Erratum: Almost Qudits in the Prepare-and-Measure Scenario [Phys. Rev. Lett. 129, 250504 (2022)].

This corrects the article DOI: 10.1103/PhysRevLett.129.250504.

Bounding the Amount of Entanglement from Witness Operators.

We present an approach to estimate the operational distinguishability between an entangled state and any separable state directly from measuring an entanglement witness. We show that this estimation also implies bounds on a variety of other well-known entanglement quantifiers. This approach for entanglement estimation is then extended to both the measurement-device-independent scenario and the fully device-independent scenario, where we obtain nontrivial but suboptimal bounds. The procedure requires no numerical optimization and is easy to compute. It offers ways for experimenters to not only detect, but also quantify, entanglement from the standard entanglement witness procedure.

Quantum Steering with Imprecise Measurements.

We study quantum steering experiments without assuming that the trusted party can perfectly control their measurement device. Instead, we introduce a scenario in which these measurements are subject to small imprecision. We show that small measurement imprecision can have a large detrimental influence in terms of false positives for steering inequalities, and that this effect can become even more relevant for high-dimensional systems. We then introduce a method for taking generic measurement imprecision into account in tests of bipartite steering inequalities. The revised steering bounds returned by this method are analytical, easily computable, and are even optimal for well-known families of arbitrary-dimensional steering tests. Furthermore, it applies equally well to generalized quantum steering scenarios, where the shared quantum state does not need to be separable, but is instead limited by some other entanglement property.

Resource-Efficient High-Dimensional Entanglement Detection via Symmetric Projections.

We introduce two families of criteria for detecting and quantifying the entanglement of a bipartite quantum state of arbitrary local dimension. The first is based on measurements in mutually unbiased bases and the second is based on equiangular measurements. Both criteria give a qualitative result in terms of the state's entanglement dimension and a quantitative result in terms of its fidelity with the maximally entangled state. The criteria are universally applicable since no assumptions on the state are required. Moreover, the experimenter can control the trade-off between resource-efficiency and noise-tolerance by selecting the number of measurements performed. For paradigmatic noise models, we show that only a small number of measurements are necessary to achieve nearly-optimal detection in any dimension. The number of global product projections scales only linearly in the local dimension, thus paving the way for detection and quantification of very high-dimensional entanglement.

Classical Cost of Transmitting a Qubit.

We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state.

Certification of non-classicality in all links of a photonic star network without assuming quantum mechanics.

Networks composed of independent sources of entangled particles that connect distant users are a rapidly developing quantum technology and an increasingly promising test-bed for fundamental physics. Here we address the certification of their post-classical properties through demonstrations of full network nonlocality. Full network nonlocality goes beyond standard nonlocality in networks by falsifying any model in which at least one source is classical, even if all the other sources are limited only by the no-signaling principle. We report on the observation of full network nonlocality in a star-shaped network featuring three independent sources of photonic qubits and joint three-qubit entanglement-swapping measurements. Our results demonstrate that experimental observation of full network nonlocality beyond the bilocal scenario is possible with current technology.

Projective Measurements Are Sufficient for Recycling Nonlocality.

Unsharp measurements are widely seen as the key resource for recycling the nonlocality of an entangled state shared between several sequential observers. Contrasting this, we here show that nonlocality can be recycled using only standard, projective, qubit measurements. Focusing on the Clauser-Horne-Shimony-Holt inequality and allowing parties to share classical randomness, we determine the optimal trade-off in the magnitude of Bell violations for a maximally entangled state. We then find that nonmaximally entangled states make possible larger sequential violations, which contrasts the standard Clauser-Horne-Shimony-Holt scenario. Furthermore, we show that nonlocality can be recycled using projective qubit measurements even when no shared classical randomness is available. We discuss the implications of our results for experimental implementations of sequential nonlocality.

Entanglement-assisted quantum communication with simple measurements.

Dense coding is the seminal example of how entanglement can boost qubit communication, from sending one bit to sending two bits. This is made possible by projecting separate particles onto a maximally entangled basis. We investigate more general communication tasks, in both theory and experiment, and show that simpler measurements enable strong and sometimes even optimal entanglement-assisted qubit communication protocols. Using only partial Bell state analysers for two qubits, we demonstrate quantum correlations that cannot be simulated with two bits of classical communication. Then, we show that there exists an established and operationally meaningful task for which product measurements are sufficient for the strongest possible quantum predictions based on a maximally entangled two-qubit state. Our results reveal that there are scenarios in which the power of entanglement in enhancing quantum communication can be harvested in simple and scalable optical experiments.

Adaptive Advantage in Entanglement-Assisted Communications.

Entanglement is known to boost the efficiency of classical communication. In distributed computation, for instance, exploiting entanglement can reduce the number of communicated bits or increase the probability to obtain a correct answer. Entanglement-assisted classical communication protocols usually consist of two successive rounds: first, a Bell test round, in which the parties measure their local shares of the entangled state, and then a communication round, where they exchange classical messages. Here, we go beyond this standard approach and investigate adaptive uses of entanglement: we allow the receiver to wait for the arrival of the sender's message before measuring their share of the entangled state. We first show that such adaptive protocols improve the success probability in random access codes. Second, we show that once adaptive measurements are used, an entanglement-assisted bit becomes a strictly stronger resource than a qubit in prepare-and-measure scenarios. We briefly discuss the extension of these ideas to scenarios involving quantum communication and we identify resource inequalities.

Entanglement Detection with Imprecise Measurements.

We investigate entanglement detection when the local measurements only nearly correspond to those intended. This corresponds to a scenario in which measurement devices are not perfectly controlled, but nevertheless operate with bounded inaccuracy. We formalize this through an operational notion of inaccuracy that can be estimated directly in the lab. To demonstrate the relevance of this approach, we show that small magnitudes of inaccuracy can significantly compromise several well-known entanglement witnesses. For two arbitrary-dimensional systems, we show how to compute tight corrections to a family of standard entanglement witnesses due to any given level of measurement inaccuracy. We also develop semidefinite programming methods to bound correlations in these scenarios.

Entanglement Swapping and Quantum Correlations via Symmetric Joint Measurements.

We use hyperentanglement to experimentally realize deterministic entanglement swapping based on quantum elegant joint measurements. These are joint projections of two qubits onto highly symmetric, isoentangled bases. We report measurement fidelities no smaller than 97.4%. We showcase the applications of these measurements by using the entanglement swapping procedure to demonstrate quantum correlations in the form of proof-of-principle violations of both bilocal Bell inequalities and more stringent correlation criteria corresponding to full network nonlocality. Our results are a foray into entangled measurements and nonlocality beyond the paradigmatic Bell state measurement and they show the relevance of more general measurements in entanglement swapping scenarios.

Testing Real Quantum Theory in an Optical Quantum Network.

Quantum theory is commonly formulated in complex Hilbert spaces. However, the question of whether complex numbers need to be given a fundamental role in the theory has been debated since its pioneering days. Recently it has been shown that tests in the spirit of a Bell inequality can reveal quantum predictions in entanglement swapping scenarios that cannot be modeled by the natural real-number analog of standard quantum theory. Here, we tailor such tests for implementation in state-of-the-art photonic systems. We experimentally demonstrate quantum correlations in a network of three parties and two independent EPR sources that violate the constraints of real quantum theory by over 4.5 standard deviations, hence disproving real quantum theory as a universal physical theory.

Full Network Nonlocality.

Networks have advanced the study of nonlocality beyond Bell's theorem. Here, we introduce the concept of full network nonlocality, which describes correlations that necessitate all links in a network to distribute nonlocal resources. Showcasing that this notion is stronger than standard network nonlocality, we prove that the most well-known network Bell test does not witness full network nonlocality. In contrast, we demonstrate that its generalization to star networks is capable of detecting full network nonlocality in quantum theory. More generally, we point out that established methods for analyzing local and theory-independent correlations in networks can be combined in order to systematically deduce sufficient conditions for full network nonlocality in any network and input-output scenario. We demonstrate the usefulness of these methods by constructing polynomial witnesses of full network nonlocality for the bilocal scenario. Then, we show that these inequalities can be violated via quantum elegant joint measurements.

Bell nonlocality in networks.

Bell's theorem proves that quantum theory is inconsistent with local physical models. It has propelled research in the foundations of quantum theory and quantum information science. As a fundamental feature of quantum theory, it impacts predictions far beyond the traditional scenario of the Einstein-Podolsky-Rosen paradox. In the last decade, the investigation of nonlocality has moved beyond Bell's theorem to consider more sophisticated experiments that involve several independent sources which distribute shares of physical systems among many parties in a network. Network scenarios, and the nonlocal correlations that they give rise to, lead to phenomena that have no counterpart in traditional Bell experiments, thus presenting a formidable conceptual and practical challenge. This review discusses the main concepts, methods, results and future challenges in the emerging topic of Bell nonlocality in networks.

Almost Qudits in the Prepare-and-Measure Scenario.

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.

Quantum theory based on real numbers can be experimentally falsified.

Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces1,2. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural3. In fact, previous studies have shown that such a 'real quantum theory' can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states4. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

Nonlocality, Steering, and Quantum State Tomography in a Single Experiment.

We investigate whether paradigmatic measurements for quantum state tomography, namely mutually unbiased bases and symmetric informationally complete measurements, can be employed to certify quantum correlations. For this purpose, we identify a simple and noise-robust correlation witness for entanglement detection, steering, and nonlocality that can be evaluated based on the outcome statistics obtained in the tomography experiment. This allows us to perform state tomography on entangled qutrits, a test of Einstein-Podolsky-Rosen steering and a Bell inequality test, all within a single experiment. We also investigate the trade-off between quantum correlations and subsets of tomographically complete measurements as well as the quantification of entanglement in the different scenarios. Finally, we perform a photonics experiment in which we demonstrate quantum correlations under these flexible assumptions, namely with both parties trusted, one party untrusted and both parties untrusted.

Bilocal Bell Inequalities Violated by the Quantum Elegant Joint Measurement.

Network Bell experiments give rise to a form of quantum nonlocality that conceptually goes beyond Bell's theorem. We investigate here the simplest network, known as the bilocality scenario. We depart from the typical use of the Bell state measurement in the network central node and instead introduce a family of symmetric isoentangled measurement bases that generalize the so-called "elegant joint measurement." This leads us to report noise-tolerant quantum correlations that elude bilocal variable models. Inspired by these quantum correlations, we introduce network Bell inequalities for the bilocality scenario and show that they admit noise-tolerant quantum violations. In contrast to many previous studies of network Bell inequalities, neither our inequalities nor their quantum violations are based on standard Bell inequalities and standard quantum nonlocality. Moreover, we pave the way for an experimental realization by presenting a simple two-qubit quantum circuit for the implementation of the elegant joint measurement and our generalization.

Semi-Device-Independent Framework Based on Restricted Distrust in Prepare-and-Measure Experiments.

A semi-device-independent framework for prepare-and-measure experiments is introduced in which an experimenter can tune the degree of distrust in the performance of the quantum devices. In this framework, a receiver operates an uncharacterized measurement device and a sender operates a preparation device that emits states with a bounded fidelity with respect to a set of target states. No assumption on Hilbert space dimension is required. The set of quantum correlations is investigated and bounded from both the interior and the exterior. Furthermore, the optimal performance of quantum state discrimination with bounded distrust is derived and applied to certification of detection efficiency. Quantum-over-classical advantages are demonstrated and the magnitude of distrust compatible with such advantages is explored. Finally, efficient schemes for semi-device-independent random number generation are developed.

Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments.

Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are crucial to many conceptual and practical aspects of quantum theory. Here, we develop their role in quantum nonlocality by (i) introducing families of Bell inequalities that are maximally violated by d-dimensional MUBs and SICs, respectively, (ii) proving device-independent certification of natural operational notions of MUBs and SICs, and (iii) using MUBs and SICs to develop optimal-rate and nearly optimal-rate protocols for device-independent quantum key distribution and device-independent quantum random number generation, respectively. Moreover, we also present the first example of an extremal point of the quantum set of correlations that admits physically inequivalent quantum realizations. Our results elaborately demonstrate the foundational and practical relevance of the two most important discrete Hilbert space structures to the field of quantum nonlocality.