The device-independent approach to physics is one where conclusions are drawn directly and solely from the observed correlations between measurement outcomes. In quantum information, this approach allows one to make strong statements about the properties of the underlying devices via the observation of Bell-inequality-violating correlations. However, since one can only perform a finite number of experimental trials, an important gap remains between the many theoretical tools developed for such purposes and the experimentally obtained raw data. In particular, a sensible way to estimate the underlying quantum distribution has so far remained elusive. Here, we propose a few methods to bridge this gap. Under the assumption that the experimental trials are independent and identically distributed, our methods allow one to achieve the analog of quantum state estimation in the device-independent setting by generating unique point estimates of the true quantum distribution. In addition, we provide strong numerical evidence showing that these estimates converge toward the underlying distribution with a rate at least as good as the relative frequencies. We further demonstrate how these estimates of the true quantum distribution can be used to provide sensible estimates of certain desired properties of the system, such as the amount of entanglement present in the measured system.
The device-independent approach to physics is one where conclusions are drawn directly and solely from the observed correlations between measurement outcomes. In quantum information, this approach allows one to make strong statements about the properties of the underlying devices via the observation of Bell-inequality-violating correlations. However, since one can only perform a finite number of experimental trials, an important gap remains between the many theoretical tools developed for such purposes and the experimentally obtained raw data. In particular, a sensible way to estimate the underlying quantum distribution has so far remained elusive. Here, we propose a few methods to bridge this gap. Under the assumption that the experimental trials are independent and identically distributed, our methods allow one to achieve the analog of quantum state estimation in the device-independent setting by generating unique point estimates of the true quantum distribution. In addition, we provide strong numerical evidence showing that these estimates converge toward the underlying distribution with a rate at least as good as the relative frequencies. We further demonstrate how these estimates of the true quantum distribution can be used to provide sensible estimates of certain desired properties of the system, such as the amount of entanglement present in the measured system.
The device-independent approach to physics is one where conclusions are drawn directly from the observed correlations between measurement outcomes. In quantum information, this approach allows one to make strong statements about the properties of the underlying systems or devices solely via the observation of Bell-inequality-violating correlations. However, since one can only perform a finite number of experimental trials, statistical fluctuations necessarily accompany any estimation of these correlations. Consequently, an important gap remains between the many theoretical tools developed for the asymptotic scenario and the experimentally obtained raw data. In particular, a physical and concurrently practical way to estimate the underlying quantum distribution has so far remained elusive. Here, we show that the natural analogs of the maximum-likelihood estimation technique and the least-square-error estimation technique in the device-independent context result in point estimates of the true distribution that are physical, unique, computationally tractable, and consistent. They thus serve as sound algorithmic tools allowing one to bridge the aforementioned gap. As an application, we demonstrate how such estimates of the underlying quantum distribution can be used to provide, in certain cases, trustworthy estimates of the amount of entanglement present in the measured system. In stark contrast to existing approaches to device-independent parameter estimations, our estimation does not require the prior knowledge of any Bell inequality tailored for the specific property and the specific distribution of interest.
Device-independent randomness generation and quantum key distribution protocols rely on a fundamental relation between the non-locality of quantum theory and its random character. This relation is usually expressed in terms of a trade-off between the probability of guessing correctly the outcomes of measurements performed on quantum systems and the amount of violation of a given Bell inequality. However, a more accurate assessment of the randomness produced in Bell experiments can be obtained if the value of several Bell expressions is simultaneously taken into account, or if the full set of probabilities characterizing the behavior of the device is considered. We introduce protocols for device-independent randomness generation secure against classical side information, that rely on the estimation of an arbitrary number of Bell expressions or even directly on the experimental frequencies of measurement outcomes. Asymptotically, this results in an optimal generation of randomness from experimental data (as measured by the min-entropy), without having to assume beforehand that the devices violate a specific Bell inequality.
For practical applications of quantum randomness generation, it is important to certify and further produce a fixed block of fresh random bits with as few trials as possible.Consequently, protocols with high finite-data efficiency are preferred.To yield such protocols with respect to quantum side information, we develop quantum probability estimation.Our approach is applicable to device-independent as well as device-dependent scenarios, and it generalizes techniques from previous works [Miller and Shi, SIAM J. Comput. 46, 1304 (2017); Arnon-Friedman et al., Nat. Commun. 9, 459 (2018)].Quantum probability estimation can adapt to changing experimental conditions, allows stopping the experiment as soon as the prespecified randomness goal is achieved, and can tolerate imperfect knowledge of the input distribution.Moreover, the randomness rate achieved at constant error is asymptotically optimal.For the device-independent scenario, our approach certifies the amount of randomness available in experimental results without first searching for relations between randomness and violations of fixed Bell inequalities.We implement quantum probability estimation for device-independent randomness generation in the CHSH Bell-test configuration, and we show significant improvements in finite-data efficiency, particularly at small Bell violations which are typical in current photonic loophole-free Bell tests.
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.
Quantum entanglement is a unique property of quantum particles. It can correlate them in such a way, that further correlations to an additional unwanted party is prevented. This renders quantum theory as the prime candidate to implement a secure distribution of an encryption key. Quantum key distribution (QKD) is dedicated to this task. A striking feature of correlations in a statistical experiment is, that they can, in principle, always be approximated. Even more so, correlations can be obtained without a detailed knowledge about the underlying physical process that generated the experimental data. This no-characterization approach is at the heart of the device-independent (DI) paradigm. Here, Bell inequalities are an imperative tool to detect nonlocal correlations, which are necessary for a DI secret key. One central subject of this thesis is multipartite DIQKD and the implications of Bell nonlocality in this context. Any DIQKD protocol involves test rounds in which the violation of a Bell inequality is checked. For multipartite DIQKD, however, not all Bell inequalities are suitable. We identify the crucial properties a Bell test requires to be a viable option. In this light, a published protocol for multipartite DIQKD is examined. We establish an incompatibility which is inherent to the proposed Bell test in combination with the required quantum states. This leads us to the conclusion, that the proposed protocol necessarily aborts. In a subsequent work, we develop a family of multipartite Bell inequalities, specifically tailored to the task of DI conference key agreement (CKA). Several features of this Bell inequality are analytically characterized. In addition, we prove its usefulness for the purposes of DICKA. To this end, semidefinite programming techniques are employed and extended to the multipartite scenario which allows us to quantify asymptotic DI conference key rates. In a second part, we explore the challenges of implementing (bipartite) long-distance DIQKD with quantum repeaters. For two repeater protocols, we describe how experimental parameters manifest themselves in DI secret-key rates. In doing so, we shed light on the fundamental differences between the usual and the DI scenario and we benchmark the threshold requirements of quantum devices to make profitable DI secret-key rates feasible. Finally, we develop a general method to describe in a non-DI setting the propagation of an important class of errors through quantum circuits in arbitrary dimensions. With this, we discuss the potential of error-corrected quantum repeaters to overcome fundamental point-to-point limitations.
Device-independent quantum key distribution allows for proving the security of a shared cryptographic key between two distant parties with potentially untrusted devices. The security proof is based on the measurement outcome statistics (correlation) of a Bell experiment, and security is guaranteed by the laws of quantum theory. While it is known that the observed correlation must be Bell nonlocal in order to prove security, recent results show that Bell nonlocality is in general not sufficient for standard device-independent quantum key distribution. In this work, we show that conversely, there is no lower bound on the amount of nonlocality that is sufficient for device-independent quantum key distribution. Even more so, we show that from certain correlations that exhibit arbitrarily small nonlocality, one can still extract unbounded device-independent key rates. Therefore, a quantitative relation between device-independent key rates and Bell nonlocality cannot be drawn in general. Our main technique comprises a rigorous connection between self-testing and device-independent quantum key distribution, applied to a recently discovered family of Bell inequalities with arbitrarily many measurement outcomes.