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A central requirement in asymmetric quantum nonlocality protocols, such as quantum steering, is the precise reconstruction of state assemblages -- statistical ensembles of quantum states correlated with remote classical signals. Here we introduce a generalized loss model for assemblage tomography that uses conical optimization techniques combined with maximum likelihood estimation. Using an evidence-based framework based on Akaike's Information Criterion, we demonstrate that our approach excels in the accuracy of reconstructions while accounting for model complexity. In comparison, standard tomographic methods fall short when applied to experimentally relevant data., Comment: 5 + 4 pages, 3 + 3 figures (Main + Supplemental Material)

Long-distance optical quantum channels are necessarily lossy, leading to errors in transmitted quantum information, entanglement degradation and, ultimately, poor protocol performance. Quantum states carrying information in the channel can be probabilistically amplified to compensate for loss, but are destroyed when amplification fails. Quantum correction of the channel itself is therefore required, but break-even performance -- where arbitrary states can be better transmitted through a corrected channel than an uncorrected one -- has so far remained out of reach. Here we perform distillation by heralded amplification to improve a noisy entanglement channel. We subsequently employ entanglement swapping to demonstrate that arbitrary quantum information transmission is unconditionally improved -- i.e. without relying on postselection or post-processing of data -- compared to the uncorrected channel. In this way, it represents realisation of a genuine quantum relay. Our channel correction for single-mode quantum states will find use in quantum repeater, communication and metrology applications., Comment: 11 pages, 6 figures, supplementary included

Whilst holding great promise for low noise, ease of operation and networking, useful photonic quantum computing has been precluded by the need for beyond-state-of-the-art components, manufactured by the millions. Here we introduce a manufacturable platform for quantum computing with photons. We benchmark a set of monolithically-integrated silicon photonics-based modules to generate, manipulate, network, and detect photonic qubits, demonstrating dual-rail photonic qubits with $99.98\% \pm 0.01\%$ state preparation and measurement fidelity, Hong-Ou-Mandel quantum interference between independent photon sources with $99.50\%\pm0.25\%$ visibility, two-qubit fusion with $99.22\%\pm0.12\%$ fidelity, and a chip-to-chip qubit interconnect with $99.72\%\pm0.04\%$ fidelity, not accounting for loss. In addition, we preview a selection of next generation technologies, demonstrating low-loss silicon nitride waveguides and components, fabrication-tolerant photon sources, high-efficiency photon-number-resolving detectors, low-loss chip-to-fiber coupling, and barium titanate electro-optic phase shifters., Comment: 8 pages, 5 figures

Interferometric phase estimation is an essential tool for precise measurements of quantities such as displacement, velocity and material properties. The lower bound on measurement uncertainty achievable with classical resources is set by the shot-noise limit (SNL) that scales asymptotically as $1/\sqrt{N}$, where $N$ is the number of resources used. The experiment of [S. Daryanoosh et al., Nat. Commun. ${\bf 9}$, 4606 (2018)] showed how to achieve the ultimate precision limit, the exact Heisenberg limit (HL), in ab-initio phase estimation with $N=3$ photon-passes, using an entangled biphoton state in combination with particular measurement techniques. The advantage of the HL over the SNL increases with the number of resources used. Here we present, and implement experimentally, a scheme for generation of the optimal $N=7$ triphoton state. We study experimentally and theoretically the generated state quality and its potential for phase estimation. We show that the expected usefulness of the prepared triphoton state for HL phase estimation is significantly degraded by even quite small experimental imperfections, such as optical mode mismatch and unwanted higher-order multi-photon terms in the states produced in parametric down-conversion., Comment: 16 pages, 10 figures (unable to contact one of the authors; revised the author list and acknowledgement)

Bell nonlocality refers to correlations between two distant, entangled particles that challenge classical notions of local causality. Beyond its foundational significance, nonlocality is crucial for device-independent technologies like quantum key distribution and randomness generation. Nonlocality quickly deteriorates in the presence of noise, and restoring nonlocal correlations requires additional resources. These often come in the form of many instances of the input state and joint measurements, incurring a significant resource overhead. Here, we experimentally demonstrate that single copies of Bell-local states, incapable of violating any standard Bell inequality, can give rise to nonlocality after being embedded into a quantum network of multiple parties. We subject the initial entangled state to a quantum channel that broadcasts part of the state to two independent receivers and certify the nonlocality in the resulting network by violating a tailored Bell-like inequality. We obtain these results without making any assumptions about the prepared states, the quantum channel, or the validity of quantum theory. Our findings have fundamental implications for nonlocality and enable the practical use of nonlocal correlations in real-world applications, even in scenarios dominated by noise., Comment: 14 pages, 6 figures. Final version published in Nature Communications

The distribution and verification of quantum nonlocality across a network of users is essential for future quantum information science and technology applications. However, beyond simple point-to-point protocols, existing methods struggle with increasingly complex state preparation for a growing number of parties. Here, we show that, surprisingly, multiparty loophole-free quantum steering, where one party simultaneously steers arbitrarily many spatially separate parties, is achievable by constructing a quantum network from a set of qubits of which only one pair is entangled. Using these insights, we experimentally demonstrate this type of steering between three parties with the detection loophole closed. With its modest and fixed entanglement requirements, this work introduces a scalable approach to rigorously verify quantum nonlocality across multiple parties, thus providing a practical tool towards developing the future quantum internet.

The ultimate random number generators are those certified to be unpredictable -- including to an adversary. The use of simple quantum processes promises to provide numbers that no physical observer could predict but, in practice, unwanted noise and imperfect devices can compromise fundamental randomness and protocol security. Certified randomness protocols have been developed which remove the need for trust in devices by taking advantage of nonlocality. Here, we use a photonic platform to implement our protocol, which operates in the quantum steering scenario where one can certify randomness in a one-sided device independent framework. We demonstrate an approach for a steering-based generator of public or private randomness, and the first generation of certified random bits, with the detection loophole closed, in the steering scenario., Comment: 7 pages, 3 figures

Violating a nonlocality inequality enables the most powerful remote quantum information tasks and fundamental tests of quantum physics. Loophole-free photonic verification of nonlocality has been achieved with polarization-entangled photon pairs, but not with states entangled in other degrees of freedom. Here we demonstrate completion of the quantum steering nonlocality task, with the detection loophole closed, when entanglement is distributed by transmitting a photon in an optical vector vortex state, formed by optical orbital angular momentum (OAM) and polarization. As well as opening up a high-efficiency encoding beyond polarization, the critically-important demonstration of vector vortex steering opens the door to new free-space and satellite-based secure quantum communication devices and device-independent protocols., Comment: 7 pages, 3 figures

Photons have been a flagship system for studying quantum mechanics, advancing quantum information science, and developing quantum technologies. Quantum entanglement, teleportation, quantum key distribution and early quantum computing demonstrations were pioneered in this technology because photons represent a naturally mobile and low-noise system with quantum-limited detection readily available. The quantum states of individual photons can be manipulated with very high precision using interferometry, an experimental staple that has been under continuous development since the 19th century. The complexity of photonic quantum computing device and protocol realizations has raced ahead as both underlying technologies and theoretical schemes have continued to develop. Today, photonic quantum computing represents an exciting path to medium- and large-scale processing. It promises to out aside its reputation for requiring excessive resource overheads due to inefficient two-qubit gates. Instead, the ability to generate large numbers of photons---and the development of integrated platforms, improved sources and detectors, novel noise-tolerant theoretical approaches, and more---have solidified it as a leading contender for both quantum information processing and quantum networking. Our concise review provides a flyover of some key aspects of the field, with a focus on experiment. Apart from being a short and accessible introduction, its many references to in-depth articles and longer specialist reviews serve as a launching point for deeper study of the field., Comment: 21 pages, 3 figures. A brief review on some topics in photonic quantum computing with lots of references to longer specialist reviews. Close to published version

Does quantum theory apply at all scales, including that of observers? New light on this fundamental question has recently been shed through a resurgence of interest in the long-standing Wigner's friend paradox. This is a thought experiment addressing the quantum measurement problem -- the difficulty of reconciling the (unitary, deterministic) evolution of isolated systems and the (non-unitary, probabilistic) state update after a measurement. Here, by building on a scenario with two separated but entangled friends introduced by Brukner, we prove that if quantum evolution is controllable on the scale of an observer, then one of 'No-Superdeterminism', 'Locality' or 'Absoluteness of Observed Events' -- that every observed event exists absolutely, not relatively -- must be false. We show that although the violation of Bell-type inequalities in such scenarios is not in general sufficient to demonstrate the contradiction between those three assumptions, new inequalities can be derived in a theory-independent manner, that are violated by quantum correlations. This is demonstrated in a proof-of-principle experiment where a photon's path is deemed an observer. We discuss how this new theorem places strictly stronger constraints on physical reality than Bell's theorem., Comment: 15 pages, 6 figures. In v4, we updated the article to the published version, as it appears in Nat. Phys

Simulations of stochastic processes play an important role in the quantitative sciences, enabling the characterisation of complex systems. Recent work has established a quantum advantage in stochastic simulation, leading to quantum devices that execute a simulation using less memory than possible by classical means. To realise this advantage it is essential that the memory register remains coherent, and coherently interacts with the processor, allowing the simulator to operate over many time steps. Here we report a multi-time-step experimental simulation of a stochastic process using less memory than the classical limit. A key feature of the photonic quantum information processor is that it creates a quantum superposition of all possible future trajectories that the system can evolve into. This superposition allows us to introduce, and demonstrate, the idea of comparing statistical futures of two classical processes via quantum interference. We demonstrate interference of two 16-dimensional quantum states, representing statistical futures of our process, with a visibility of 0.96 $\pm$ 0.02., Comment: 9 pages, 5 figures

Stochastic processes underlie a vast range of natural and social phenomena. Some processes such as atomic decay feature intrinsic randomness, whereas other complex processes, e.g. traffic congestion, are effectively probabilistic because we cannot track all relevant variables. To simulate a stochastic system's future behaviour, information about its past must be stored and thus memory is a key resource. Quantum information processing promises a memory advantage for stochastic simulation that has been validated in recent proof-of-concept experiments. Yet, in all past works, the memory saving would only become accessible in the limit of a large number of parallel simulations, because the memory registers of individual quantum simulators had the same dimensionality as their classical counterparts. Here, we report the first experimental demonstration that a quantum stochastic simulator can encode the relevant information in fewer dimensions than any classical simulator, thereby achieving a quantum memory advantage even for an individual simulator. Our photonic experiment thus establishes the potential of a new, practical resource saving in the simulation of complex systems.

Finding exponential separation between quantum and classical information tasks is like striking gold in quantum information research. Such an advantage is believed to hold for quantum computing but is proven for quantum communication complexity. Recently, a novel quantum resource called the quantum switch---which creates a coherent superposition of the causal order of events, known as quantum causality---has been harnessed theoretically in a new protocol providing provable exponential separation. We experimentally demonstrate such an advantage by realizing a superposition of communication directions for a two-party distributed computation. Our photonic demonstration employs $d$-dimensional quantum systems, qudits, up to $d=2^{13}$ dimensions and demonstrates a communication complexity advantage, requiring less than $(0.696 \pm 0.006)$ times the communication of any causally ordered protocol. These results elucidate the crucial role of the coherence of communication direction in achieving the exponential separation for the one-way processing task, and open a new path for experimentally exploring the fundamentals and applications of advanced features of indefinite causal structures., Comment: Accepted by Phys. Rev. Lett

With quantum resources a precious commodity, their efficient use is highly desirable. Quantum autoencoders have been proposed as a way to reduce quantum memory requirements. Generally, an autoencoder is a device that uses machine learning to compress inputs, that is, to represent the input data in a lower-dimensional space. Here, we experimentally realize a quantum autoencoder, which learns how to compress quantum data using a classical optimization routine. We demonstrate that when the inherent structure of the data set allows lossless compression, our autoencoder reduces qutrits to qubits with low error levels. We also show that the device is able to perform with minimal prior information about the quantum data or physical system and is robust to perturbations during its optimization routine., Comment: Supplementary information included in the document

The set of all qubit states that can be steered to by measurements on a correlated qubit is predicted to form an ellipsoid---called the quantum steering ellipsoid---in the Bloch ball. This ellipsoid provides a simple visual characterisation of the initial 2-qubit state, and various aspects of entanglement are reflected in its geometric properties. We experimentally verify these properties via measurements on many different polarisation-entangled photonic qubit states. Moreover, for pure 3-qubit states, the volumes of the two quantum steering ellipsoids generated by measurements on the first qubit are predicted to satisfy a tight monogamy relation, which is strictly stronger than the well-known monogamy of entanglement for concurrence. We experimentally verify these predictions, using polarisation and path entanglement. We also show experimentally that this monogamy relation can be violated by a mixed entangled state, which nevertheless satisfies a weaker monogamy relation.

There has been a concerted effort to identify problems computable with quantum technology which are intractable with classical technology or require far fewer resources to compute. Recently, randomness processing in a Bernoulli factory has been identified as one such task. Here, we report two quantum photonic implementations of a Bernoulli factory, one utilising quantum coherence and single-qubit measurements and the other which uses quantum coherence and entangling measurements of two qubits. We show that the former consumes three orders of magnitude fewer resources than the best known classical method, while entanglement offers a further five-fold reduction. These concepts may provide a means for quantum enhanced-performance in the simulation of stochastic processes and sampling tasks., Comment: 7 pages, 5 figures, and 4 pages of supplementary material

Einstein-Podolsky-Rosen steering is a quantum phenomenon wherein one party influences, or steers, the state of a distant party's particle beyond what could be achieved with a separable state, by making measurements on one half of an entangled state. This type of quantum nonlocality stands out through its asymmetric setting, and even allows for cases where one party can steer the other, but where the reverse is not true. A series of experiments have demonstrated one-way steering in the past, but all were based on significant limiting assumptions. These consisted either of restrictions on the type of allowed measurements, or of assumptions about the quantum state at hand, by mapping to a specific family of states and analysing the ideal target state rather than the real experimental state. Here, we present the first experimental demonstration of one-way steering free of such assumptions. We achieve this using a new sufficient condition for non-steerability, and, although not required by our analysis, using a novel source of extremely high-quality photonic Werner states., Comment: Supplemental Material included in the document

A Bell test is a randomized trial that compares experimental observations against the philosophical worldview of local realism. A Bell test requires spatially distributed entanglement, fast and high-efficiency detection and unpredictable measurement settings. Although technology can satisfy the first two of these requirements, the use of physical devices to choose settings in a Bell test involves making assumptions about the physics that one aims to test. Bell himself noted this weakness in using physical setting choices and argued that human `free will' could be used rigorously to ensure unpredictability in Bell tests. Here we report a set of local-realism tests using human choices, which avoids assumptions about predictability in physics. We recruited about 100,000 human participants to play an online video game that incentivizes fast, sustained input of unpredictable selections and illustrates Bell-test methodology. The participants generated 97,347,490 binary choices, which were directed via a scalable web platform to 12 laboratories on five continents, where 13 experiments tested local realism using photons, single atoms, atomic ensembles, and superconducting devices. Over a 12-hour period on 30 November 2016, participants worldwide provided a sustained data flow of over 1,000 bits per second to the experiments, which used different human-generated data to choose each measurement setting. The observed correlations strongly contradict local realism and other realistic positions in bipartite and tripartite scenarios. Project outcomes include closing the `freedom-of-choice loophole' (the possibility that the setting choices are influenced by `hidden variables' to correlate with the particle properties), the utilization of video-game methods for rapid collection of human generated randomness, and the use of networking techniques for global participation in experimental science., Comment: This version includes minor changes resulting from reviewer and editorial input. Abstract shortened to fit within arXiv limits

The use of quantum resources can provide measurement precision beyond the shot-noise limit (SNL). The task of ab initio optical phase measurement---the estimation of a completely unknown phase---has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able---even in principle---to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme uses two photonic qubits, one double passed, so that, for a successful coincidence detection, the number of photon-passes is $N=3$. We achieve a precision that is within $4\%$ of the HL, surpassing the best precision theoretically achievable with simpler techniques with $N=3$. This work represents a fundamental achievement of the ultimate limits of metrology, and the scheme can be extended to higher $N$ and other physical systems., Comment: (12 pages, 6 figures), typos corrected

We derive and experimentally investigate a strong uncertainty relation valid for any $n$ unitary operators, which implies the standard uncertainty relation as a special case, and which can be written in terms of geometric phases. It is saturated by every pure state of any $n$-dimensional quantum system, generates a tight overlap uncertainty relation for the transition probabilities of any $n+1$ pure states, and gives an upper bound for the out-of-time-order correlation function. We test these uncertainty relations experimentally for photonic polarisation qubits, including the minimum uncertainty states of the overlap uncertainty relation, via interferometric measurements of generalised geometric phases., Comment: 5 pages of main text, 5 pages of Supplemental Material. Clarifications added in this updated version

How much information do we need about a process' past to faithfully simulate its future? The statistical complexity is a prominent quantifier of structure for stochastic processes. Quantum machines, however, can simulate classical stochastic processes while storing significantly less information than their optimal classical counterparts. This implies qualitative divergences between classical and quantum statistical complexity. Here, we develop error-tolerant techniques to witness such divergences, enabling us to account for the inevitable imperfections in realising quantum stochastic simulators with present-day quantum technology. We apply these tools to experimentally verify the quantum memory advantage in simulating an Ising spin chain, even when accounting for experimental distortion. This then leads us to observe a recently conjectured effect, the ambiguity of simplicity$\unicode{x2013}$the notion that the relative complexity of two different processes can depend on whether we model the process using classical or quantum means of information processing., Comment: 9 pages, 6 figures

Einstein-Podolsky-Rosen steering refers to the quantum phenomenon whereby the state of a system held by one party can be "steered" into different states at the will of another, distant, party by performing different local measurements. Although steering has been demonstrated in a number of experiments involving qubits, the question of which two-qubit states are steerable remains an open theoretical problem. Here, we derive a necessary condition for any two-qubit state to be steerable when the steering party suffers from a given probability of qubit loss. Our main result finds application in one-way steering demonstrations that rely upon loss. Specifically, we apply it to a recent experiment on one-way steering with projective measurements and POVMs, reported by Wollmann et. al. [Phys. Rev. Lett., 116, 160403 (2016)]., Comment: 7 pages, 1 figure

Interferometric phase measurement is widely used to precisely determine quantities such as length, speed, and material properties. Without quantum correlations, the best phase sensitivity $\Delta\varphi$ achievable using $n$ photons is the shot noise limit (SNL), $\Delta\varphi=1/\sqrt{n}$. Quantum-enhanced metrology promises better sensitivity, but despite theoretical proposals stretching back decades, no measurement using photonic (i.e. definite photon number) quantum states has truly surpassed the SNL. Rather, all such demonstrations --- by discounting photon loss, detector inefficiency, or other imperfections --- have considered only a subset of the photons used. Here, we use an ultra-high efficiency photon source and detectors to perform unconditional entanglement-enhanced photonic interferometry. Sampling a birefringent phase shift, we demonstrate precision beyond the SNL without artificially correcting our results for loss and imperfections. Our results enable quantum-enhanced phase measurements at low photon flux and open the door to the next generation of optical quantum metrology advances., Comment: 5 pages,3 figures