1.3 Some general remarks

At this point, it is worth noting that for all the three properties discussed above—negativity (and hence dimension), entanglement depth, and reference-state fidelity—their DI certification can be achieved via the characterization of some convex set in the space of correlation vectors . More precisely, for negativity, by turning the objective function of Equations 9a into the constraint [17]we obtain an SDP that characterizes the set of correlations attainable by quantum states having a negativity upper-bound by . Then, Equation 10 can be understood as a separating hyperplane relevant for witnessing a negativity larger than .

On the other hand, if we drop the constraint of Equation 9b, but imposes additional positive-partial-transposition constraints, then we obtain an SDP characterization of the set having a bounded amount of entanglement depth [19] (see constraints of Equations 44b, c and d below). In this case, the first two inequalities of Equation 14 serve as the corresponding witness for entanglement depth. Finally, by demanding together with Equation 9c, we obtain an SDP characterization of the set associated with a swapped state [86] with a -fidelity upper bounded by . In fact, SDP characterization can also be obtained for a number of other properties, including genuine negativity [17], steering robustness [22], entanglement robustness [23], and (measurement) incompatibility robustness [22, 28].

2.2 Methodologies for hypothesis testing

Having understood how DI certification can be achieved from a given correlation , we now proceed to discuss the more realistic setting involving only a finite number of experimental trials. For concreteness, the following presentation assumes an analysis based on the data collected from trials in a Bell test. Below, we explain our approaches to the problem based on hypothesis testing. Our first step is to formulate a null hypothesis based on the desired property to be certified. For example, to certify that the underlying state has a negativity larger than , we formulate the (converse) null hypothesis:

2.2.1 Martingale-based protocol

We shall start with the martingale-based protocol, pioneered by Gill in [43, 61], for testing against LHV theories, and further developed in [60, 90]. The protocol relies on the observation of the (super)martingale structure in some random variables of interest. To employ the martingale-based protocol, one has to fix a Bell function in advance. Ideally, should be chosen such that the Bell-like inequalitymay be violated by some quantum correlation (cf. Equation 6) to be prepared in an experiment.

Let be the value realized for the random variables of the measurement outcomes and settings at the -th experimental trial and , the corresponding value of Bell function for that trial. Moreover, let . Then, from the observed average value of over trials, i.e., , the following -value upper bound is known [90] to hold whenever :where, for simplicity, we have suppressed the dependency of on (and on ), whereas the minimum and maximum values of over all possible values of areIt is worth noting that the martingale-based -value upper bound of Equation 25 improves over the upper bound given in [48, 60, 61]. In Figure 1, we provide a pseudocode to explain the steps involved in applying the martingale-based protocol for DI certification.

FIGURE 1

Let be the expectation value of when we replace by some capable of violating the Bell-like inequality in Equation 24. Then, in the i.i.d. setting, where the experimental data follow the distributions given by , the corresponding asymptotic confidence-gain rate can be deduced from Equation 23 and Equation 25 as

2.2.2 The prediction-based ratio (PBR) protocol

The other hypothesis-testing protocol that we consider in this work is based on the so-called PBR protocol proposed in [60] (see also [90]). In contrast with a martingale-based protocol, the PBR protocol does not need to presuppose any Bell-like inequality for determining a -value bound. Instead, for the data collected in trials, one may start by using the first trials from to estimate the relative frequencywhere and counts among these trials the total number of times the specific combination of measurement settings and outcomes occurs.

The key idea of the PBR protocol is to use this relative frequency to obtain an optimized Bell-like inequality³ and apply that to from . To this end, we minimize the Kullback–Leibler (KL) divergence [91] from a regularized relative frequency (explained below) to the set of correlations compatible with :

An important point to note now is that if the composite null hypothesis is associated with a convex set that admits an SDP characterization (as discussed in Section 2.1.3) like the kind proposed in [17, 22, 66, 67, 68]; then, Equation 29 is a conic program (see [37]) and thus efficiently solvable using a solver like [93].

The unique [37] minimizer can then be used to define the non-negative prediction-based ratio (PBR),which gives the optimized Bell-like inequality . Next, we compute the test statistic aswhere the product is only carried out over the remaining trials. Using arguments completely analogous to those given in [60] for , it can then be shown that the following upper bound on the -value holds:

Several remarks are now in order. First, if none of the entries in vanishes, one could also use directly in the optimization problem of Equation 29. However, for a small , a vanishing entry in is almost bound to happen, we thus follow [60] and mix with the uniform distribution to obtain

Next, notice that typically cannot be cast in the form of Equation 6. Consequently, we observe empirically that obtained by solving Equation 29 with in place of gives evidently suboptimal performance (see, e.g., Supplementary Figures S1, S4 and S6). As such, we shall first regularize [37] to some outer approximation of the quantum set by solving Equation 29 with replaced by . In our work, is the level- outer approximation of the quantum set introduced in [17]. However, one may also consider other approximations [22, 66]. Since all these outer approximations admit SDP characterization, this regularization process is a conic program (see [37]). The resulting minimizer, which we call the regularized relative frequency, is then fed into Equation 29 to obtain the desired PBR.

Another important feature of the PBR protocol is that the optimized inequality characterized by can be updated as more data are incorporated into the analysis. In principle, one can update as frequently as one desires. However, this is neither necessary nor efficient. As such, we work with blocks of trials. The first block of data is used exclusively for producing the first regularized relative frequency, the first PBR , and by applying to the second block of , we obtain the first test statistic:where for all (if N_tot is divisible by N_blk). In the next iteration, we determine the PBR by solving Equation 29 using from the first two blocks and apply this updated PBR to the third block of to get, for ,where . These steps may then be repeated iteratively until all the data have been consumed in one way or another in the computation of for . For a schematic illustration of this procedure, see Figure 2. Importantly, once the test statistic for each iteration is determined, we can obtain the corresponding -value bound using Equation 32. For the readers’ convenience, we also provide in Figure 3 a pseudocode to explain the steps involved in applying the PBR protocol for DI certification.

FIGURE 2

FIGURE 3

Finally, note that for an ideal Bell test giving the correlation and a composite hypothesis associated with , the PBR protocol has the asymptotic confidence-gain ratewhich may be obtained by solving Equation 29 with replaced by . The proof is again completely analogous to that given for in [60] and is thus omitted.

3 Results

3.1 Device-independent certification with a confidence interval

We are now ready to present our simulations results involving a finite number of trials. Throughout this section, the results presented for finite trials consist of an average over 30 complete Bell tests, each involving trials, with the trials partitioned into blocks of size . Moreover, we always consider a uniform distribution for (possibly a restricted set of) measurement settings. In each Bell test, we then simulate the raw data using the function from the Lightspeed MATLAB toolbox [93]. For the certification with finite data, we set a confidence level of . We also present some related confidence-gain rates in the respective subsections.

3.1.1 Negativity and dimension certification

3.1.1.1 Negativity certification

Our first example consists of a Bell test based on the quantum strategy presented in Equations 15a, b, c, which leads to a CHSH Bell value of . Using Equation 10, we know that the resulting quantum correlation gives a tight negativity lower bound of for a Bell state. From the numerically simulated data, we then perform composite hypothesis testing for Null Hypothesis 1 with .

Specifically, for the martingale-based protocol, we use Equation 25 with the CHSH Bell expression of Equation 4. In this case, for the chosen , while it follows from Equations 4, 10 and 24 thatOn the other hand, for the PBR protocol, the optimizing distribution for the -iteration can be obtained by solving (cf. Equation 29)where seeks for the argument minimizing the expression in Equation 38a, is the regularized frequency obtained for the same iteration, and each also appears as an optimization variable in the moment matrix . Then, the PBR used in the computation of can be evaluated by replacing and in Equation 30, respectively, by and .

Figure 4 shows the average amount of certifiable negativity from these two methods as a function of the number of trials employed. From the figure, it is clear that for certifying the underlying negativity using the data arising from , the performance of the two protocols is similar. In fact, even though the martingale-based protocol appears to have a slight advantage over the PBR protocol for this certification task for small ’s, our computations of the asymptotic gain-rates and show that they, in fact, agree (for all these values of that we have considered), up to a numerical precision of . In addition, in both cases, we see that with approximately and trials, we can already certify, respectively, more than 80% and 90% of the underlying negativity with a confidence . In Supplementary Section 1.1, we provide some additional plots showing how the -value bound changes with for several values of .

FIGURE 4

These results clearly suggest that the CHSH Bell function of Equation 4 is optimal for certifying the underlying negativity of using the martingale-based protocol. Indeed, a separate computation of Equation 29 and Equation 30 using in place of show that, within a precision of , the optimized Bell-like inequality for is equivalent to Equation 10. How would things change if we perform DI negativity certification using the data generated from the partially entangled state , Equation 17a? To this end, consider the quantum strategy of Equations 17a, b, whose resulting correlation gives the maximal Bell CHSH violation for , as well as the maximal violation of the tilted CHSH Bell inequality of Equation 19. Then, instead of repeating the same analysis, we show in Figure 5 the confidence-gain rates due to both protocols for certifying several given fractions of the underlying negativity. From the plots shown, it is evident that asymptotically, the martingale-based protocol employing the CHSH Bell function is far from optimal for certifying the underlying negativity of . Indeed, the PBR protocol could identify some other Bell-like inequality that gives a much better confidence-gain rate, especially for the correlations arising from that is weakly entangled (small ). To a large extent, this can be understood by noting that the negativity lower bound of Equation 10 due to its CHSH Bell violation is generally far from tight for these states; see Supplementary Figure S3.

FIGURE 5

3.1.1.2 Dimension certification via negativity certification

As mentioned in Section 2.1.2.1, the correlation is insufficient to demonstrate any nontrivial dimension bound. Let us consider, instead, a correlation derived by locally measuring the partially entangled two-qutrit state:with the local measurementswhere , , and is the set of computational basis states. It is known [67, 71] that this strategy gives the maximal CGLMP Bell-inequality violation of . Moreover, the negativity of can be easily evaluated to yield .

Next, we use the data numerically simulated from to perform hypothesis testing for Null Hypothesis 1 but now with . For the PBR protocol, the computation proceeds in exactly the same way as described above (see the paragraph containing Equation 38). However, for the martingale-based protocol, since we do not have an explicit expression like that shown in Equation 37 for the CGLMP Bell expression, we compute an upper bound on for each given value of according towhere the CGLMP Bell coefficients are defined in Equation 11. Meanwhile, since and , we again have for .

Figure 6 shows that with approximately trials, we can already certify a negativity lower bound of 0.9. On the other hand, if we want to certify that we need at least a two-qutrit state to produce the observed data (arising from ), it suffices to certify that the underlying negativity is strictly larger than 0.50, which happens already with approximately 1,500 trials. Could other two-qutrit states provide a more favorable correlation in this regard? To gain insight into the problem, we consider the following one-parameter family of two-qutrit statesand numerically maximize their CGLMP Bell-inequality violation using the heuristic algorithm given in [94]. We denote the corresponding correlation by , compute the corresponding asymptotic confidence-gain rate for both protocols, and plot the results in Figure 7.

FIGURE 6

FIGURE 7

Interestingly, even though Figure 6 suggests that the CGLMP Bell function is very effective in providing a good -value bound against Null Hypothesis 1, Figure 7 clearly shows that, asymptotically, it is not optimal. The results shown in Figure 7 further suggest that among the family of two-qutrit states given in Equation 41, the qutrit signature of , cf. Equation 39a, could even be the most prominent, when it comes to its DI certification using these hypothesis-testing techniques.

3.1.2 Entanglement depth certification

Next, we consider the tripartite correlation that results from locally measuring the -eigenvalue observableson the Greenberger–Horne–Zeilinger (GHZ) state [95, 96]:

It is easy to verify that leads to a violation of the Mermin Bell inequality, as shown in Equations 12 and 14, giving the algebraic maximum of . For our simulations, we assume a uniform distribution over all measurement settings that satisfy . Then, we test the data against the following composite hypotheses:

3.1.3 Fidelity certification

Our last examples concern the DI certification of a lower bound on the fidelity of the swapped state with respect to the target state of Equation 17a. To this end, we use the same set of data generated for the analysis in Section 3.1.1.1 and consider the following null hypothesis:

FIGURE 8

FIGURE 9

3.1.4 Properties certification via Bell-value certification

The advantage of a fidelity certification based on the SWAP method [86, 88] is that the technique is applicable to a general Bell scenario. However, in the simplest CHSH Bell scenario, it is known that a much tighter lower bound on the Bell-state fidelity can be obtained by considering a more general extraction map. Specifically, Kaniewski showed in [97] thatwhere , are local extraction maps acting, respectively, on Alice’s and Bob’s subsystem, while is the threshold CHSH value, for which the fidelity bound becomes trivial.

To take the advantage of Equation 53, we can first perform hypothesis testing based on the following null hypothesis.

FIGURE 10

FIGURE 11

4 Discussion

Tomography and witnesses are two commonly employed toolkits for certifying the desirable properties of quantum devices [98]. In recent years, the device-independent paradigm has offered an appealing alternative to these conventional means as it involves only a minimal set of assumptions. Nonetheless, many DI certification schemes, e.g., [16–19, 21–27, 29, 30], implicitly assumes that the underlying quantum correlation (or the actual Bell-inequality violation due to ) is known. In practice, this is unrealistic for two reasons: 1) we always have access to only a finite amount of experimental data, and 2) actual experimental trials are typically not independent and identically distributed (i.i.d.).

To this end, very specialized tools have been developed for the task of randomness generation, quantum key distributions, and the self-testing [86, 99, 100] of quantum states. Among them, the possibility of using hypothesis testing (based on the PBR protocol [60]) for self-testing with finite data was first discussed in [86] (see also [99] for a different approach). Meanwhile, it is long known [43, 61, 90] that hypothesis testing in a Bell test can also be carried out using a martingale-based protocol. Here, we demonstrate the viability and versatility of such hypothesis-testing-based approaches for the general problem of DI certification.

Central to our finding is the observation that many desirable quantum properties that one wishes to certify can be characterized by (the complement of) some convex set in the space of correlation vectors . In other words, if a given lies outside , a Bell-like inequality can be provided to witness this fact. This separating hyperplane then provides the basis for our martingale-based protocol for DI certification. On the other hand, if itself admits a semidefinite programming characterization like the kind proposed in [17, 22, 66–68], then the problem of minimizing the statistical distance to can be cast as a conic program, which can readily be solved using existing solvers, such as [92]. In turn, the PBR protocol provides an optimized Bell-like inequality that facilitates the corresponding hypothesis testing.

In this paper, we explain in detail how the two aforementioned hypothesis-testing protocols can be adapted for the DI certification of desirable properties. Specifically, we illustrate how we can use them to perform the DI certification of the underlying negativity [64], local Hilbert space dimension [15], entanglement depth [76]; [77], and fidelity to some target two-qubit entangled pure state . In each of these examples, we further demonstrate how the certifiable property (with a confidence of 99%) varies with the number of experimental trials involved; see Figures 4, 6, 8. Even though we have focused on certifying desirable properties of quantum states, as explained above, the protocols can also be applied to certify desirable properties of the measurement devices, such as their measurement incompatibility [22, 23, 26, 28] or their similarity to some target measurements [86] or instruments [27], etc. Note, however, that the usefulness of our protocols relies on the possibility of certifying the desired property from a Bell inequality-violating correlation. To this end, we remind that determining the complete list of quantum properties certifiable in a device-independent manner remains, to our knowledge, an open problem.

In the i.i.d. setting, the PBR protocol is known to be asymptotically optimal (in terms of its confidence-gain rate). However, we see from Figures 4, 6, 8 that for a relatively small number of trials and with the right choice of the Bell function, the martingale-based protocol performs equally well, if not better. A similar observation was also noted in [101] where the authors therein compare the PBR method with the Chernoff–Hoeffding bound in determining the success probability of Bernoulli trials. In our case, this is not surprising as the PBR method does not presuppose a Bell-like inequality but rather sacrifices some of the data to determine one. Indeed, if we equip the PBR protocol with the optimized Bell-like inequality right from the beginning, its performance is, as expected, no worse than the martingale-based protocol; see Supplementary Figures S1, S2, S4-S7 for some explicit examples.

Meanwhile, we also see from Figures 5, 7, 9 that for several cases that we have investigated, one’s intuitive choice of the Bell function for the martingale-based method can lead to a relatively poor confidence-gain rate and hence impairs its efficiency to produce a good -value bound; see Supplementary Figures S5 and S7. For example, even though the titled CHSH Bell inequality of Equation 19 is known to self-test all entangled two-qubit pure states , this choice of the Bell function in the martingale-based method leads to a worse performance (for bounding the target-state fidelity) compared with using the CHSH Bell function, which, in turn, gives a suboptimal performance compared with that derived from the PBR protocol; see Figure 9. At this point, it is worth reiterating that both protocols do not require the assumption that the experimental trials are i.i.d., even though we have only given, for simplicity, examples with i.i.d. trials.

Several research directions naturally follow from the present work. First, there are the scalability questions: 1) how do the number of measurement bases and 2) the number of samples scale with the complexity (say, dimension) of the measured system? The former is again closely related to the general viability of the device-independent certification approach, where our understanding is far from complete. As for the latter, we remark that it is indeed one of the goals of the present work to shed light on the sample complexity of our hypothesis-testing-based approaches. In some cases, such as the certification from the GHZ correlation, we see that hundreds of trials suffice, but in some others, several tens of thousands may be required to give a satisfactory level of certification. Still, some general understanding of how the sample size scales with the properties to be certified and the confidence level will be surely welcome.

Second, for experimental trials expected to deviate significantly from being i.i.d., one should choose a much smaller block size than the size adopted in our analysis using the PBR protocol. Intuitively, we should choose so that the trials do not differ significantly within each block of data. In fact, for testing against LHV theories, some guidelines have been provided in [60] on how we should choose . A similar analysis for other DI certifications is clearly desirable. Next, even though our hypothesis-testing-based approaches enable rigorous DI certification with a confidence interval, by virtue of the techniques involved, one can only make a relatively weak certification; out of the many experimental trials, we can be sure that at least one consists of a setup that exhibits the desired property (say, with 99% confidence). This is evidently far from satisfactory. A preferable certification scheme should allow one to comment on the general or average behavior of all the measured samples, as has been achieved in [100, 102] for self-testing.

Given that self-testing with a high fidelity is technically challenging, it is still of interest to devise a general recipe for certifying the average behavior of other more specific properties (such as entanglement and steerability), which may already be sufficient for the specific information processing task at hand. However, note that the rejection of a null hypothesis on the average behavior (e.g., average negativity ) necessarily entails the rejection of the corresponding null hypothesis for all trials (e.g., in every trial). Thus, we may expect a tradeoff when switching from the current kind of hypothesis testing to that for an average behavior.

In addition, it is worth noting that if the i.i.d. assumption is somehow granted, then our protocols also certify the quality of the setup for every single runs, including those that have not been measured. In this case, once a sufficiently small -value bound is obtained, one can stop measuring the rest of the systems and use them, instead, for the information processing tasks of interest. Of course, since the i.i.d. assumption is generally not warranted, a protocol that achieves certification for some fraction of the copies while leaving the rest useful for subsequent tasks will be desirable. This has been considered for one-shot distillable entanglement in [20] and the self-testing fidelity in [102]. Again, a general treatment will be more than welcome (see, e.g., [103]).

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