Publications
274 results found
Self CN, Khosla KE, Smith AWR, et al., 2021, Variational quantum algorithm with information sharing, npj Quantum Information, Vol: 7, ISSN: 2056-6387
We introduce an optimisation method for variational quantum algorithms and experimentally demonstrate a 100-fold improvement in efficiency compared to naive implementations. The effectiveness of our approach is shown by obtaining multi-dimensional energy surfaces for small molecules and a spin model. Our method solves related variational problems in parallel by exploiting the global nature of Bayesian optimisation and sharing information between different optimisers. Parallelisation makes our method ideally suited to the next generation of variational problems with many physical degrees of freedom. This addresses a key challenge in scaling-up quantum algorithms towards demonstrating quantum advantage for problems of real-world interest.
Smith AWR, Gray J, Kim MS, 2021, Efficient quantum state sample tomography with basis-dependent neural networks, PRX Quantum, Vol: 2, Pages: 1-15, ISSN: 2691-3399
We use a metalearning neural-network approach to analyze data from a measured quantum state. Once our neural network has been trained, it can be used to efficiently sample measurements of the state in measurement bases not contained in the training data. These samples can be used to calculate expectation values and other useful quantities. We refer to this process as “state sample tomography.” We encode the state’s measurement outcome distributions using an efficiently parameterized generative neural network. This allows each stage in the tomography process to be performed efficiently even for large systems. Our scheme is demonstrated on recent IBM Quantum devices, producing a model for a six-qubit state’s measurement outcomes with a predictive accuracy (classical fidelity) greater than 95% for all test cases using only 100 random measurement settings as opposed to the 729 settings required for standard full tomography using local measurements. This reduction in the required number of measurements scales favorably, with training data in 200 measurement settings, yielding a predictive accuracy greater than 92% for a ten-qubit state where 59 049 settings are typically required for full local measurement-based quantum state tomography. A reduction in the number of measurements by a factor, in this case, of almost 600 could allow for estimations of expectation values and state fidelities in practicable times on current quantum devices.
Toros M, Van De Kamp TW, Marshman RJ, et al., 2021, Relative acceleration noise mitigation for nanocrystal matter-wave interferometry: Applications to entangling masses via quantum gravity, Physical Review Special Topics: Physics Education Research, Vol: 3, Pages: 1-14, ISSN: 1554-9178
Matter-wave interferometers with large momentum transfers, irrespective of specific implementations, will face a universal dephasing due to relative accelerations between the interferometric mass and the associated apparatus. Here we propose a solution that works even without actively tracking the relative accelerations: putting both the interfering mass and its associated apparatus in a freely falling capsule, so that the strongest inertial noise components vanish due to the equivalence principle. In this setting, we investigate two of the most important remaining noise sources: (a) the noninertial jitter of the experimental setup and (b) the gravity-gradient noise. We show that the former can be reduced below desired values by appropriate pressures and temperatures, while the latter can be fully mitigated in a controlled environment. We finally apply the analysis to a recent proposal for testing the quantum nature of gravity [S. Bose et al., Phys. Rev. Lett. 119, 240401 (2017)] through the entanglement of two masses undergoing interferometry. We show that the relevant entanglement witnessing is feasible with achievable levels of relative acceleration noise.
Im D-G, Lee C-H, Kim Y, et al., 2021, Optimal teleportation via noisy quantum channels without additional qubit resources, npj Quantum Information, Vol: 7, Pages: 1-7, ISSN: 2056-6387
Quantum teleportation exemplifies how the transmission of quantum information starkly differs from that of classical information and serves as a key protocol for quantum communication and quantum computing. While an ideal teleportation protocol requires noiseless quantum channels to share a pure maximally entangled state, the reality is that shared entanglement is often severely degraded due to various decoherence mechanisms. Although the quantum noise induced by the decoherence is indeed a major obstacle to realizing a near-term quantum network or processor with a limited number of qubits, the methodologies considered thus far to address this issue are resource-intensive. Here, we demonstrate a protocol that allows optimal quantum teleportation via noisy quantum channels without additional qubit resources. By analyzing teleportation in the framework of generalized quantum measurement, we optimize the teleportation protocol for noisy quantum channels. In particular, we experimentally demonstrate that our protocol enables to teleport an unknown qubit even via a single copy of an entangled state under strong decoherence that would otherwise preclude any quantum operation. Our work provides a useful methodology for practically coping with decoherence with a limited number of qubits and paves the way for realizing noisy intermediate-scale quantum computing and quantum communication.
Thekkadath GS, Sempere-Llagostera S, Bell BA, et al., 2021, Single-shot discrimination of coherent states beyond the standard quantum limit, OPTICS LETTERS, Vol: 46, Pages: 2565-2568, ISSN: 0146-9592
- Author Web Link
- Cite
- Citations: 2
Chevalier H, Paige AJ, Kwon H, et al., 2021, Violating the Leggett-Garg inequalities with classical light, Physical Review A: Atomic, Molecular and Optical Physics, Vol: 103, Pages: 1-9, ISSN: 1050-2947
In an endeavor to better define the distinction between classical macroscopic and quantum microscopic regimes, the Leggett-Garg inequalities were established as a test of macroscopic-realistic theories, which are commonly thought to be a suitable class of descriptions for classical dynamics. The relationship between their violation and nonclassicality is however not obvious. We show that classical states of light, which in the quantum optical sense are any convex sums of coherent states, may not satisfy the Leggett-Garg inequalities. After introducing a simple Mach-Zehnder setup and showing how to obtain a violation with a single photon using negative measurements, we focus on classical states of light, in particular those of low average photon number. We demonstrate how one can still perform negative measurements with an appropriate assignment of variables, and show that the inequalities are violable with coherent states. Finally, we abandon the initial phase reference and demonstrate that the violation is still possible, in particular with thermal states of light, and we investigate the effect of intermediate dephasing.
Lie SH, Kwon H, Kim MS, et al., 2021, Quantum one-time tables for unconditionally secure qubit- commitment, Quantum, Vol: 5, Pages: 1-17, ISSN: 2521-327X
The commodity-based cryptography is an alternative approach to realize conventionally impossible cryptographic primitives such as unconditionally secure bit-commitment by consuming pre-established correlation between distrustful participants. A unit of such classical correlation is known as the one-time table (OTT). In this paper, we introduce a new example besides quantum key distribution in which quantum correlation is useful for cryptography. We propose a scheme for unconditionally secure qubit-commitment, a quantum cryptographic primitive forbidden by the recently proven no-masking theorem in the standard model, based on the consumption of the quantum generalization of the OTT, the bipartite quantum state we named quantum one-time tables (QOTT). The construction of the QOTT is based on the newly analyzed internal structure of quantum masker and the quantum secret sharing schemes. Our qubit-commitment scheme is shown to be universally composable. We propose to measure the randomness cost of preparing a (Q)OTT in terms of its entropy, and show that the QOTT with superdense coding can increase the security level with half the cost of OTTs for unconditionally secure bit-commitment. The QOTT exemplifies an operational setting where neither maximally classically correlated state nor maximally entangled state, but rather a well-structured partially entangled mixed state is more valuable resource.
Kim MS, 2021, A massive test, NATURE PHYSICS, Vol: 17, Pages: 14-15, ISSN: 1745-2473
Martinetz L, Hornberger K, Millen J, et al., 2020, Quantum electromechanics with levitated nanoparticles, npj Quantum Information, Vol: 6, ISSN: 2056-6387
Preparing and observing quantum states of nanoscale particles is a challenging task with great relevance for quantum technologies and tests of fundamental physics. In contrast to atomic systems with discrete transitions, nanoparticles exhibit a practically continuous absorption spectrum and thus their quantum dynamics cannot be easily manipulated. Here, we demonstrate that charged nanoscale dielectrics can be artificially endowed with a discrete level structure by coherently interfacing their rotational and translational motion with a superconducting qubit. We propose a pulsed scheme for the generation and read-out of motional quantum superpositions and entanglement between several levitated nanoparticles, providing an all-electric platform for networked hybrid quantum devices.
Paige AJ, Kwon H, Simsek S, et al., 2020, Quantum delocalized interactions, Physical Review Letters, Vol: 125, Pages: 240406 – 1-240406 – 6, ISSN: 0031-9007
Classical mechanics obeys the intuitive logic that a physical event happens at a definite spatial point. Entanglement, however, breaks this logic by enabling interactions without a specific location. In this work we study these delocalized interactions. These are quantum interactions that create less locational information than would be possible classically, as captured by the disturbance induced on some spatial superposition state. We introduce quantum games to capture the effect and demonstrate a direct operational use for quantum concurrence in that it bounds the nonclassical performance gain. We also find a connection with quantum teleportation, and demonstrate the games using an IBM quantum processor.
Oh C, Kwon H, Jiang L, et al., 2020, Field-gradient measurement using a Stern-Gerlach atomic interferometer with butterfly geometry, Physical Review A: Atomic, Molecular and Optical Physics, Vol: 102, Pages: 053321 – 1-053321 – 8, ISSN: 1050-2947
Atomic interferometers have been studied as a promising device for precise sensing of external fields. Among various configurations, a particular configuration with a butterfly-shaped geometry has been designed to sensitively probe field gradients. We introduce a Stern-Gerlach (SG) butterfly interferometer by incorporating magnetic field in the conventional butterfly-shaped configuration. Atomic trajectories of the interferometer can be flexibly adjusted by controlling magnetic fields to increase the sensitivity of the interferometer, while the conventional butterfly interferometer using Raman transitions can be understood as a special case. We also show that the SG interferometer can keep high contrast against a misalignment in position and momentum caused by the field gradient.
Nehra R, Eaton M, Gonzalez-Arciniegas C, et al., 2020, Generalized overlap quantum state tomography, PHYSICAL REVIEW RESEARCH, Vol: 2
- Author Web Link
- Cite
- Citations: 6
Kwon H, Paige AJ, Kim MS, 2020, Condition on the Rényi entanglement entropy under stochastic local manipulation, Physical Review Letters, Vol: 125, Pages: 100502 – 1-100502 – 7, ISSN: 0031-9007
The Rényi entanglement entropy (REE) is an entanglement quantifier considered as a natural generalization of the entanglement entropy. When it comes to stochastic local operations and classical communication (SLOCC), however, only a limited class of the REEs satisfy the monotonicity condition, while their statistical properties beyond mean values have not been fully investigated. Here, we establish a general condition that the probability distribution of the REE of any order obeys under SLOCC. The condition is obtained by introducing a family of entanglement monotones that contain the higher-order moments of the REEs. The contribution from the higher-order moments imposes a strict limitation on entanglement distillation via SLOCC. We find that the upper bound on success probabilities for entanglement distillation exponentially decreases as the amount of raised entanglement increases, which cannot be captured from the monotonicity of the REE. Based on the strong restriction on entanglement transformation under SLOCC, we design a new method to estimate entanglement in quantum many-body systems from experimentally observable quantities.
Ma Y, Khosla K, Stickler B, et al., 2020, Quantum persistent tennis racket dynamics of nanorotors, Physical Review Letters, Vol: 125, ISSN: 0031-9007
Classical rotations of asymmetric rigid bodies are unstable around the axis of intermediate moment of inertia, causing a flipping of rotor orientation. This effect, known as the tennis racket effect, quickly averages to zero in classical ensembles since the flipping period varies significantly upon approaching the separatrix. Here, we explore the quantum rotations of rapidly spinning thermal asymmetric nanorotors and show that classically forbidden tunnelling gives rise to persistent tennis racket dynamics, in stark contrast to the classical expectation. We characterise this effect, demonstrating that quantum coherent flipping dynamics can persist even in the regime where millions of angular momentum states are occupied. This persistent flipping offers a promising route for observing and exploiting quantum effects in rotational degrees of freedom for molecules and nanoparticles.
Clarke J, Sahium P, Khosla K, et al., 2020, Generating mechanical and optomechanical entanglement via pulsed interaction and measurement, New Journal of Physics, Vol: 22, Pages: 1-32, ISSN: 1367-2630
Entanglement generation at a macroscopic scale o ers an exciting avenue to de-velop new quantum technologies and study fundamental physics on a tabletop.Cavity quantum optomechanics provides an ideal platform to generate and exploitsuch phenomena owing to the precision of quantum optics combined with recent ex-perimental advances in optomechanical devices. In this work, we propose schemesoperating outside the resolved-sideband regime, to prepare and verify both optical-mechanical and mechanical-mechanical entanglement. Our schemes employ pulsedinteractions with a duration much less than the mechanical period and, togetherwith homodyne measurements, can both generate and characterize these types ofentanglement. To improve the performance of our schemes, a precooling stagecomprising prior pulses can be utilized to increase the amount of entanglementprepared, and local optical squeezers may be used to provide resilience againstopen-system dynamics. The entanglement generated by our schemes is quanti edusing the logarithmic negativity and is analysed with respect to the strength of thepulsed optomechanical interactions for realistic experimental scenarios includingmechanical decoherence and optical loss. Two separate schemes for mechanicalentanglement generation are introduced and compared: one scheme based on anoptical interferometric design, and the other comprising sequential optomechani-cal interactions. The pulsed nature of our protocols provides more direct access tothese quantum correlations in the time domain, with applications including quan-tum metrology and tests of quantum decoherence. By considering a parameter setbased on recent experiments, the feasibility to generate signi cant entanglementwith our schemes, even with large optical losses, is demonstrated.
Ma Y, Armata F, Khosla KE, et al., 2020, Optical squeezing for an optomechanical system without quantizing the mechanical motion, Physical Review & Research International, Vol: 2, ISSN: 2231-1815
Witnessing quantumness in mesoscopic objects is an important milestone for both quantum technologies and foundational reasons. Cavity optomechanics offers the ideal system to achieve this by combing high-precision optical measurements with mechanical oscillators. However, mechanical quantumness can only be established if the behavior is incompatible with any classical description of an oscillator. After explicitly considering classical and hybrid quantum-classical descriptions of an optomechanical system, we rule out squeezing of the optical field as such a witness by showing it is also predicted without quantizing the mechanical oscillator.
Paige AJ, Plato ADK, Kim MS, 2020, Classical and nonclassical time dilation for quantum clocks, Physical Review Letters, Vol: 124, Pages: 160602-1-160602-6, ISSN: 0031-9007
Proper time, ideal clocks, and boosts are well understood classically, but subtleties arise in quantum physics. We show that quantum clocks set in motion via momentum boosts do not witness classical time dilation. However, using velocity boosts we find the ideal behavior in both cases, where the quantum clock and classical observer are set in motion. Without internal state-dependent forces additional effects arise. As such, we derive observed frequency shifts in ion trap atomic clocks, indicating a small additional shift, and also show the emergence of nonideal behavior in a theoretical clock model.
Jin X-M, Kim MS, Smith BJ, 2019, Quantum photonics: feature introduction, PHOTONICS RESEARCH, Vol: 7, Pages: QP1-QP2, ISSN: 2327-9125
- Author Web Link
- Cite
- Citations: 2
Smith A, Kim MS, Pollmann F, et al., 2019, Simulating quantum many-body dynamics on a current digital quantum computer, npj Quantum Information, Vol: 5, Pages: 1-13, ISSN: 2056-6387
Universal quantum computers are potentially an ideal setting for simulating many-body quantum dynamics that is out of reach for classical digital computers. We use state-of-the-art IBM quantum computers to study paradigmatic examples of condensed matter physics—we simulate the effects of disorder and interactions on quantum particle transport, as well as correlation and entanglement spreading. Our benchmark results show that the quality of the current machines is below what is necessary for quantitatively accurate continuous-time dynamics of observables and reachable system sizes are small comparable to exact diagonalization. Despite this, we are successfully able to demonstrate clear qualitative behaviour associated with localization physics and many-body interaction effects.
Aqua Z, Kim MS, Dayan B, 2019, Generation of optical Fock and W states with single-atom-based bright quantum scissors, Photonics Research, Vol: 7, Pages: A45-A45, ISSN: 2327-9125
We introduce a multi-step protocol for optical quantum state engineering that performs as “bright quantum scissors,” namely truncating an arbitrary input quantum state to have at least a certain number of photons. The protocol exploits single-photon pulses and is based on the effect of single-photon Raman interaction, which is implemented with a single three-level Λ system (e.g., a single atom) Purcell-enhanced by a single-sided cavity. A single step of the protocol realizes the inverse of the bosonic annihilation operator. Multiple iterations of the protocol can be used to deterministically generate a chain of single photons in a W state. Alternatively, upon appropriate heralding, the protocol can be used to generate Fock-state optical pulses. This protocol could serve as a useful and versatile building block for the generation of advanced optical quantum states that are vital for quantum communication, distributed quantum information processing, and all-optical quantum computing.
Paige A, Yadin B, Kim M, 2019, Quantum correlations for anonymous metrology, Quantum, Vol: 3, Pages: 1-14, ISSN: 2521-327X
We introduce the task of anonymous metrology, in which a physical parameter of an object may be determined without revealing the object's location. Alice and Bob share a correlated quantum state, with which one of them probes the object. Upon receipt of the quantum state, Charlie is then able to estimate the parameter without knowing who possesses the object. We show that quantum correlations are resources for this task when Alice and Bob do not trust the devices in their labs. The anonymous metrology protocol moreover distinguishes different kinds of quantum correlations according to the level of desired security: discord is needed when the source of states is trustworthy, otherwise entanglement is necessary.
Kwon H, Kim M, 2019, Fluctuation theorems for a quantum channel, Physical Review X, Vol: 9, ISSN: 2160-3308
We establish the general framework of quantum fluctuation theorems by finding the symmetry between the forward and backward transitions of any given quantum channel. The Petz recovery map is adopted as the reverse quantum channel, and the notion of entropy production in thermodynamics is extended to the quantum regime. Our result shows that the fluctuation theorems, which are normally considered for thermodynamic processes, can be a powerful tool to study the detailed statistics of quantum systems as well as the effect of coherence transfer in an arbitrary non-equilibrium quantum process. We introduce a complex-valued entropy production to fully understand the relation between the forward and backward processes through the quantum channel. We find the physical meaning of the imaginary part of entropy production to witness the broken symmetry of the quantum channel. We also show that the imaginary part plays a crucial role in deriving the second law from the quantum fluctuation theorem. The dissipation and fluctuation of various quantum resources including quantum free energy, asymmetry and entanglement can be coherently understood in our unified framework. Our fluctuation theorem can be applied to a wide range of physical systems and dynamics to query the reversibility of a quantum state for the given quantum processing channel involving coherence and entanglement.
Milz S, Kim MS, Pollock FA, et al., 2019, Completely positive divisibility does not mean markovianity, Physical Review Letters, Vol: 123, Pages: 040401-1-040401-6, ISSN: 0031-9007
In the classical domain, it is well known that divisibility does not imply that a stochastic process is Markovian. However, for quantum processes, divisibility is often considered to be synonymous with Markovianity. We show that completely positive divisible quantum processes can still involve non-Markovian temporal correlations, that we then fully classify using the recently developed process tensor formalism, which generalizes the theory of stochastic processes to the quantum domain.
Zhang A, Xu H, Xie J, et al., 2019, Experimental test of contextuality in quantum and classical systems, Physical Review Letters, Vol: 122, ISSN: 0031-9007
Contextuality is considered as an intrinsic signature of nonclassicality and a crucial resource for achieving unique advantages of quantum information processing. However, recently, there have been debates on whether classical fields may also demonstrate contextuality. Here, we experimentally configure a contextuality test for optical fields, adopting various definitions of measurement events, and analyze how the definitions affect the emergence of nonclassical correlations. The heralded single-photon state, which is a typical nonclassical light field, manifests contextuality in our setup; whereas contextuality for classical coherent fields strongly depends on the specific definition of measurement events, which is equivalent to filtering the nonclassical component of the input state. Our results highlight the importance of the definition of measurement events to demonstrate contextuality, and they link the contextual correlations to nonclassicality defined by quasiprobabilities in phase space.
Banchi L, Kolthammer WS, Kim MS, 2018, Multiphoton tomography with linear optics and photon counting, Physical Review Letters, Vol: 121, ISSN: 0031-9007
Determining an unknown quantum state from an ensemble of identical systems is a fundamental, yet experimentally demanding, task in quantum science. Here we study the number of measurement bases needed to fully characterize an arbitrary multimode state containing a definite number of photons, or an arbitrary mixture of such states. We show this task can be achieved using only linear optics and photon counting, which yield a practical though nonuniversal set of projective measurements. We derive the minimum number of measurement settings required and numerically show that this lower bound is saturated with random linear optics configurations, such as when the corresponding unitary transformation is Haar random. Furthermore, we show that for N photons, any unitary 2N design can be used to derive an analytical, though nonoptimal, state reconstruction protocol.
Yadin B, Binder FC, Thompson J, et al., 2018, Operational resource theory of continuous-variable nonclassicality, Physical Review X, Vol: 8, ISSN: 2160-3308
Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P representation—a state lacking a positive P function is said to be nonclassical. In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed forward. Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements. These measures lead to fundamental constraints on the manipulation of nonclassicality, especially its concentration into subsystems, that apply to generic multimode non-Gaussian states. Special cases of our framework include no-go results in the concentration of squeezing and a complete hierarchy of nonclassicality for single-mode Gaussian states.
Tang H, Di Franco C, Shi ZY, et al., 2018, Experimental quantum fast hitting on hexagonal graphs, Nature Photonics, Vol: 12, Pages: 754-758, ISSN: 1749-4885
Quantum walks are powerful kernels in quantum computing protocols, and possess strong capabilities in speeding up various simulation and optimization tasks. One striking example is provided by quantum walkers evolving on glued trees1, which demonstrate faster hitting performances than classical random walks. However, their experimental implementation is challenging, as this involves highly complex arrangements of an exponentially increasing number of nodes. Here, we propose an alternative structure with a polynomially increasing number of nodes. We successfully map such graphs on quantum photonic chips using femtosecond-laser direct writing techniques in a geometrically scalable fashion. We experimentally demonstrate quantum fast hitting by implementing two-dimensional quantum walks on graphs with up to 160 nodes and a depth of eight layers, achieving a linear relationship between the optimal hitting time and the network depth. Our results open up a scalable path towards quantum speed-up in classically intractable complex problems.
Lubasch M, Valido AA, Renema JJ, et al., 2018, Tensor network states in time-bin quantum optics, Physical Review A, Vol: 97, ISSN: 1050-2947
The current shift in the quantum optics community towards experiments with many modes and photons necessitates new classical simulation techniques that efficiently encode many-body quantum correlations and go beyond the usual phase-space formulation. To address this pressing demand we formulate linear quantum optics in the language of tensor network states. We extensively analyze the quantum and classical correlations of time-bin interference in a single fiber loop. We then generalize our results to more complex time-bin quantum setups and identify different classes of architectures for high-complexity and low-overhead boson sampling experiments.
Kwon H, Jeong H, Jennings D, et al., 2018, Clock-work trade-off relation for coherence in quantum thermodynamics, Physical Review Letters, Vol: 120, ISSN: 0031-9007
In thermodynamics, quantum coherences—superpositions between energy eigenstates—behave in distinctly nonclassical ways. Here we describe how thermodynamic coherence splits into two kinds—“internal” coherence that admits an energetic value in terms of thermodynamic work, and “external” coherence that does not have energetic value, but instead corresponds to the functioning of the system as a quantum clock. For the latter form of coherence, we provide dynamical constraints that relate to quantum metrology and macroscopicity, while for the former, we show that quantum states exist that have finite internal coherence yet with zero deterministic work value. Finally, under minimal thermodynamic assumptions, we establish a clock–work trade-off relation between these two types of coherences. This can be viewed as a form of time-energy conjugate relation within quantum thermodynamics that bounds the total maximum of clock and work resources for a given system.
Rigovacca L, Kolthammer WS, Franco CD, et al., 2018, Optical nonclassicality test based on third-order intensity correlations, Physical Review A, Vol: 97, ISSN: 1050-2947
We develop a nonclassicality criterion for the interference of three delayed,but otherwise identical, light fields in a three-mode Bell interferometer. Wedo so by comparing the prediction of quantum mechanics with those of aclassical framework in which independent sources emit electric fields withrandom phases. In particular, we evaluate third-order correlations among output intensities as a function of the delays, and show how the presence of a correlation revival for small delays cannot be explained by the classical model of light. The observation of a revival is thus a nonclassicality signature, which can be achieved only by sources with a photon-number statistics that is highly sub-Poissonian. Our analysis provides strong evidence for the nonclassicality of the experiment discussed in [Menssen et al., PRL, 118, 153603 (2017)], and shows how a collective "triad" phase affects the interference of any three or more light fields, irrespective of their quantum or classical character.
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.