Stefano Pironio

Violations of Bell inequalities are often regarded as evidence that quantum nonlocality guarantees intrinsic randomness, effectively playing the role of a “dice” at the heart of device-independent (DI) cryptographic protocols. Yet the precise connection between nonlocality and randomness is more nuanced. We first show that there exist nontrivial Bell inequalities that are maximally violated by quantum correlations while certifying no randomness for any fixed input pair, rendering them ineffective for a large class of standard DI schemes. Moreover, we construct maximally nonlocal quantum correlations that remain deterministic for every fixed input pair, in the sense that for any chosen inputs they can be decomposed into strategies with fixed outputs. Conversely, we show when all input pairs are used for randomness generation, any amount of quantum nonlocality suffices to certify randomness, implying that no form of bound randomness exists in quantum nonlocality: every nonlocal behavior can be useful for DI randomness generation under an appropriately designed protocol. Building on this, we introduce the average guessing probability over all inputs, in contrast to the hitherto considered fixed-input guessing probability, as a faithful and monotonic quantifier of nonlocality. Using this measure, we prove that, contrary to recent findings in PRL 134, 090201, the detection efficiency threshold for certifying randomness is never lower than that required for detecting nonlocality. Finally, we analytically compute the average guessing probability by a quantum adversary in the standard CHSH test and show how this leads to improved generation rates in state-of-the-art amplification protocols. Together, our results precisely delineate the limits of determinism compatible with quantum nonlocality and establish average guessing probability as the correct operational bridge between nonlocality and DI randomness.

Quantum communication is often investigated in scenarios where only the dimension of Hilbert space is known. However, assigning a precise dimension is often an approximation of what is actually a higher-dimensional process. Here, we introduce and investigate quantum information encoded in carriers that nearly, but not entirely, correspond to standard qudits. We demonstrate the relevance of this concept for semi-device-independent quantum information by showing how small higher-dimensional components can significantly compromise the conclusions of established protocols. Then we provide a general method, based on semidefinite relaxations, for bounding the set of almost qudit correlations, and apply it to remedy the demonstrated issues. This method also offers a novel systematic approach to the well-known task of device-independent tests of classical and quantum dimensions with unentangled devices. Finally, we also consider viewing almost qubit systems as a physical resource available to the experimenter and determine the optimal quantum protocol for the well-known Random Access Code.

**merged with** Thomas Van Himbeeck and Stefano Pironio. Correlations and Randomness Generation based on an Energy Constraint In a previous paper, we introduced a semi-device-independent scheme consisting of an untrusted source sending quantum states to an untrusted measuring device, with the sole assumption that the average energy of the states emitted by the source is bounded. Given this energy constraint, we showed that certain correlations between the source and the measuring device can only occur if the outcomes of the measurement are non-deterministic, i.e., these correlations certify the presence of randomness. In the present paper, we go further and show how to quantify the randomness as a function of the correlations and prove the soundness of a QRNG protocol exploiting this relation. For this purpose, we introduce (1) a semidefinite characterization of the set of quantum correlations, (2) an algorithm to lower-bound the Shannon entropy as a function of the correlations and (3) a proof of soundness using finite trials compatible with our energy assumption.

Device-independent cryptographic protocols are by definition more secure than their device-dependent counterparts, since they do not rely on any assumptions regarding the internal workings of the apparatus used to implement them. Thus far, the device-independent approach has been successfully applied to problems such as quantum-key distribution and randomness generation, but it is not a priori clear whether it can be applied to protocols in the mistrustful cryptography class, where the parties do not trust one another. In this work we show that for bit-commitment and coin flipping a device-independent treatment is possible.

Device-independent quantum key distribution aims to provide key distribution schemes whose security is based on the laws of quantum physics but which does not require any assumptions about the internal working of the quantum devices used in the protocol. We provide a general security proof valid for a large class of device-independent quantum key distribution protocols in a model in which the raw key elements are generated by causally independent measurement processes. The validity of this independence condition may be justifiable in a variety of implementations and is necessarily satisfied in a physical realization where the raw key is generated by N separate pairs of devices.

Performing a measurement on one half of an entangled pair of systems remotely prepares the other half in an ensemble of quantum states. Now consider any set of ensembles that mix to the same density operator. The Schrödinger--HJW theorem states that one can remotely prepare any chosen ensemble from this set by a choice of measurement performed on one half of a fixed entangled state. We generalise this result to show that any set of ensembles can be remotely prepared if one is allowed to post-select on the outcome of the preparing measurement. The probability that the remote preparation is successful is then lower bounded in terms of the distance between the ensembles. In a prepare-and-measure quantum key distribution protocol the distance between the ensembles represents the amount of information that is leaked about the choice of ensemble, e.g. the choice of basis in the BB84 protocol. Using our generalised result, we can prove the security of such protocols from only the fundamental assumption of how much information is leaked about the choice of ensemble/basis, i.e. from bounded basis-dependency.

Our motivation is to exploit the partial commutation structure between the variables in non-commutative polynomial optimisation problems to boost the performance. We provide an efficient normal form for free words in partially commuting letters based on the maximal cliques of the non-commutation graph between the letters. We adapt several non-commutative computations to the partially commuting setting exploiting this additional structure. In particular, we provide an algorithm to compute Grobner bases for polynomial ideals in partially commuting variables that overcomes some difficulties appearing in the non-commutative cases: sometimes infinite Grobner basis can be avoided using the normal form based on these cliques.

Losses in the transmission channel, which increase with distance, pose a major obstacle to photonics demonstrations of quantum nonlocality and its applications to device-independent protocols such as device-independent quantum key distribution. Recently, Chaturvedi, Viola, and Pawlowski (CVP) arXiv:2211.14231 introduced a variation of standard Bell experiments, which we call routed Bell experiments, with the goal of extending the range over which quantum nonlocality can be demonstrated. In these experiments, in some of the rounds, photons from the source are routed by an actively controlled switch to a nearby test device instead of the distant one. CVP showed that there are quantum correlations in routed Bell experiments such that the outcomes of the remote device cannot be classically predetermined, even when its detection efficiency is arbitrarily low. In our work, we show that the correlations considered by CVP, though they cannot be classically predetermined, do not require the transmission of quantum systems to the remote device. This leads us to properly define the concept of 'short-range' and 'long-range' quantum correlations in routed Bell experiments. We then explore the conditions under which short-range quantum correlations can be ruled out. We find that routed Bell experiments do allow for reducing the detection efficiency threshold but the improvements are smaller than those suggested by CVP's analysis. We then investigate DIQKD protocols based on the routed setup. We show how to analyze the security of these protocols and compute lower bounds on the key rates using non-commutative polynomial optimization and the Brown-Fawzi-Fawzi method. We determine lower bounds on the asymptotic key rates of several simple two-qubit routed DIQKD protocols based on CHSH or BB84 correlations and compare their performance to standard protocols. We find that in an ideal case routed DIQKD protocols can significantly improve detection efficiency requirements, by up to 30%, compared to their non-routed counterparts. Notably, the routed BB84 protocol achieves a positive key rate with a detection efficiency as low as 50% for the distant device, the minimal threshold for any DIQKD protocol featuring two untrusted measurements. However, the advantages we find are highly sensitive to noise and losses affecting the short-range correlations involving the additional test device.