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.

Proofs of Network Quantum Nonlocality in Continuous Families of Distributions.

The study of nonlocality in scenarios that depart from the bipartite Einstein-Podolsky-Rosen setup is allowing one to uncover many fundamental features of quantum mechanics. Recently, an approach to building network-local models based on machine learning led to the conjecture that the family of quantum triangle distributions of [Renou et al., Phys. Rev. Lett. 123, 140401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140401] did not admit triangle-local models in a larger range than the original proof. We prove part of this conjecture in the affirmative. Our approach consists of reducing the family of original, four-outcome distributions to families of binary-outcome ones, and then using the inflation technique to prove that these families of binary-outcome distributions do not admit triangle-local models. This constitutes the first successful use of inflation in a proof of quantum nonlocality in networks for distributions whose nonlocality could not be proved with alternative methods. Moreover, we provide a method to extend proofs of network nonlocality in concrete distributions of a parametrized family to continuous ranges of the parameter. In the process, we produce a large collection of network Bell inequalities for the triangle scenario with binary outcomes, which are of independent interest.

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.

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.

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.

The Relativity of Indeterminacy.

A long-standing tradition, largely present in both the physical and the philosophical literature, regards the advent of (special) relativity-with its block-universe picture-as the failure of any indeterministic program in physics. On the contrary, in this paper, we note that upholding reasonable principles of finiteness of information hints at a picture of the physical world that should be both relativistic and indeterministic. We thus rebut the block-universe picture by assuming that fundamental indeterminacy itself should also be regarded as a relative property when considered in a relativistic scenario. We discuss the consequence that this view may have when correlated randomness is introduced, both in the classical case and in the quantum one.

Indeterminism in physics and intuitionistic mathematics.

Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to "speak" of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We suggest that intuitionistic mathematics provides such a language and we illustrate it in simple terms.

Simultaneous coherence enhancement of optical and microwave transitions in solid-state electronic spins.

Solid-state electronic spins are extensively studied in quantum information science, as their large magnetic moments offer fast operations for computing1 and communication2-4, and high sensitivity for sensing5. However, electronic spins are more sensitive to magnetic noise, but engineering of their spectroscopic properties, for example, using clock transitions and isotopic engineering, can yield remarkable spin coherence times, as for electronic spins in GaAs6, donors in silicon7-11 and vacancy centres in diamond12,13. Here we demonstrate simultaneously induced clock transitions for both microwave and optical domains in an isotopically purified 171Yb3+:Y2SiO5 crystal, reaching coherence times of greater than 100 µs and 1 ms in the optical and microwave domains, respectively. This effect is due to the highly anisotropic hyperfine interaction, which makes each electronic-nuclear state an entangled Bell state. Our results underline the potential of 171Yb3+:Y2SiO5 for quantum processing applications relying on both optical and spin manipulation, such as optical quantum memories4,14, microwave-to-optical quantum transducers15,16, and single-spin detection17, while they should also be observable in a range of different materials with anisotropic hyperfine interactions.

Spectral noise in frequency conversion from the visible to the telecommunication C-band.

We report a detailed study of the noise properties of a visible-to-telecom photon frequency converter based on difference frequency generation (DFG). The device converts 580 nm photons to 1541 nm using a strong pump laser at 930 nm, in a periodically poled lithium niobate ridge waveguide. The converter reaches a maximum device efficiency of 46 % (internal efficiency of 67%) at a pump power of 250 mW. The noise produced by the pump laser is investigated in detail by recording the noise spectra both in the telecom and visible regimes and measuring the power dependence of the noise rates. The noise spectrum in the telecom is very broadband, as expected from previous work on similar DFG converters. However, we also observe several narrow dips in the telecom spectrum, with corresponding peaks appearing in the 580 nm noise spectrum. These features are explained by sum frequency generation of the telecom noise at wavelengths given by the phase-matching condition of different spatial modes in the waveguide. The proposed noise model is in good agreement with all the measured data, including the power dependence of the noise rates, both in the visible and telecom regimes. These results are applicable to the class of DFG converters where the pump laser wavelength is in between the input and target wavelength.

Genuine Quantum Nonlocality in the Triangle Network.

Quantum networks allow in principle for completely novel forms of quantum correlations. In particular, quantum nonlocality can be demonstrated here without the need of having various input settings, but only by considering the joint statistics of fixed local measurement outputs. However, previous examples of this intriguing phenomenon all appear to stem directly from the usual form of quantum nonlocality, namely via the violation of a standard Bell inequality. Here we present novel examples of "quantum nonlocality without inputs," which we believe represent a new form of quantum nonlocality, genuine to networks. Our simplest examples, for the triangle network, involve both entangled states and joint entangled measurements. A generalization to any odd-cycle network is also presented.

Limits on Correlations in Networks for Quantum and No-Signaling Resources.

A quantum network consists of independent sources distributing entangled states to distant nodes which can then perform entangled measurements, thus establishing correlations across the entire network. But how strong can these correlations be? Here we address this question, by deriving bounds on possible quantum correlations in a given network. These bounds are nonlinear inequalities that depend only on the topology of the network. We discuss in detail the notably challenging case of the triangle network. Moreover, we conjecture that our bounds hold in general no-signaling theories. In particular, we prove that our inequalities for the triangle network hold when the sources are arbitrary no-signaling boxes which can be wired together. Finally, we discuss an application of our results for the device-independent characterization of the topology of a quantum network.

Entanglement 25 Years after Quantum Teleportation: Testing Joint Measurements in Quantum Networks.

Twenty-five years after the invention of quantum teleportation, the concept of entanglement gained enormous popularity. This is especially nice to those who remember that entanglement was not even taught at universities until the 1990s. Today, entanglement is often presented as a resource, the resource of quantum information science and technology. However, entanglement is exploited twice in quantum teleportation. Firstly, entanglement is the "quantum teleportation channel", i.e., entanglement between distant systems. Second, entanglement appears in the eigenvectors of the joint measurement that Alice, the sender, has to perform jointly on the quantum state to be teleported and her half of the "quantum teleportation channel", i.e., entanglement enabling entirely new kinds of quantum measurements. I emphasize how poorly this second kind of entanglement is understood. In particular, I use quantum networks in which each party connected to several nodes performs a joint measurement to illustrate that the quantumness of such joint measurements remains elusive, escaping today's available tools to detect and quantify it.

Bell Inequalities with One Bit of Communication.

We study Bell scenarios with binary outcomes supplemented by one bit of classical communication. We developed a method to find facet inequalities for such scenarios even when direct facet enumeration is not possible, or at least difficult. Using this method, we partially solved the scenario where Alice and Bob choose between three inputs, finding a total of 668 inequivalent facet inequalities (with respect to relabelings of inputs and outputs). We also show that some of these inequalities are constructed from facet inequalities found in scenarios without communication, that is, the well-known Bell inequalities.

From quantum foundations to applications and back.

Quantum non-locality has been an extremely fruitful subject of research, leading the scientific revolution towards quantum information science, in particular, to device-independent quantum information processing. We argue that the time is ripe to work on another basic problem in the foundations of quantum physics, the quantum measurement problem, which should produce good physics in theoretical, mathematical, experimental and applied physics. We briefly review how quantum non-locality contributed to physics (including some outstanding open problems) and suggest ways in which questions around macroscopic quantumness could equally contribute to all aspects of physics.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.

Why Bohmian Mechanics? One- and Two-Time Position Measurements, Bell Inequalities, Philosophy, and Physics.

In Bohmian mechanics, particles follow continuous trajectories, so two-time position correlations have been well defined. However, Bohmian mechanics predicts the violation of Bell inequalities. Motivated by this fact, we investigate position measurements in Bohmian mechanics by coupling the particles to macroscopic pointers. This explains the violation of Bell inequalities despite two-time position correlations. We relate this fact to so-called surrealistic trajectories that, in our model, correspond to slowly moving pointers. Next, we emphasize that Bohmian mechanics, which does not distinguish between microscopic and macroscopic systems, implies that the quantum weirdness of quantum physics also shows up at the macro-scale. Finally, we discuss the fact that Bohmian mechanics is attractive to philosophers but not so much to physicists and argue that the Bohmian community is responsible for the latter.

Robust Macroscopic Quantum Measurements in the Presence of Limited Control and Knowledge.

Quantum measurements have intrinsic properties that seem incompatible with our everyday-life macroscopic measurements. Macroscopic Quantum Measurement (MQM) is a concept that aims at bridging the gap between well-understood microscopic quantum measurements and macroscopic classical measurements. In this paper, we focus on the task of the polarization direction estimation of a system of N spins 1/2 particles and investigate the model some of us proposed in Barnea et al., 2017. This model is based on a von Neumann pointer measurement, where each spin component of the system is coupled to one of the three spatial component directions of a pointer. It shows traits of a classical measurement for an intermediate coupling strength. We investigate relaxations of the assumptions on the initial knowledge about the state and on the control over the MQM. We show that the model is robust with regard to these relaxations. It performs well for thermal states and a lack of knowledge about the size of the system. Furthermore, a lack of control on the MQM can be compensated by repeated "ultra-weak" measurements.

Multimode and Long-Lived Quantum Correlations Between Photons and Spins in a Crystal.

The realization of quantum networks and quantum repeaters remains an outstanding challenge in quantum communication. These rely on the entanglement of remote matter systems, which in turn requires the creation of quantum correlations between a single photon and a matter system. A practical way to establish such correlations is via spontaneous Raman scattering in atomic ensembles, known as the Duan-Lukin-Cirac-Zoller (DLCZ) scheme. However, time multiplexing is inherently difficult using this method, which leads to low communication rates even in theory. Moreover, it is desirable to find solid-state ensembles where such matter-photon correlations could be generated. Here we demonstrate quantum correlations between a single photon and a spin excitation in up to 12 temporal modes, in a ^{151}Eu^{3+}-doped Y_{2}SiO_{5} crystal, using a novel DLCZ approach that is inherently multimode. After a storage time of 1 ms, the spin excitation is converted into a second photon. The quantum correlation of the generated photon pair is verified by violating a Cauchy-Schwarz inequality. Our results show that solid-state rare-earth-ion-doped crystals could be used to generate remote multimode entanglement, an important resource for future quantum networks.

Quantifying Photonic High-Dimensional Entanglement.

High-dimensional entanglement offers promising perspectives in quantum information science. In practice, however, the main challenge is to devise efficient methods to characterize high-dimensional entanglement, based on the available experimental data which is usually rather limited. Here we report the characterization and certification of high-dimensional entanglement in photon pairs, encoded in temporal modes. Building upon recently developed theoretical methods, we certify an entanglement of formation of 2.09(7) ebits in a time-bin implementation, and 4.1(1) ebits in an energy-time implementation. These results are based on very limited sets of local measurements, which illustrates the practical relevance of these methods.