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The quantum switch, the canonical example of a process with indefinite causal order, has been claimed to provide various advantages over processes with definite causal orders for some particular tasks in the field of quantum metrology. In this work, we argue that some of these advantages in fact do not hold if a fairer comparison is made. To this end, we consider a framework that allows for a proper comparison between the performance, quantified by the quantum Fisher information, of different classes of indefinite causal order processes and that of causal strategies on a given metrological task. More generally, by considering the recently proposed classes of circuits with classical or quantum control of the causal order, we come up with different examples where processes with indefinite causal order offer (or not) an advantage over processes with definite causal order, qualifying the interest of indefinite causal order regarding quantum metrology. As it turns out, for a range of examples, the class of quantum circuits with quantum control of causal order, which are known to be physically realizable, is shown to provide a strict advantage over causally ordered quantum circuits as well as over the class of quantum circuits with causal superposition. Thus, the consideration of this class provides new evidence that indefinite causal order strategies can strictly outperform definite causal order strategies in quantum metrology.

Causal nonseparability is the property underlying quantum processes incompatible with a definite causal order. So far it has remained a central open question as to whether any process with a clear physical realisation can violate a causal inequality, so that its causal nonseparability can be certified in a device-independent way, as originally conceived. Here we present a method solely based on the observed correlations, which certifies the causal nonseparability of all the processes that can induce a causally nonseparable distributed measurement in a scenario with trusted quantum input states, as defined in [Dourdent et al., Phys. Rev. Lett. 129, 090402 (2022)]. This notably includes the celebrated quantum switch. This device-independent certification is achieved by introducing a network of untrusted operations, allowing one to self-test the quantum inputs on which the effective distributed measurement induced by the process is performed., Comment: 7+15 pages, 7 figures

We present the experimental measurement, on a quantum processor, of a series of polynomial lower bounds that converge to the quantum Fisher information (QFI), a fundamental quantity for certifying multipartite entanglement that is useful for metrological applications. We combine advanced methods of the randomized measurement toolbox to obtain estimators that are robust against drifting errors caused uniquely during the randomized measurement protocol. We estimate the QFI for Greenberg-Horne-Zeilinger states, observing genuine multipartite entanglement. Then, we prepare the ground state of the transverse field Ising model at the critical point using a variational circuit. We estimate its QFI and investigate the interplay between state optimization and noise induced by increasing the circuit depth., Comment: 24 pages, 13 figures

We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state of interest. This method involves performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multi-copy observables and present performance guarantees in terms of variance bounds which depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through numerical experiments estimating polynomial approximations of the von Neumann entropy and quantum state fidelities.

The operator entanglement (OE) is a key quantifier of the complexity of a reduced density matrix. In out-of-equilibrium situations, e.g. after a quantum quench of a product state, it is expected to exhibit an entanglement barrier. The OE of a reduced density matrix initially grows linearly as entanglement builds up between the local degrees of freedom, it then reaches a maximum, and ultimately decays to a small finite value as the reduced density matrix converges to a simple stationary state through standard thermalization mechanisms. Here, by performing a new data analysis of the published experimental results of [Brydges et al., Science 364, 260 (2019)], we obtain the first experimental measurement of the OE of a subsystem reduced density matrix in a quantum many-body system. We employ the randomized measurements toolbox and we introduce and develop a new efficient method to post-process experimental data in order to extract higher-order density matrix functionals and access the OE. The OE thus obtained displays the expected barrier as long as the experimental system is large enough. For smaller systems, we observe a barrier with a double-peak structure, whose origin can be interpreted in terms of pairs of quasi-particles being reflected at the boundary of the qubit chain. As $U(1)$ symmetry plays a key role in our analysis, we introduce the notion of symmetry resolved operator entanglement (SROE), in addition to the total OE. To gain further insights into the SROE, we provide a thorough theoretical analysis of this new quantity in chains of non-interacting fermions, which, in spite of their simplicity, capture most of the main features of OE and SROE. In particular, we uncover three main physical effects: the presence of a barrier in any charge sector, a time delay for the onset of the growth of SROE, and an effective equipartition between charge sectors., Comment: 13 + 24 pages with 3 + 7 figures

Quantum mechanics allows processes to be superposed, leading to a genuinely quantum lack of causal structure. For example, the process known as the quantum switch applies two operations ${\cal A}$ and ${\cal B}$ in a superposition of the two possible orders, ${\cal A}$ before ${\cal B}$ and ${\cal B}$ before ${\cal A}$. Experimental implementations of the quantum switch have been challenged by some on the grounds that the operations ${\cal A}$ and ${\cal B}$ were implemented more than once, thereby simulating indefinite causal order rather than actually implementing it. Motivated by this debate, we consider a situation in which the quantum operations are physically described by a light-matter interaction model. While for our model the two processes are indistinguishable in the infinite energy regime, restricting the energy available for the implementation of the operations introduces imperfections, which allow one to distinguish processes using different number of operations. We consider such an energetically-constrained scenario and compare the quantum switch to one of its natural simulations, where each operation is implemented twice. Considering a commuting-vs-anticommuting unitary discrimination task, we find that within our model the quantum switch performs better, for some fixed amount of energy, than its simulation. In addition to the known computational or communication advantages of causal superpositions, our work raises new questions about their potential energetic advantages., Comment: Updated version with a slightly modified abstract and conclusion emphasizing more on the exact assumptions and outcomes of the work. The Fig1 has a rewritten legend. Minor details were also added on various parts

We use hyper-entanglement to experimentally realize deterministic entanglement swapping based on quantum Elegant Joint Measurements. These are joint projections of two qubits onto highly symmetric, iso-entangled, 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.

It has been shown that it is theoretically possible for there to exist quantum and classical processes in which the operations performed by separate parties do not occur in a well-defined causal order. A central question is whether and how such processes can be realised in practice. In order to provide a rigorous argument for the notion that certain such processes have a realisation in standard quantum theory, the concept of time-delocalised quantum subsystem has been introduced. In this paper, we show that realisations on time-delocalised subsystems exist for all unitary extensions of tripartite processes. Remarkably, this class contains processes that violate causal inequalities, i.e., that can generate correlations that witness the incompatibility with definite causal order in a device-independent manner. We consider a known striking example of such a tripartite classical process that has a unitary extension, and study its realisation on time-delocalised subsystems. We then discuss the question of what a violation of causal inequalities implies in this setting, and argue that it is indeed a meaningful concept to show the absence of a definite causal order between the variables of interest., Comment: V2: Various improvements, argument on causal inequalities assumptions corrected and significantly strengthened. V3: Close to published version

While the standard formulation of quantum theory assumes a fixed background causal structure, one can relax this assumption within the so-called process matrix framework. Remarkably, some processes, termed causally nonseparable, are incompatible with a definite causal order. We explore a form of certification of causal nonseparability in a semi-device-independent scenario where the involved parties receive trusted quantum inputs, but whose operations are otherwise uncharacterised. Defining the notion of causally nonseparable distributed measurements, we show that certain causally nonseparable processes which cannot violate any causal inequality, including the canonical example of the quantum switch, can generate noncausal correlations in such a scenario. Moreover, by imposing some further natural structure to the untrusted operations, we show that all bipartite causally nonseparable process matrices can be certified with trusted quantum inputs., Comment: 6 + 17 pages, 1 figure. v2: updated to published version

The quantum Fisher information (QFI) is a fundamental quantity of interest in many areas from quantum metrology to quantum information theory. It can in particular be used as a witness to establish the degree of multi-particle entanglement in quantum many-body-systems. In this work, we use polynomials of the density matrix to construct monotonically increasing lower bounds that converge to the QFI. Using randomized measurements we propose a protocol to accurately estimate these lower bounds in state-of-art quantum technological platforms. We estimate the number of measurements needed to achieve a given accuracy and confidence level in the bounds, and present two examples of applications of the method in quantum systems made of coupled qubits and collective spins., Comment: 7 + 10 pages with 3 + 3 figures. Accepted version with additional analytical expressions for statistical errors of arbitrary order polynomials of the density matrix estimated via classical shadows. Code available at https://github.com/bvermersch/RandomMeas

We introduce a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). As standard completely positive trace preserving maps are known not to be appropriate to represent coherently controlled quantum channels, we propose to instead use purified channels, an extension of Stinespring's dilation. We characterise the observational equivalence of purified channels in various coherent-control contexts, paving the way towards a faithful representation of quantum channels under coherent control., Comment: 34 pages

Quantum supermaps are transformations that map quantum operations to quantum operations. It is known that quantum supermaps which respect a definite, predefined causal order between their input operations correspond to fixed-order quantum circuits. A systematic understanding of the physical interpretation of more general types of quantum supermaps--in particular, those incompatible with a definite causal structure--is however lacking. Here we identify two new types of circuits that naturally generalise the fixed-order case and that likewise correspond to distinct classes of quantum supermaps, which we fully characterise. We first introduce "quantum circuits with classical control of causal order", in which the order of operations is still well-defined, but not necessarily fixed in advance: it can in particular be established dynamically, in a classically-controlled manner, as the circuit is being used. We then consider "quantum circuits with quantum control of causal order", in which the order of operations is controlled coherently. The supermaps described by these classes of circuits are physically realisable, and the latter encompasses all known examples of physically realisable processes with indefinite causal order, including the celebrated "quantum switch". Interestingly, it also contains new examples arising from the combination of dynamical and coherent control of causal order, and we detail explicitly one such process. Nevertheless, we show that quantum circuits with quantum control of causal order can only generate "causal" correlations, compatible with a well-defined causal order. We furthermore extend our considerations to probabilistic circuits that produce also classical outcomes, and we demonstrate by an example how our characterisations allow us to identify new advantages for quantum information processing tasks that could be demonstrated in practice., Comment: 34+17 pages, 13 figures. V3: corrected the statement of Proposition 7, specifying that the process matrix $W$ itself has to be positive semidefinite

Device independent protocols based on Bell nonlocality, such as quantum key distribution and randomness generation, must ensure no adversary can have prior knowledge of the measurement outcomes. This requires a measurement independence assumption: that the choice of measurement is uncorrelated with any other underlying variables that influence the measurement outcomes. Conversely, relaxing measurement independence allows for a fully `causal' simulation of Bell nonlocality. We construct the most efficient such simulation, as measured by the mutual information between the underlying variables and the measurement settings, for the Clauser-Horne-Shimony-Holt (CHSH) scenario, and find that the maximal quantum violation requires a mutual information of just $\sim 0.080$ bits. Any physical device built to implement this simulation allows an adversary to have full knowledge of a cryptographic key or `random' numbers generated by a device independent protocol based on violation of the CHSH inequality. We also show that a previous model for the CHSH scenario, requiring only $\sim 0.046$ bits to simulate the maximal quantum violation, corresponds to the most efficient `retrocausal' simulation, in which future measurement settings necessarily influence earlier source variables. This may be viewed either as an unphysical limitation of the prior model, or as an argument for retrocausality on the grounds of its greater efficiency. Causal and retrocausal models are also discussed for maximally entangled two-qubit states, as well as superdeterministic, one-sided and zigzag causal models., Comment: 16 pages, 7 figures

In quantum communication networks, wires represent well-defined trajectories along which quantum systems are transmitted. In spite of this, trajectories can be used as a quantum control to govern the order of different noisy communication channels, and such a control has been shown to enable the transmission of information even when quantum communication protocols through well-defined trajectories fail. This result has motivated further investigations on the role of the superposition of trajectories in enhancing communication, which revealed that the use of quantum control of parallel communication channels, or of channels in series with quantum-controlled operations, can also lead to communication advantages. Building upon these findings, here we experimentally and numerically compare different ways in which two trajectories through a pair of noisy channels can be superposed. We observe that, within the framework of quantum interferometry, the use of channels in series with quantum-controlled operations generally yields the largest advantages. Our results contribute to clarify the nature of these advantages in experimental quantum-optical scenarios, and showcase the benefit of an extension of the quantum communication paradigm in which both the information exchanged and the trajectory of the information carriers are quantum., Comment: 20 pages, 11 figures. Accepted version

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 iso-entangled measurement bases that generalise 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 realisation by presenting a simple two-qubit quantum circuit for the implementation of the Elegant Joint Measurement and our generalisation.

Models for quantum computation with circuit connections subject to the quantum superposition principle have been recently proposed. There, a control quantum system can coherently determine the order in which a target quantum system undergoes $N$ gate operations. This process, known as the quantum $N$-switch, is a resource for several information-processing tasks. In particular, it provides a computational advantage -- over fixed-gate-order quantum circuits -- for phase-estimation problems involving $N$ unknown unitary gates. However, the corresponding algorithm requires an experimentally unfeasible target-system dimension (super)exponential in $N$. Here, we introduce a promise problem for which the quantum $N$-switch gives an equivalent computational speed-up with target-system dimension as small as 2 regardless of $N$. We use state-of-the-art multi-core optical-fiber technology to experimentally demonstrate the quantum $N$-switch with $N=4$ gates acting on a photonic-polarization qubit. This is the first observation of a quantum superposition of more than $N=2$ temporal orders, demonstrating its usefulness for efficient phase-estimation., Comment: Main text: 9 pages, 3 figures; total 15 pages, 5 figures

The scientific method relies on facts, established through repeated measurements and agreed upon universally, independently of who observed them. In quantum mechanics, the objectivity of observations is not so clear, most dramatically exposed in Eugene Wigner's eponymous thought experiment where two observers can experience seemingly different realities. The question whether these realities can be reconciled in an observer-independent way has long remained inaccessible to empirical investigation, until recent no-go-theorems constructed an extended Wigner's friend scenario with four observers that allows us to put it to the test. In a state-of-the-art 6-photon experiment, we realise this extended Wigner's friend scenario, experimentally violating the associated Bell-type inequality by 5 standard deviations. If one holds fast to the assumptions of locality and free-choice, this result implies that quantum theory should be interpreted in an observer-dependent way., Comment: 5+5 pages, 6 figures

A completely depolarising quantum channel always outputs a fully mixed state and thus cannot transmit any information. In a recent Letter [D. Ebler et al., Phys. Rev. Lett. 120, 120502 (2018)], it was however shown that if a quantum state passes through two such channels in a quantum superposition of different orders - a setup known as the "quantum switch" - then information can nevertheless be transmitted through the channels. Here, we show that a similar effect can be obtained when one coherently controls between sending a target system through one of two identical depolarising channels. Whereas it is tempting to attribute this effect in the quantum switch to the indefinite causal order between the channels, causal indefiniteness plays no role in this new scenario. This raises questions about its role in the corresponding effect in the quantum switch. We study this new scenario in detail and we see that, when quantum channels are controlled coherently, information about their specific implementation is accessible in the output state of the joint control-target system. This allows two different implementations of what is usually considered to be the same channel to therefore be differentiated. More generally, we find that to completely describe the action of a coherently controlled quantum channel, one needs to specify not only a description of the channel (e.g., in terms of Kraus operators), but an additional "transformation matrix" depending on its implementation., Comment: 14 pages, 2 figures

The concept of causal nonseparability has been recently introduced, in opposition to that of causal separability, to qualify physical processes that locally abide by the laws of quantum theory, but cannot be embedded in a well-defined global causal structure. While the definition is unambiguous in the bipartite case, its generalisation to the multipartite case is not so straightforward. Two seemingly different generalisations have been proposed, one for a restricted tripartite scenario and one for the general multipartite case. Here we compare the two, showing that they are in fact inequivalent. We propose our own definition of causal (non)separability for the general case, which---although a priori subtly different---turns out to be equivalent to the concept of "extensible causal (non)separability" introduced before, and which we argue is a more natural definition for general multipartite scenarios. We then derive necessary, as well as sufficient conditions to characterise causally (non)separable processes in practice. These allow one to devise practical tests, by generalising the tool of witnesses of causal nonseparability.

A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an "anomalous weak value", is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures., Comment: 15 pages

A tight information-theoretic measurement uncertainty relation is experimentally tested with neutron spin-1/2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e., positive-operator valued measures), as predicted theoretically., Comment: 6 pages, 3 figures

We introduce Bell inequalities based on covariance, one of the most common measures of correlation. Explicit examples are discussed, and violations in quantum theory are demonstrated. A crucial feature of these covariance Bell inequalities is their nonlinearity; this has nontrivial consequences for the derivation of their local bound, which is not reached by deterministic local correlations. For our simplest inequality, we derive analytically tight bounds for both local and quantum correlations. An interesting application of covariance Bell inequalities is that they can act as "shared randomness witnesses": specifically, the value of the Bell expression gives device-independent lower bounds on both the dimension and the entropy of the shared random variable in a local model., Comment: 15 pages, 3 figures

The study of correlations with no definite causal order has revealed a rich structure emerging when more than two parties are involved. This motivates the consideration of multipartite "noncausal" correlations that cannot be realised even if noncausal resources are made available to a smaller number of parties. Here we formalise this notion: genuinely N-partite noncausal correlations are those that cannot be produced by grouping N parties into two or more subsets, where a causal order between the subsets exists. We prove that such correlations can be characterised as lying outside a polytope, whose vertices correspond to deterministic strategies and whose facets define what we call "2-causal" inequalities. We show that genuinely multipartite noncausal correlations arise within the process matrix formalism, where quantum mechanics holds locally but no global causal structure is assumed, although for some inequalities no violation was found. We further introduce two refined definitions that allow one to quantify, in different ways, to what extent noncausal correlations correspond to a genuinely multipartite resource., Comment: 18 pages, 5 figures, 1 supplementary CDF file

The existence of a global causal order between events places constraints on the correlations that parties may share. Such "causal correlations" have been the focus of recent attention, driven by the realization that some extensions of quantum mechanics may violate so-called causal inequalities. In this paper we study causal correlations from an entropic perspective, and we show how to use this framework to derive entropic causal inequalities. We consider two different ways to derive such inequalities. Firstly, we consider a method based on the causal Bayesian networks describing the causal relations between the parties. In contrast to the Bell-nonlocality scenario, where this method has previously been shown to be ineffective, we show that it leads to several interesting entropic causal inequalities. Secondly, we consider an alternative method based on counterfactual variables that has previously been used to derive entropic Bell inequalities. We compare the inequalities obtained via these two methods and discuss their violation by noncausal correlations. As an application of our approach, we derive bounds on the quantity of information - which is more naturally expressed in the entropic framework - that parties can communicate when operating in a definite causal order., Comment: 21 pages, 3 figures; updated to published version

Quantum mechanics admits correlations that cannot be explained by local realistic models. Those most studied are the standard local hidden variable models, which satisfy the well-known Bell inequalities. To date, most works have focused on bipartite entangled systems. Here, we consider correlations between three parties connected via two independent entangled states. We investigate the new type of so-called "bilocal" models, which correspondingly involve two independent hidden variables. Such models describe scenarios that naturally arise in quantum networks, where several independent entanglement sources are employed. Using photonic qubits, we build such a linear three-node quantum network and demonstrate non-bilocal correlations by violating a Bell-like inequality tailored for bilocal models. Furthermore, we show that the demonstration of non-bilocality is more noise-tolerant than that of standard Bell non-locality in our three-party quantum network., Comment: 5 pages, with 3 pages supplementary material

We consider the most general correlations that can be obtained by a group of parties whose causal relations are well-defined, although possibly probabilistic and dependent on past parties' operations. We show that, for any fixed number of parties and inputs and outputs for each party, the set of such correlations forms a convex polytope, whose vertices correspond to deterministic strategies, and whose (nontrivial) facets define so-called causal inequalities. We completely characterize the simplest tripartite polytope in terms of its facet inequalities, propose generalizations of some inequalities to scenarios with more parties, and show that our tripartite inequalities can be violated within the process matrix formalism, where quantum mechanics is locally valid but no global causal structure is assumed., Comment: 14 pages and 1 supplementary CDF file

Information-theoretic definitions for the noise associated with a quantum measurement and the corresponding disturbance to the state of the system have recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401 (2014)]. These definitions are invariant under relabelling of measurement outcomes, and lend themselves readily to the formulation of state-independent uncertainty relations both for the joint estimate of observables (noise-noise relations) and the noise-disturbance tradeoff. Here we derive such relations for incompatible qubit observables, which we prove to be tight in the case of joint estimates, and present progress towards fully characterising the noise-disturbance tradeoff. In doing so, we show that the set of obtainable noise-noise values for such observables is convex, whereas the conjectured form for the set of obtainable noise-disturbance values is not. Furthermore, projective measurements are not optimal with respect to the joint-measurement noise or noise-disturbance tradeoffs. Interestingly, it seems that four-outcome measurements are needed in the former case, whereas three-outcome measurements are optimal in the latter., Comment: Minor changes, corresponds to final published version. 14 pages, 5 figures

It was recently realised that quantum theory allows for so-called causally nonseparable processes, which are incompatible with any definite causal order. This was first suggested on a rather abstract level by the formalism of process matrices, which only assumes that quantum theory holds locally in some observers' laboratories, but does not impose a global causal structure; it was then shown, on a more practical level, that the quantum switch---a new resource for quantum computation that goes beyond causally ordered circuits---provided precisely a physical example of a causally nonseparable process. To demonstrate that a given process is causally nonseparable, we introduced in [Ara\'ujo et al., New J. Phys. 17, 102001 (2015)] the concept of witnesses of causal nonseparability. Here we present a shorter introduction to this concept, and concentrate on some explicit examples to show how to construct and use such witnesses in practice., Comment: 15 pages, 7 figures

The well-known Robertson-Schr\"odinger uncertainty relations have state-dependent lower bounds which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state., Comment: 14 pages, 3 figures

In a scenario where two parties share, act on and exchange some physical resource, the assumption that the parties' actions are ordered according to a definite causal structure yields constraints on the possible correlations that can be established. We show that the set of correlations that are compatible with a definite causal order forms a polytope, whose facets define causal inequalities. We fully characterize this causal polytope in the simplest case of bipartite correlations with binary inputs and outputs. We find two families of nonequivalent causal inequalities; both can be violated in the recently introduced framework of process matrices, which extends the standard quantum formalism by relaxing the implicit assumption of a fixed causal structure. Our work paves the way to a more systematic investigation of causal inequalities in a theory-independent way, and of their violation within the framework of process matrices., Comment: 7 + 4 pages, 2 figures

In a quantum network, distant observers sharing physical resources emitted by independent sources can establish strong correlations, which defy any classical explanation in terms of local variables. We discuss the characterization of nonlocal correlations in such a situation, when compared to those that can be generated in networks distributing independent local variables. We present an iterative procedure for constructing Bell inequalities tailored for networks: starting from a given network, and a corresponding Bell inequality, our technique provides new Bell inequalities for a more complex network, involving one additional source and one additional observer. The relevance of our method is illustrated on a variety of networks, for which we demonstrate significant quantum violations., Comment: 8 pages, 2 figures. Comments welcome

Our common understanding of the physical world deeply relies on the notion that events are ordered with respect to some time parameter, with past events serving as causes for future ones. Nonetheless, it was recently found that it is possible to formulate quantum mechanics without any reference to a global time or causal structure. The resulting framework includes new kinds of quantum resources that allow performing tasks - in particular, the violation of causal inequalities - which are impossible for events ordered according to a global causal order. However, no physical implementation of such resources is known. Here we show that a recently demonstrated resource for quantum computation - the quantum switch - is a genuine example of "indefinite causal order". We do this by introducing a new tool - the causal witness - which can detect the causal nonseparability of any quantum resource that is incompatible with a definite causal order. We show however that the quantum switch does not violate any causal nequality., Comment: 15 + 12 pages, 5 figures. Published version

Quantum mechanics is an outstandingly successful description of nature, underpinning fields from biology through chemistry to physics. At its heart is the quantum wavefunction, the central tool for describing quantum systems. Yet it is still unclear what the wavefunction actually is: does it merely represent our limited knowledge of a system, or is it an element of reality? Recent no-go theorems argued that if there was any underlying reality to start with, the wavefunction must be real. However, that conclusion relied on debatable assumptions, without which a partial knowledge interpretation can be maintained to some extent. A different approach is to impose bounds on the degree to which knowledge interpretations can explain quantum phenomena, such as why we cannot perfectly distinguish non-orthogonal quantum states. Here we experimentally test this approach with single photons. We find that no knowledge interpretation can fully explain the indistinguishability of non-orthogonal quantum states in three and four dimensions. Assuming that some underlying reality exists, our results strengthen the view that the entire wavefunction should be real. The only alternative is to adopt more unorthodox concepts such as backwards-in-time causation, or to completely abandon any notion of objective reality., Comment: 7 pages, 4 figures

We study the extent to which \psi-epistemic models for quantum measurement statistics---models where the quantum state does not have a real, ontic status---can explain the indistinguishability of nonorthogonal quantum states. This is done by comparing the overlap of any two quantum states with the overlap of the corresponding classical probability distributions over ontic states in a \psi-epistemic model. It is shown that in Hilbert spaces of dimension $d \geq 4$, the ratio between the classical and quantum overlaps in any \psi-epistemic model must be arbitrarily small for certain nonorthogonal states, suggesting that such models are arbitrarily bad at explaining the indistinguishability of quantum states. For dimensions $d$ = 3 and 4, we construct explicit states and measurements that can be used experimentally to put stringent bounds on the ratio of classical-to-quantum overlaps in \psi-epistemic models, allowing one in particular to rule out maximally \psi-epistemic models more efficiently than previously proposed., Comment: 8 pages

The quantification of the "measurement uncertainty" aspect of Heisenberg's Uncertainty Principle---that is, the study of trade-offs between accuracy and disturbance, or between accuracies in an approximate joint measurement on two incompatible observables---has regained a lot of interest recently. Several approaches have been proposed and debated. In this paper we consider Ozawa's definitions for inaccuracies (as root-mean-square errors) in approximate joint measurements, and study how these are constrained in different cases, whether one specifies certain properties of the approximations---namely their standard deviations and/or their bias---or not. Extending our previous work [C. Branciard, Proc. Natl. Acad. Sci. U.S.A. 110, 6742 (2013)], we derive new error-trade-off relations, which we prove to be tight for pure states. We show explicitly how all previously known relations for Ozawa's inaccuracies follow from ours. While our relations are in general not tight for mixed states, we show how these can be strengthened and how tight relations can still be obtained in that case., Comment: 18 pages

Standard quantum key distribution (QKD) protocols typically assume that the distant parties share a common reference frame. In practice, however, establishing and maintaining a good alignment between distant observers is rarely a trivial issue, which may significantly restrain the implementation of long-distance quantum communication protocols. Here we propose simple QKD protocols that do not require the parties to share any reference frame, and study their security and feasibility in both the usual device-dependent case--in which the two parties use well characterized measurement devices--as well as in the device-independent case--in which the measurement devices can be untrusted, and the security relies on the violation of a Bell inequality. To illustrate the practical relevance of these ideas, we present a proof-of-principle demonstration of our protocols using polarization entangled photons distributed over a coiled 10-km-long optical fiber. We consider two situations, in which either the fiber spool freely drifts, or randomly chosen polarization transformations are applied. The correlations obtained from measurements allow, with high probability, to generate positive asymptotic secret key rates in both the device-dependent and device-independent scenarios (under the fair-sampling assumption for the latter case)., Comment: 12 pages, 11 figures

Quantum physics constrains the accuracy of joint measurements of incompatible observables. Here we test tight measurement-uncertainty relations using single photons. We implement two independent, idealized uncertainty-estimation methods, the 3-state method and the weak-measurement method, and adapt them to realistic experimental conditions. Exceptional quantum state fidelities of up to 0.99998(6) allow us to verge upon the fundamental limits of measurement uncertainty., Comment: 13 pages, 12 figures

This note is a reply to M. Navascu\'es' claim that "all entangled states violate Leggett's crypto-nonlocality" [arXiv:1303.5124v2]. I argue that such a conclusion can only be reached if one introduces additional assumptions that further restrict Leggett's notion of "crypto-nonlocality". If a contrario one sticks only to Leggett's original axioms, there exist entangled states whose correlations are always compatible with Leggett's crypto-nonlocality---which is thus a genuinely different concept from quantum separability. I clarify in this note the relation between these two notions, together also with Bell's assumption of local causality., Comment: 3.5 pages. v2: typo corrected. Published version (titled "Bell's local causality, Leggett's crypto-nonlocality and quantum separability are genuinely different concepts") concentrates on clarifying the differences between these 3 notions, rather than on the claims of arXiv:1303.5124v2 (which has since been updated)

Heisenberg's uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: although Heisenberg's first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg's intuition. Even if two incompatible observables cannot be measured together, one can still approximate their joint measurement, at the price of introducing some errors with respect to the ideal measurement of each of them. We present a new, tight relation characterizing the optimal trade-off between the error on one observable versus the error on the other. As a particular case, our approach allows us to characterize the disturbance of an observable induced by the approximate measurement of another one; we also derive a stronger error-disturbance relation for this scenario., Comment: 13 pages (7+6), 4 figures. Title changed in the published version to "Error-tradeoff and error-disturbance relations for incompatible quantum measurements", to comply with editorial policies

Simulation tasks are insightful tools to compare information-theoretic resources. Considering a generalization of usual Bell scenarios where external quantum inputs are provided to the parties, we show that any entangled quantum state exhibits correlations that cannot be simulated using only shared randomness and classical communication, even when the amount and rounds of classical communication involved are unrestricted. We indeed construct explicit Bell-like inequalities that are necessarily satisfied by such classical resources but nevertheless violated by correlations obtainable from entangled quantum states, when measured a single copy at a time., Comment: 9 pages, 1 figure. Version published in NJP

The problem of demonstrating entanglement is central to quantum information processing applications. Resorting to standard entanglement witnesses requires one to perfectly trust the implementation of the measurements to be performed on the entangled state, which may be an unjustified assumption. Inspired by the recent work of F. Buscemi [Phys. Rev. Lett. 108, 200401 (2012)], we introduce the concept of Measurement-Device-Independent Entanglement Witnesses (MDI-EWs), which allow one to demonstrate entanglement of all entangled quantum states with untrusted measurement apparatuses. We show how to systematically obtain such MDI-EWs from standard entanglement witnesses. Our construction leads to MDI-EWs that are loss-tolerant, and can be implemented with current technology., Comment: 5 pages

The (non-)local content in the sense of Elitzur, Popescu, and Rohrlich (EPR2) [Phys. Lett. A 162, 25 (1992)] is a natural measure for the (non-)locality of quantum states. Its computation is in general difficult, even in low dimensions, and is one of the few open questions about pure two-qubit states. We present a complete solution to this long-lasting problem., Comment: 9 pages, 3 figures

Entanglement appears under two different forms in quantum theory, namely as a property of states of joint systems and as a property of measurement eigenstates in joint measurements. By combining these two aspects of entanglement, it is possible to generate nonlocality between particles that never interacted, using the protocol of entanglement swapping. We show that even in the more constraining bilocal scenario where distant sources of particles are assumed to be independent, i.e. to share no prior randomness, this process can be simulated classically with bounded communication, using only 9 bits in total. Our result thus provides an upper bound on the nonlocality of the process of entanglement swapping., Comment: 6 pages, 1 figure

Full-correlation Bell-like inequalities represent an important subclass of Bell-like inequalities that have found applications in both a better understanding of fundamental physics and in quantum information science. Loosely speaking, these are inequalities where only measurement statistics involving all parties play a role. In this paper, we provide a framework for the study of a large family of such inequalities that are symmetrical with respect to arbitrary permutations of the parties. As an illustration of the power of our framework, we derive (i) a new family of Svetlichny inequalities for arbitrary numbers of parties, settings and outcomes, (ii) a new family of two-outcome device-independent entanglement witnesses for genuine n-partite entanglement and (iii) a new family of two-outcome Tsirelson inequalities for arbitrary numbers of parties and settings. We also discuss briefly the application of these new inequalities in the characterization of quantum correlations., Comment: 9 pages, 1 figure

Entanglement swapping is a process by which two initially independent quantum systems can become entangled and generate nonlocal correlations. To characterize such correlations, we compare them to those predicted by bilocal models, where systems that are initially independent are described by uncorrelated states. We extend in this paper the analysis of bilocal correlations initiated in [Phys. Rev. Lett. 104, 170401 (2010)]. In particular, we derive new Bell-type inequalities based on the bilocality assumption in different scenarios, we study their possible quantum violations, and analyze their resistance to experimental imperfections. The bilocality assumption, being stronger than Bell's standard local causality assumption, lowers the requirements for the demonstration of quantumness in entanglement swapping experiments.

Bell tests---the experimental demonstration of a Bell inequality violation---are central to understanding the foundations of quantum mechanics, underpin quantum technologies, and are a powerful diagnostic tool for technological developments in these areas. To date, Bell tests have relied on careful calibration of the measurement devices and alignment of a shared reference frame between the two parties---both technically demanding tasks in general. Surprisingly, we show that neither of these operations are necessary, violating Bell inequalities with near certainty with (i) unaligned, but calibrated, measurement devices, and (ii) uncalibrated and unaligned devices. We demonstrate generic quantum nonlocality with randomly chosen local measurements on a singlet state of two photons implemented with reconfigurable integrated optical waveguide circuits based on voltage-controlled phase shifters. The observed results demonstrate the robustness of our schemes to imperfections and statistical noise. This new approach is likely to have important applications in both fundamental science and in quantum technologies, including device independent quantum key distribution., Comment: 7 pages, 7 figures

Quantum steering allows two parties to verify shared entanglement even if one measurement device is untrusted. A conclusive demonstration of steering through the violation of a steering inequality is of considerable fundamental interest and opens up applications in quantum communication. To date all experimental tests with single photon states have relied on post-selection, allowing untrusted devices to cheat by hiding unfavourable events in losses. Here we close this "detection loophole" by combining a highly efficient source of entangled photon pairs with superconducting transition edge sensors. We achieve an unprecedented ~62% conditional detection efficiency of entangled photons and violate a steering inequality with the minimal number of measurement settings by 48 standard deviations. Our results provide a clear path to practical applications of steering and to a photonic loophole-free Bell test., Comment: Preprint of 7 pages, 3 figures; the definitive version is published in Nature Communications, see below. Also, see related experimental work by A. J. Bennet et al., arXiv:1111.0739 and B. Wittmann et al., arXiv:1111.0760