In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the 'uncertainty regions' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius s s 1 . + ( ) Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r s , ( ) where r s s 1  ( ) as s .  ¥

Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of preassigned accuracy can be obtained straightforwardly. Our method also works for postive-operator-valued measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional spin squeezing criteria fail because of detector noise.

Device-independent quantum key distribution (DIQKD) is the art of using untrusted devices to distribute secret keys in an insecure network. It thus represents the ultimate form of cryptography, offering not only information-theoretic security against channel attacks, but also against attacks exploiting implementation loopholes. In recent years, much progress has been made towards realising the first DIQKD experiments, but current proposals are just out of reach of today’s loophole-free Bell experiments. Here, we significantly narrow the gap between the theory and practice of DIQKD with a simple variant of the original protocol based on the celebrated Clauser-Horne-Shimony-Holt (CHSH) Bell inequality. By using two randomly chosen key generating bases instead of one, we show that our protocol significantly improves over the original DIQKD protocol, enabling positive keys in the high noise regime for the first time. We also compute the finite-key security of the protocol for general attacks, showing that approximately 108–1010 measurement rounds are needed to achieve positive rates using state-of-the-art experimental parameters. Our proposed DIQKD protocol thus represents a highly promising path towards the first realisation of DIQKD in practice.

Device-independent quantum key distribution (DIQKD) enables the generation of secret keys over an untrusted channel using uncharacterized and potentially untrusted devices1–9. The proper and secure functioning of the devices can be certified by a statistical test using a Bell inequality10–12. This test originates from the foundations of quantum physics and also ensures robustness against implementation loopholes13, thereby leaving only the integrity of the users’ locations to be guaranteed by other means. The realization of DIQKD, however, is extremely challenging—mainly because it is difficult to establish high-quality entangled states between two remote locations with high detection efficiency. Here we present an experimental system that enables for DIQKD between two distant users. The experiment is based on the generation and analysis of event-ready entanglement between two independently trapped single rubidium atoms located in buildings 400 metre apart14. By achieving an entanglement fidelity of $$ {\mathcal F} \,\ge 0.892(23)$$ ℱ ≥ 0.892 ( 23 ) and implementing a DIQKD protocol with random key basis15, we observe a significant violation of a Bell inequality of S = 2.578(75)—above the classical limit of 2—and a quantum bit error rate of only 0.078(9). For the protocol, this results in a secret key rate of 0.07 bits per entanglement generation event in the asymptotic limit, and thus demonstrates the system’s capability to generate secret keys. Our results of secure key exchange with potentially untrusted devices pave the way to the ultimate form of quantum secure communications in future quantum networks.