We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell’s S or Eberhard’s J. Further improvement was obtained by using the Wilks likelihood ratio test based on the four tetranomially distributed vectors of counts of the four different outcome combinations, one 4-vector for each of the four setting combinations. The methodology was illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST, and in Vienna, respectively, and also to the earlier (1998) Innsbruck experiment of Weihs et al. and the recent (2022) Munich experiment of Zhang et al., which investigates the use of a loophole-free Bell experiment as part of a protocol for device-independent quantum key distribution (DIQKD).
We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell’s S or Eberhard’s J. Further improvement was obtained by using the Wilks likelihood ratio test based on the four tetranomially distributed vectors of counts of the four different outcome combinations, one 4-vector for each of the four setting combinations. The methodology was illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST, and in Vienna, respectively, and also to the earlier (1998) Innsbruck experiment of Weihs et al. and the recent (2022) Munich experiment of Zhang et al., which investigates the use of a loophole-free Bell experiment as part of a protocol for device-independent quantum key distribution (DIQKD).
We present an eliminative analysis of no-signaling equalities in the NIST loophole-free Bell test (Shalm et al., 2015).Standard analyses examine outcome correlations to compute CHSH S values. We instead examine whether one party'sdetection rate depends on the other party's setting choice. Under quantum mechanics, these marginal probabilities must beindependent of the remote setting. Using a six-gate streaming pipeline on 356 million event-level trials, we apply theexperiment's own timing calibration (phase filter: pk=90, radius=4) to select genuine entangled photon detections inPockels cell slots 5-7. In the phase-filtered data, Alice's marginal detection rate for setting a=2 shows a stationary deviation:z = 2.41, permutation p = 0.017, 95% bootstrap CI [1.44e-6, 1.31e-5] excluding zero. The combined chi-squared is 9.40 (4df), one tick below the p < 0.05 threshold of 9.49. The deviation is temporally stationary across 100 time windows spanningthe full experimental run, with near-zero autocorrelation and near-zero linear trend. Without phase filtering, all trialsproduce a null result (p = 0.40), consistent with noise dilution from background detector clicks. A second NIST run using adifferent random number generator (ClassicalRNG XOR) lacks detection-level data and cannot be phase-filtered,precluding direct replication with matching detection definitions.
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.
Device-Independent Quantum Key Distribution (DIQKD) certifies security from the violation of a Bell inequality alone, but its practical margin against noise is narrow and is usually quoted only through the Bell violation itself. We build a simulator of the event-ready DIQKD experiment and use it to trace readout noise through the Bell test into classical post-processing. For a symmetric outcome-flip channel of strength e, we reproduce the simulated violation window e<5.86%, to within 0.11 percentage points. Combining these with a multi-pass Cascade reconciliation and the Devetak-Winter bound gives r<=0.162 bits per round, and shows that a positive key rate requires e<0.89 - roughly one-sixth of the window in which a Bell violation is still observed. The gap quantifies how far certified nonlocality overstates the usable noise budget of a DIQKD link.
In device-independent quantum key distribution (DIQKD) the security is not based on any assumptions about the intrinsic properties of the devices and the quantum signals but on the violation of a Bell inequality. We introduce a DIQKD scenario in which an optimal Bell inequality is constructed from the performed measurement data rather than fixing beforehand a specific Bell inequality. Our method can be employed in a general way, for any number of measurement settings and any number of outcomes. We provide an implementable DIQKD protocol and perform finite-size security key analysis for collective attacks. We compare our approach with related procedures in the literature and analyze the robustness of our protocol. We also study the performance of our method in several Bell scenarios as well as for random measurement settings.
Last year the first experimental tests closing the detection loophole (also referred to as the fair sampling loophole) were performed by two experimental groups \cite{Zeilinger}, \cite{Kwiat}. To violate Bell-type inequalities (the Eberhard inequality in the first test and the Clauser-Horne inequality in the second test), one has to optimize a number of parameters involved in the experiment (angles of polarization beam splitters and quantum state parameters). Although these are technicalities, their optimal determination plays an important role in approaching statistically significant violations of the inequalities. In this paper we study this problem for the Eberhard inequality in very detail by using the advanced method of numerical optimization, the Nelder-Mead method. First of all, we improve the the results of optimization for the original Eberhard model \cite{Eberhard} and the Gustina et al. work \cite{Zeilinger} ("Vienna-13 experiment") by using the model of this experiment presented in Kofler et al. \cite{Zeilinger1}. We also take into account the well known fact that detectors can have different efficiencies and perform the corresponding optimization. In previous studies the objective function had the meaning of the mathematical expectation. However, it is also useful to investigate the possible level of variability of the results, expressed in terms of standard deviation. In this paper we consider the optimization of parameters for the Eberhard inequality using coefficient of variation taking into account possible random fluctuations in the setup of angles during the experiment.
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.