Logic artificial intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and be solved by checking their satisfiability (SAT). Recently, SAT solvers have become a sophisticated and powerful computational tool capable, among other things, of solving longstanding mathematical conjectures. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logicbased algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the stateoftheart algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI improves significantly the resolution of this problem, paving the path to develop more formalbased approaches in the context of quantum physics experiments.
Learning Interpretable Representations of Entanglement in Quantum Optics Experiments using Deep Generative Models
Daniel FlamShepherd, Tony Wu, Xuemei Gu, Alba CerveraLierta, M. Krenn, Alan AspuruGuzik
Quantum physics experiments produce interesting phenomena such as interference or entanglement, which is a core property of numerous future quantum technologies. The complex relationship between a quantum experiment's structure and its entanglement properties is essential to fundamental research in quantum optics but is difficult to intuitively understand. We present the first deep generative model of quantum optics experiments where a variational autoencoder (QOVAE) is trained on a dataset of experimental setups. In a series of computational experiments, we investigate the learned representation of the QOVAE and its internal understanding of the quantum optics world. We demonstrate that the QOVAE learns an intrepretable representation of quantum optics experiments and the relationship between experiment structure and entanglement. We show the QOVAE is able to generate novel experiments for highly entangled quantum states with specific distributions that match its training data. Importantly, we are able to fully interpret how the QOVAE structures its latent space, finding curious patterns that we can entirely explain in terms of quantum physics. The results demonstrate how we can successfully use and understand the internal representations of deep generative models in a complex scientific domain. The QOVAE and the insights from our investigations can be immediately applied to other physical systems throughout fundamental scientific research.
Conceptual Understanding through Efficient Automated Design of Quantum Optical Experiments
Mario Krenn, Jakob S. Kottmann, Nora Tischler, Alan AspuruGuzik
Physical Review X
11(3)
031044
(2021)

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Artificial intelligence (AI) is a potentially disruptive tool for physics and science in general. One crucial question is how this technology can contribute at a conceptual level to help acquire new scientific understanding. Scientists have used AI techniques to rediscover previously known concepts. So far, no examples of that kind have been reported that are applied to open problems for getting new scientific concepts and ideas. Here, we present THESEUS, an algorithm that can provide new conceptual understanding, and we demonstrate its applications in the field of experimental quantum optics. To do so, we make four crucial contributions. (i) We introduce a graphbased representation of quantum optical experiments that can be interpreted and used algorithmically. (ii) We develop an automated design approach for new quantum experiments, which is orders of magnitude faster than the best previous algorithms at concrete design tasks for experimental configuration. (iii) We solve several crucial open questions in experimental quantum optics which involve practical blueprints of resource states in photonic quantum technology and quantum states and transformations that allow for new foundational quantum experiments. Finally, and most importantly, (iv) the interpretable representation and enormous speedup allow us to produce solutions that a human scientist can interpret and gain new scientific concepts from outright. We anticipate that THESEUS will become an essential tool in quantum optics for developing new experiments and photonic hardware. It can further be generalized to answer open questions and provide new concepts in a large number of other quantum physical questions beyond quantum optical experiments. THESEUS is a demonstration of explainable AI (XAI) in physics that shows how AI algorithms can contribute to science on a conceptual level.
Deep molecular dreaming: inverse machine learning for denovo molecular design and interpretability with surjective representations
Cynthia Shen, Mario Krenn, Sagi Eppel, Alán AspuruGuzik
Machine Learning: Science and Technology
3
(2021)

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Computerbased denovo design of functional molecules is one of the most prominent challenges in cheminformatics today. As a result, generative and evolutionary inverse designs from the field of artificial intelligence have emerged at a rapid pace, with aims to optimize molecules for a particular chemical property. These models 'indirectly' explore the chemical space; by learning latent spaces, policies, and distributions, or by applying mutations on populations of molecules. However, the recent development of the SELFIES (Krenn 2020 Mach. Learn.: Sci. Technol. 1 045024) string representation of molecules, a surjective alternative to SMILES, have made possible other potential techniques. Based on SELFIES, we therefore propose PASITHEA, a direct gradientbased molecule optimization that applies inceptionism (Mordvintsev 2015) techniques from computer vision. PASITHEA exploits the use of gradients by directly reversing the learning process of a neural network, which is trained to predict realvalued chemical properties. Effectively, this forms an inverse regression model, which is capable of generating molecular variants optimized for a certain property. Although our results are preliminary, we observe a shift in distribution of a chosen property during inversetraining, a clear indication of PASITHEA's viability. A striking property of inceptionism is that we can directly probe the model's understanding of the chemical space on which it is trained. We expect that extending PASITHEA to larger datasets, molecules and more complex properties will lead to advances in the design of new functional molecules as well as the interpretation and explanation of machine learning models.
Quantum computeraided design of quantum optics hardware
Jakob S. Kottmann, Mario Krenn, Thi Ha Kyaw, Sumner AlperinLea, Alan AspuruGuzik
Quantum Science and Technology
6(3)
035010
(2021)

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The parameters of a quantum system grow exponentially with the number of involved quantum particles. Hence, the associated memory requirement to store or manipulate the underlying wavefunction goes well beyond the limit of the best classical computers for quantum systems composed of a few dozen particles, leading to serious challenges in their numerical simulation. This implies that the verification and design of new quantum devices and experiments are fundamentally limited to small system size. It is not clear how the full potential of large quantum systems can be exploited. Here, we present the concept of quantum computer designed quantum hardware and apply it to the field of quantum optics. Specifically, we map complex experimental hardware for highdimensional, manybody entangled photons into a gatebased quantum circuit. We show explicitly how digital quantum simulation of Boson sampling experiments can be realized. We then illustrate how to design quantumoptical setups for complex entangled photonic systems, such as highdimensional GreenbergerHorneZeilinger states and their derivatives. Since photonic hardware is already on the edge of quantum supremacy and the development of gatebased quantum computers is rapidly advancing, our approach promises to be a useful tool for the future of quantum device design.
Scientific intuition inspired by machine learninggenerated hypotheses
Pascal Friederich, Mario Krenn, Isaac Tamblyn, Alan AspuruGuzik
Machine Learning  Science and Technology
2(2)
025027
(2021)

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Machine learning with application to questions in the physical sciences has become a widely used tool, successfully applied to classification, regression and optimization tasks in many areas. Research focus mostly lies in improving the accuracy of the machine learning models in numerical predictions, while scientific understanding is still almost exclusively generated by human researchers analysing numerical results and drawing conclusions. In this work, we shift the focus on the insights and the knowledge obtained by the machine learning models themselves. In particular, we study how it can be extracted and used to inspire human scientists to increase their intuitions and understanding of natural systems. We apply gradient boosting in decision trees to extract humaninterpretable insights from big data sets from chemistry and physics. In chemistry, we not only rediscover widely know rules of thumb but also find new interesting motifs that tell us how to control solubility and energy levels of organic molecules. At the same time, in quantum physics, we gain new understanding on experiments for quantum entanglement. The ability to go beyond numerics and to enter the realm of scientific insight and hypothesis generation opens the door to use machine learning to accelerate the discovery of conceptual understanding in some of the most challenging domains of science.
Beyond generative models: superfast traversal, optimization, novelty, exploration and discovery (STONED) algorithm for molecules using SELFIES
AkshatKumar Nigam, Robert Pollice, Mario Krenn, Gabriel dos Passos Gomes, Alan AspuruGuzik
Chemical Science
12(20)
70797090
(2021)

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Inverse design allows the generation of molecules with desirable physical quantities using property optimization. Deep generative models have recently been applied to tackle inverse design, as they possess the ability to optimize molecular properties directly through structure modification using gradients. While the ability to carry out direct property optimizations is promising, the use of generative deep learning models to solve practical problems requires large amounts of data and is very timeconsuming. In this work, we propose STONED  a simple and efficient algorithm to perform interpolation and exploration in the chemical space, comparable to deep generative models. STONED bypasses the need for large amounts of data and training times by using string modifications in the SELFIES molecular representation. First, we achieve nontrivial performance on typical benchmarks for generative models without any training. Additionally, we demonstrate applications in highthroughput virtual screening for the design of drugs, photovoltaics, and the construction of chemical paths, allowing for both property and structurebased interpolation in the chemical space. Overall, we anticipate our results to be a stepping stone for developing more sophisticated inverse design models and benchmarking tools, ultimately helping generative models achieve wider adoption.
Experimental highdimensional GreenbergerHorneZeilinger entanglement with superconducting transmon qutrits
Alba CerveraLierta, M. Krenn, Alan AspuruGuzik, Alexey Galda
DataDriven Strategies for Accelerated Materials Design
Robert Pollice, Gabriel dos Passos Gomes, Matteo Aldeghi, Riley J. Hickman, M. Krenn, Cyrille Lavigne, Michael LindnerD’Addario, AkshatKumar Nigam, Cher Tian Ser, et al.
Computeraided design of molecules has the potential to disrupt the field of drug and material discovery. Machine learning, and deep learning, in particular, have been topics where the field has been developing at a rapid pace. Reinforcement learning is a particularly promising approach since it allows for molecular design without prior knowledge. However, the search space is vast and efficient exploration is desirable when using reinforcement learning agents. In this study, we propose an algorithm to aid efficient exploration. The algorithm is inspired by a concept known in the literature as curiosity. We show on three benchmarks that a curious agent finds better performing molecules. This indicates an exciting new research direction for reinforcement learning agents that can explore the chemical space out of their own motivation. This has the potential to eventually lead to unexpected new molecules that no human has thought about so far.
Compact Greenberger—Horne—Zeilinger state generation via frequency combs and graph theory
Designing new experiments in physics is a challenge for humans; therefore, computers have become a tool to expand scientists' capabilities and to provide creative solutions. This Perspective article examines computerinspired designs in quantum physics that led to laboratory experiments and inspired new scientific insights.
The design of new devices and experiments has historically relied on the intuition of human experts. Now, design inspirations from computers are increasingly augmenting the capability of scientists. We briefly overview different fields of physics that rely on computerinspired designs using a variety of computational approaches based on topological optimization, evolutionary strategies, deep learning, reinforcement learning or automated reasoning. Then we focus specifically on quantum physics. When designing new quantum experiments, there are two challenges: quantum phenomena are unintuitive, and the number of possible configurations of quantum experiments explodes exponentially. These challenges can be overcome by using computerdesigned quantum experiments. We focus on the most mature and practical approaches to find new complex quantum experiments, which have subsequently been realized in the lab. These methods rely on a highly efficient topological search, which can inspire new scientific ideas. We review several extensions and alternatives based on various optimization and machine learning techniques. Finally, we discuss what can be learned from the different approaches and outline several future directions.
Selfreferencing embedded strings (SELFIES): A 100% robust molecular string representation
Mario Krenn, Florian Häse, AkshatKumar Nigam, Pascal Friederich, Alan AspuruGuzik
Machine Learning: Science and Technology
1
(2020)

Journal
Path identity as a source of highdimensional entanglement
Jaroslav Kysela, Manuel Erhard, Armin Hochrainer, Mario Krenn, Anton Zeilinger
Proceedings of the National Academy of Sciences of the United States of America
117
(2020)

Journal
ComputerInspired Concept for HighDimensional Multipartite Quantum Gates
Xiaoqin Gao, Manuel Erhard, Anton Zeilinger, Mario Krenn
An open question in quantum optics is how to manipulate and control complex quantum states in an experimentally feasible way. Here we present concepts for transformations of highdimensional multiphotonic quantum systems. The proposals rely on two new ideas: (i) a novel highdimensional quantum nondemolition measurement, (ii) the encoding and decoding of the entire quantum transformation in an ancillary state for sharing the necessary quantum information between the involved parties. Many solutions can readily be performed in laboratories around the world and thereby we identify important pathways for experimental research in the near future. The concepts have been found using the computer algorithm MELVIN for designing computerinspired quantum experiments. As opposed to the field of machine learning, here the human learns new scientific concepts by interpreting and analyzing the results presented by the machine. This demonstrates that computer algorithms can inspire new ideas in science, which has a widely unexplored potential that goes far beyond experimental quantum information science.
Since its discovery, quantum entanglement has challenged some of the best established views of the world: locality and reality. Quantum technologies promise to revolutionize computation, communication, metrology and imaging. Here we review conceptual and experimental advances in complex entangled systems involving many multilevel quantum particles. We provide an overview of the latest technological developments in the generation and manipulation of highdimensionally entangled photonic systems encoded in various discrete degrees of freedom such as path, transverse spatial modes or timefrequency bins. This overview should help to transfer various physical principles for the generation and manipulation from one degree of freedom to another and thus inspire new technical developments. We also show how purely academic questions and curiosity led to new technological applications. Fundamental research provides the necessary knowledge for upcoming technologies, such as a prospective quantum internet or the quantum teleportation of all information stored in a quantum system. Finally, we discuss some important problems in the area of highdimensional entanglement and give a brief outlook on possible future developments.
The study of higherdimensional quantum states has seen numerous conceptual and technological developments. This review discusses various techniques for the generation and processing of qudits, which are stored in the momentum, path, time/frequencybins, or the orbital angular momentum of photons.
Phenomenology of complex structured light in turbulent air
Xuemei Gu, Lijun Chen, Mario Krenn
Optics Express
28(8)
1103311050
(2020)

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The study of light propagation has been a cornerstone of progress in physics and technology. Recently, advances in control and shaping of light have created significant interest in the propagation of complex structures of light  particularly under realistic terrestrial conditions. While theoretical understanding of this research question has significantly grown over the last two decades, outdoor experiments with complex light structures are rare, and comparisons with theory have been nearly lacking. Such situations show a significant gap between theoretical models of atmospheric light behaviour and current experimental effort. Here, in an attempt to reduce this gap, we describe an interesting result of atmospheric models that are feasible for empirical observation. We analyze in detail light propagation in different spatial bases and present results of the theory that the influence of atmospheric turbulence is basisdependent. Concretely, light propagating as eigenstate in one complete basis is more strongly influenced by atmosphere than light propagating in a different, complete basis. We obtain these results by exploiting a family of the continuously adjustable, complete basis of spatial modesthe InceGauss modes. Our concrete numerical results will hopefully inspire experimental efforts and bring the theoretical and empirical study of complex light patterns in realistic scenarios closer together. Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License.
The sounds of science—a symphony for many instruments and voices
Gerianne Alexander, Roland E Allen, Anthony Atala, Warwick P Bowen, Alan A Coley, John B Goodenough, Mikhail I Katsnelson, Eugene V Koonin, Mario Krenn, et al.
We introduce the concept of hypergraphs to describe quantum optical experiments with probabilistic multiphoton sources. Every hyperedge represents a correlated photon source, and every vertex stands for an optical output path. Such a general graph description provides new insights for producing complex highdimensional multiphoton quantum entangled states, which go beyond limitations imposed by pair creation via spontaneous parametric downconversion. Furthermore, the properties of hypergraphs can be investigated experimentally. For example, the NPcomplete problem of deciding whether a hypergraph has a perfect matching can be answered by experimentally detecting multiphoton events in quantum experiments. By introducing complex weights in hypergraphs, we show a general manyparticle quantum interference and manipulating entanglement in a pictorial way. Our work paves the path for the development of multiphoton highdimensional state generation and might inspire new applications of quantum computations using hypergraph mappings.
Predicting research trends with semantic and neural networks with an application in quantum physics
Mario Krenn, Anton Zeilinger
Proceedings of the National Academy of Sciences of the United States of America
117(4)
19101916
(2020)

Journal
The vast and growing number of publications in all disciplines of science cannot be comprehended by a single human researcher. As a consequence, researchers have to specialize in narrow subdisciplines, which makes it challenging to uncover scientific connections beyond the own field of research. Thus, access to structured knowledge from a large corpus of publications could help push the frontiers of science. Here, we demonstrate a method to build a semantic network from published scientific literature, which we call SEMNET. We use SEMNET to predict future trends in research and to inspire personalized and surprising seeds of ideas in science. We apply it in the discipline of quantum physics, which has seen an unprecedented growth of activity in recent years. In SEMNET, scientific knowledge is represented as an evolving network using the content of 750,000 scientific papers published since 1919. The nodes of the network correspond to physical concepts, and links between two nodes are drawn when two concepts are concurrently studied in research articles. We identify influential and prizewinning research topics from the past inside SEMNET, thus confirming that it stores useful semantic knowledge. We train a neural network using states of SEMNET of the past to predict future developments in quantum physics and confirm highquality predictions using historic data. Using network theoretical tools, we can suggest personalized, outofthebox ideas by identifying pairs of concepts, which have unique and extremal semantic network properties. Finally, we consider possible future developments and implications of our findings.
Augmenting Genetic Algorithms with Deep Neural Networks for Exploring the Chemical Space
AkshatKumar Nigam, Pascal Friederich, Mario Krenn, Alán AspuruGuzik
Challenges in natural sciences can often be phrased as optimization problems. Machine learning techniques have recently been applied to solve such problems. One example in chemistry is the design of tailormade organic materials and molecules, which requires efficient methods to explore the chemical space. We present a genetic algorithm (GA) that is enhanced with a neural network (DNN) based discriminator model to improve the diversity of generated molecules and at the same time steer the GA. We show that our algorithm outperforms other generative models in optimization tasks. We furthermore present a way to increase interpretability of genetic algorithms, which helped us to derive design principles.
2019
Quantum Optical Experiments Modeled by Long ShortTerm Memory
Thomas Adler, Manuel Erhard, M. Krenn, Johannes Brandstetter, Johannes Kofler, Sepp Hochreiter
Quantum teleportation allows a "disembodied" transmission of unknown quantum states between distant quantum systems. Yet, all teleportation experiments to date were limited to a twodimensional subspace of quantized multiple levels of the quantum systems. Here, we propose a scheme for teleportation of arbitrarily highdimensional photonic quantum states and demonstrate an example of teleporting a qutrit. Measurements over a complete set of 12 qutrit states in mutually unbiased bases yield a teleportation fidelity of 0.75(1), which is well above both the optimal singlecopy qutrit stateestimation limit of 1/2 and maximal qubitqutrit overlap of 2/3, thus confirming a genuine and nonclassical threedimensional teleportation. Our work will enable advanced quantum technologies in high dimensions, since teleportation plays a central role in quantum repeaters and quantum networks.
Questions on the Structure of Perfect Matchings Inspired by Quantum Physics
Mario Krenn, Xuemei Gu, Daniel Soltesz
Proceedings of the 2nd Croatian Combinatorial Days
(2019)

Journal
Quantum experiments and graphs. III. Highdimensional and multiparticle
entanglement
Xuemei Gu, Lijun Chen, Anton Zeilinger, Mario Krenn
Quantum entanglement plays an important role in quantum information processes, such as quantum computation and quantum communication. Experiments in laboratories are unquestionably crucial to increase our understanding of quantum systems and inspire new insights into future applications. However, there are no general recipes for the creation of arbitrary quantum states with many particles entangled in high dimensions. Here we exploit a recent connection between quantum experiments and graph theory and answer this question for a plethora of classes of entangled states. We find experimental setups for GreenbergerHorneZeilinger states, W states, general Dicke states, and asymmetrically highdimensional multipartite entangled states. This result sheds light on the producibility of arbitrary quantum states using photonic technology with probabilistic pair sources and allows us to understand the underlying technological and fundamental properties of entanglement.
Quantum experiments and graphs II: Quantum interference, computation,
and state generation
Xuemei Gu, Manuel Erhard, Anton Zeilinger, Mario Krenn
Proceedings of the National Academy of Sciences of the United States of America
116(10)
41474155
(2019)

Journal
We present an approach to describe stateoftheart photonic quantum experiments using graph theory. There, the quantum states are given by the coherent superpositions of perfect matchings. The crucial observation is that introducing complex weights in graphs naturally leads to quantum interference. This viewpoint immediately leads to many interesting results, some of which we present here. First, we identify an experimental unexplored multiphoton interference phenomenon. Second, we find that computing the results of such experiments is #Phard, which means it is a classically intractable problem dealing with the computation of a matrix function Permanent and its generalization Hafnian. Third, we explain how a recent nogo result applies generally to linear optical quantum experiments, thus revealing important insights into quantum state generation with current photonic technology. Fourth, we show how to describe quantum protocols such as entanglement swapping in a graphical way. The uncovered bridge between quantum experiments and graph theory offers another perspective on a widely used technology and immediately raises many followup questions.
Arbitrary ddimensional Pauli X gates of a flying qudit
Xiaoqin Gao, Mario Krenn, Jaroslav Kysela, Anton Zeilinger
Highdimensional degrees of freedom of photons can encode more quantum information than their twodimensional counterparts. While the increased information capacity has advantages in quantum applications (such as quantum communication), controlling and manipulating these systems has been challenging. Here we show a method to perform deterministic arbitrary highdimensional Pauli X gates for single photons carrying orbital angular momentum. The X gate consists of a cyclic permutation of qudit basis vectors and, together with the Z gate, forms the basis for performing arbitrary transformations. The proposed experimental setups only use two basic optical elements such as mode sorters and mode shifters and thus could be implemented in any system where these experimental tools are available. Furthermore the number of involved interferometers scales logarithmically with the dimension, which is important for practical implementation.
Quantum entanglement is important for emerging quantum technologies such as quantum computation and secure quantum networks. To boost these technologies, a race is currently ongoing to increase the number of particles in multiparticle entangled states, such as GreenbergerHorneZeilinger (GHZ) states. An alternative route is to increase the number of entangled quantum levels. Here, we overcome present experimental and technological challenges to create a threeparticle GHZ state entangled in three levels for every particle. The resulting qutritentangled states are able to carry more information than entangled states of qubits. Our method, inspired by the computer algorithm Melvin, relies on a new multiport that coherently manipulates several photons simultaneously in higher dimensions. The realization required us to develop a new highbrightness fourphoton source entangled in orbital angular momentum. Our results allow qualitatively new refutations of localrealistic world views. We also expect that they will open up pathways for a further boost to quantum technologies.
On small beams with large topological charge: II. Photons, electrons and gravitational waves
Mario Krenn, Anton Zeilinger
New Journal of Physics
20
063006
(2018)

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Beams of light with a large topological charge significantly change their spatial structure when they are focused strongly. Physically, it can be explained by an emerging electromagnetic field component in the direction of propagation, which is neglected in the simplified scalar wave picture in optics. Here we ask: is this a specific photonic behavior, or can similar phenomena also be predicted for other species of particles? We show that the same modification of the spatial structure exists for relativistic electrons as well as for focused gravitational waves. However, this is for different physical reasons: for electrons, which are described by the Dirac equation, the spatial structure changes due to a spinorbit coupling in the relativistic regime. In gravitational waves described with linearized general relativity, the curvature of spacetime between the transverse and propagation direction leads to the modification of the spatial structure. Thus, this universal phenomenon exists for both massive and massless elementary particles with spin 1 /2,1 and 2. It would be very interesting whether other types of particles such as composite systems (neutrons or C60) or neutrinos show a similar behavior and how this phenomenon can be explained in a unified physical way.
Twisted photons: new quantum perspectives in high dimensions
Manuel Erhard, Robert Fickler, Mario Krenn, Anton Zeilinger
Twisted photons can be used as alphabets to encode information beyond one bit per single photon. This ability offers great potential for quantum information tasks, as well as for the investigation of fundamental questions. In this review article, we give a brief overview of the theoretical differences between qubits and higher dimensional systems, qudits, in different quantum information scenarios. We then describe recent experimental developments in this field over the past three years. Finally, we summarize some important experimental and theoretical questions that might be beneficial to understand better in the near future.
Gouy Phase Radial Mode Sorter for Light: Concepts and Experiments
Xuemei Gu, Mario Krenn, Manuel Erhard, Anton Zeilinger
We present an in principle lossless sorter for radial modes of light, using accumulated Gouy phases. The experimental setups have been found by a computer algorithm, and can be intuitively understood in a geometric way. Together with the ability to sort angularmomentum modes, we now have access to the complete twodimensional transverse plane of light. The device can readily be used in multiplexing classical information. On a quantum level, it is an analog of the SternGerlach experimentsignificant for the discussion of fundamental concepts in quantum physics. As such, it can be applied in highdimensional and multiphotonic quantum experiments.
Active learning machine learns to create new quantum experiments
Alexey A. Melnikov, Hendrik Poulsen Nautrup, Mario Krenn, Vedran Dunjko, Markus Tiersch, Anton Zeilinger, Hans J. Briegel
Proceedings of the National Academy of Sciences of the United States of America
115(6)
12211226
(2018)

Journal
How useful can machine learning be in a quantum laboratory? Here we raise the question of the potential of intelligent machines in the context of scientific research. A major motivation for the present work is the unknown reachability of various entanglement classes in quantum experiments. We investigate this question by using the projective simulation model, a physicsoriented approach to artificial intelligence. In our approach, the projective simulation system is challenged to design complex photonic quantum experiments that produce highdimensional entangled multiphoton states, which are of high interest in modern quantum experiments. The artificial intelligence system learns to create a variety of entangled states and improves the efficiency of their realization. In the process, the system autonomously (re)discovers experimental techniques which are only now becoming standard in modern quantum optical experimentsa trait which was not explicitly demanded from the system but emerged through the process of learning. Such features highlight the possibility that machines could have a significantly more creative role in future research.
2017
Generation of the complete fourdimensional Bell basis
Feiran Wang, Manuel Erhard, Amin Babazadeh, Mehul Malik, Mario Krenn, Anton Zeilinger
The Bell basis is a distinctive set of maximally entangled twoparticle quantum states that forms the foundation for many quantum protocols such as teleportation, dense coding, and entanglement swapping. While the generation, manipulation, and measurement of twolevel quantum states are well understood, the same is not true in higher dimensions. Here we present the experimental generation of a complete set of Bell states in a fourdimensional Hilbert space, comprising 16 orthogonal entangled Belllike states encoded in the orbital angular momentum of photons. The states are created by the application of generalized highdimensional Pauli gates on an initial entangled state. Our results pave the way for the application of highdimensional quantum states in complex quantum protocols such as quantum dense coding. (c) 2017 Optical Society of America
Quantum Experiments and Graphs: Multiparty States as Coherent
Superpositions of Perfect Matchings
We show a surprising link between experimental setups to realize highdimensional multipartite quantum states and graph theory. In these setups, the paths of photons are identified such that the photonsource information is never created. We find that each of these setups corresponds to an undirected graph, and every undirected graph corresponds to an experimental setup. Every term in the emerging quantum superposition corresponds to a perfect matching in the graph. Calculating the final quantum state is in the #Pcomplete complexity class, thus it cannot be done efficiently. To strengthen the link further, theorems from graph theorysuch as Hall's marriage problemare rephrased in the language of pair creation in quantum experiments. We show explicitly how this link allows one to answer questions about quantum experiments (such as which classes of entangled states can be created) with graph theoretical methods, and how to potentially simulate properties of graphs and networks with quantum experiments (such as critical exponents and phase transitions).
HighDimensional SinglePhoton Quantum Gates: Concepts and Experiments
Amin Babazadeh, Manuel Erhard, Feiran Wang, Mehul Malik, Rahman Nouroozi, Mario Krenn, Anton Zeilinger
Transformations on quantum states form a basic building block of every quantum information system. From photonic polarization to twolevel atoms, complete sets of quantum gates for a variety of qubit systems are well known. For multilevel quantum systems beyond qubits, the situation is more challenging. The orbital angular momentum modes of photons comprise one such highdimensional system for which generation and measurement techniques are well studied. However, arbitrary transformations for such quantum states are not known. Here we experimentally demonstrate a fourdimensional generalization of the Pauli X gate and all of its integer powers on single photons carrying orbital angular momentum. Together with the wellknown Z gate, this forms the first complete set of highdimensional quantum gates implemented experimentally. The concept of the X gate is based on independent access to quantum states with different parities and can thus be generalized to other photonic degrees of freedom and potentially also to other quantum systems.
Quantum gate description for induced coherence without induced emission
and its applications
We introduce unitary quantum gates for photon pair creation in spontaneous parametric downconversion nonlinear crystals (NLs) and for photon path alignment. These are the two key ingredients for the method of induced coherence without induced emission and many ensuing variations thereof. The difficulty in doing so stems from an apparent mixing of the mode picture (such as the polarization of photons) and the Fock picture (such as the existence of the photons). We illustrate utility of these gates by obtaining quantum circuits for the experimental setups of the frustrated generation of photon pairs, identification of a pointlike object with undetected photons, and creation of a Bell state. We also introduce an effective nonunitary description for the action of NLs in experiments where all the NLs are pumped coherently. As an example, by using this simplifying picture, we show how NLs can be used to create superposition of given quantum states in a modular fashion.
Orbital angular momentum of photons and the entanglement of LaguerreGaussian modes
Mario Krenn, Mehul Malik, Manuel Erhard, Anton Zeilinger
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
375(2087)
20150442
(2017)

Journal
The identification of orbital angular momentum (OAM) as a fundamental property of a beam of light nearly 25 years ago has led to an extensive body of research around this topic. The possibility that single photons can carry OAM has made this degree of freedom an ideal candidate for the investigation of complex quantum phenomena and their applications. Research in this direction has ranged from experiments on complex forms of quantum entanglement to the interaction between light and quantum states of matter. Furthermore, the use of OAM in quantum information has generated a lot of excitement, as it allows for encoding large amounts of information on a single photon. Here, we explain the intuition that led to the first quantum experiment with OAM 15 years ago. We continue by reviewing some key experiments investigating fundamental questions on photonic OAMand the first steps to applying these properties in novel quantum protocols. At the end, we identify several interesting open questions that could form the subject of future investigations with OAM.
This article is part of the themed issue 'Optical orbital angular momentum'.
Entanglement by Path Identity
Mario Krenn, Armin Hochrainer, Mayukh Lahiri, Anton Zeilinger
Quantum entanglement is one of the most prominent features of quantum mechanics and forms the basis of quantum information technologies. Here we present a novel method for the creation of quantum entanglement in multipartite and highdimensional systems. The two ingredients are (i) superposition of photon pairs with different origins and (ii) aligning photons such that their paths are identical. We explain the experimentally feasible creation of various classes of multiphoton entanglement encoded in polarization as well as in highdimensional Hilbert spacesstarting only from nonentangled photon pairs. For two photons, arbitrary highdimensional entanglement can be created. The idea of generating entanglement by path identity could also apply to quantum entities other than photons. We discovered the technique by analyzing the output of a computer algorithm. This shows that computer designed quantum experiments can be inspirations for new techniques.
Quantifying high dimensional entanglement with two mutually unbiased bases
We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases where the second basis is not characterized beyond being unbiased, thus enabling entanglement quantification with minimal assumptions. Furthermore, we show that it is feasible to experimentally implement our method with readily available equipment and even conservative estimates of physical parameters.
2016
Twisted light transmission over 143 km
Mario Krenn, Johannes Handsteiner, Matthias Fink, Robert Fickler, Rupert Ursin, Mehul Malik, Anton Zeilinger
Proceedings of the National Academy of Sciences of the United States of America
113(48)
1364813653
(2016)

Journal
Spatial modes of light can potentially carry a vast amount of information, making them promising candidates for both classical and quantum communication. However, the distribution of such modes over large distances remains difficult. Intermodal coupling complicates their use with common fibers, whereas freespace transmission is thought to be strongly influenced by atmospheric turbulence. Here, we show the transmission of orbital angular momentum modes of light over a distance of 143 km between two Canary Islands, which is 50x greater than the maximum distance achieved previously. As a demonstration of the transmission quality, we use superpositions of these modes to encode a short message. At the receiver, an artificial neural network is used for distinguishing between the different twisted light superpositions. The algorithm is able to identify different mode superpositions with an accuracy of more than 80% up to the third mode order and decode the transmitted message with an error rate of 8.33%. Using our data, we estimate that the distribution of orbital angular momentum entanglement over more than 100 km of free space is feasible. Moreover, the quality of our freespace link can be further improved by the use of stateoftheart adaptive optics systems.
Quantum optical rotatory dispersion
Nora Tischler, Mario Krenn, Robert Fickler, Xavier Vidal, Anton Zeilinger, Gabriel MolinaTerriza
Science Advances
2(10)
e1601306
(2016)

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The phenomenon of molecular optical activity manifests itself as the rotation of the plane of linear polarization when light passes through chiral media. Measurements of optical activity and its wavelength dependence, that is, optical rotatory dispersion, can reveal information about intricate properties of molecules, such as the threedimensional arrangement of atoms comprising a molecule. Given a limited probe power, quantum metrology offers the possibility of outperforming classical measurements. This has particular appeal when samples may be damaged by high power, which is a potential concern for chiroptical studies. We present the first experiment in which multiwavelength polarizationentangled photon pairs are used to measure the optical activity and optical rotatory dispersion exhibited by a solution of chiral molecules. Our work paves the way for quantumenhanced measurements of chirality, with potential applications in chemistry, biology, materials science, and the pharmaceutical industry. The scheme that we use for probing wavelength dependence not only allows one to surpass the information extracted per photon in a classical measurement but also can be used for more general differential measurements.
Cyclic transformation of orbital angular momentum modes
Florian Schlederer, Mario Krenn, Robert Fickler, Mehul Malik, Anton Zeilinger
New Journal of Physics
18
043019
(2016)

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The spatial modes of photons are one realization of a QuDit, a quantum system that is described in a Ddimensional Hilbert space. In order to perform quantum information tasks with QuDits, a general class of Ddimensional unitary transformations is needed. Among these, cyclic transformations are an important special case required in many highdimensional quantum communication protocols. In this paper, we experimentally demonstrate a cyclic transformation in the highdimensional space of photonic orbital angular momentum (OAM). Using simple linear optical components, we show a successful fourfold cyclic transformation of OAM modes. Interestingly, our experimental setup was found by a computer algorithm. In addition to the fourcyclic transformation, the algorithm also found extensions to higherdimensional cycles in a hybrid space of OAM and polarization. Besides being useful for quantum cryptography with QuDits, cyclic transformations are key for the experimental production of highdimensional maximally entangled Bellstates.
Multiphoton entanglement in high dimensions
Mehul Malik, Manuel Erhard, Marcus Huber, Mario Krenn, Robert Fickler, Anton Zeilinger
Forming the backbone of quantum technologies today, entanglement(1,2) has been demonstrated in physical systems as diverse as photons(3), ions(4) and superconducting circuits(5). Although steadily pushing the boundary of the number of particles entangled, these experiments have remained in a twodimensional space for each particle. Here we show the experimental generation of the first multiphoton entangled state where both the number of particles and dimensions are greater than two. Two photons in our state reside in a threedimensional space, whereas the third lives in two dimensions. This asymmetric entanglement structure(6) only appears in multiparticle entangled states with d > 2(6). Our method relies on combining two pairs of photons, highdimensionally entangled in their orbital angular momentum(7). In addition, we show how this state enables a new type of 'layered' quantum communication protocol. Entangled states such as these serve as a manifestation of the complex dance of correlations that can exist within quantum mechanics.
Automated Search for new Quantum Experiments
Mario Krenn, Mehul Malik, Robert Fickler, Radek Lapkiewicz, Anton Zeilinger
Quantum mechanics predicts a number of, at first sight, counterintuitive phenomena. It therefore remains a question whether our intuition is the best way to find new experiments. Here, we report the development of the computer algorithm MELVIN which is able to find new experimental implementations for the creation and manipulation of complex quantum states. Indeed, the discovered experiments extensively use unfamiliar and asymmetric techniques which are challenging to understand intuitively. The results range from the first implementation of a highdimensional GreenbergerHorneZeilinger state, to a vast variety of experiments for asymmetrically entangled quantum statesa feature that can only exist when both the number of involved parties and dimensions is larger than 2. Additionally, new types of highdimensional transformations are found that perform cyclic operations. MELVIN autonomously learns from solutions for simpler systems, which significantly speeds up the discovery rate of more complex experiments. The ability to automate the design of a quantum experiment can be applied to many quantum systems and allows the physical realization of quantum states previously thought of only on paper.
On small beams with large topological charge
Mario Krenn, Nora Tischler, Anton Zeilinger
New Journal of Physics
18
033012
(2016)

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Light beams can carry a discrete, in principle unbounded amount of angular momentum. Examples of such beams, the LaguerreGauss modes, are frequently expressed as solutions of the paraxial wave equation. The paraxial wave equation is a smallangle approximation of the Helmholtz equation, and is commonly used in beam optics. There, the LaguerreGauss modes have welldefined orbital angular momentum (OAM). The paraxial solutions predict that beams with large OAM could be used to resolve arbitrarily small distancesa dubious situation. Here we show how to solve that situation by calculating the properties of beams free from the paraxial approximation. We find the surprising result that indeed one can resolve smaller distances with larger OAM, although with decreased visibility. If the visibility is kept constant (for instance at the Rayleigh criterion, the limit where two points are reasonably distinguishable), larger OAM does not provide an advantage. The drop in visibility is due to a field in the direction of propagation, which is neglected within the paraxial limit. Our findings have implications for imaging techniques and raise questions on the difference between photonic and matter waves, which we briefly discuss in the conclusion.
2015
Physical meaning of the radial index of LaguerreGauss beams
The LaguerreGauss modes are a class of fundamental and wellstudied optical fields. These stable shapeinvariant photons, exhibiting circularcylindrical symmetry, are familiar from laser optics, micromechanical manipulation, quantum optics, communication, and foundational studies in both classical optics and quantum physics. They are characterized, chiefly, by two mode numbers: the azimuthal index indicating the orbital angular momentum of the beam, which itself has spawned a burgeoning and vibrant subfield, and the radial index, which up until recently has largely been ignored. In this paper we develop a differential operator formalism for dealing with the radial modes in both the position and momentum representations and, more importantly, give the meaning of this quantum number in terms of a welldefined physical parameter: the intrinsic hyperbolic momentum charge.
Twisted photon entanglement through turbulent air across Vienna
Mario Krenn, Johannes Handsteiner, Matthias Fink, Robert Fickler, Anton Zeilinger
Proceedings of the National Academy of Sciences of the United States of America
112(46)
1419714201
(2015)

Journal
Photons with a twisted phase front can carry a discrete, in principle, unbounded amount of orbital angular momentum (OAM). The large state space allows for complex types of entanglement, interesting both for quantum communication and for fundamental tests of quantum theory. However, the distribution of such entangled states over large distances was thought to be infeasible due to influence of atmospheric turbulence, indicating a serious limitation on their usefulness. Here we show that it is possible to distribute quantum entanglement encoded in OAM over a turbulent intracity link of 3 km. We confirm quantum entanglement of the first two higherorder levels (with OAM=+/ 1h and +/ 2h). They correspond to four additional quantum channels orthogonal to all that have been used in longdistance quantum experiments so far. Therefore, a promising application would be quantum communication with a large alphabet. We also demonstrate that our link allows access to up to 11 quantum channels of OAM. The restrictive factors toward higher numbers are technical limitations that can be circumvented with readily available technologies.
2014
Communication with spatially modulated light through turbulent air across Vienna
Mario Krenn, Robert Fickler, Matthias Fink, Johannes Handsteiner, Mehul Malik, Thomas Scheidl, Rupert Ursin, Anton Zeilinger
New Journal of Physics
16
113028
(2014)

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Transverse spatial modes of light offer a large state space with interesting physical properties. For exploiting these special modes in future longdistance experiments, the modes will have to be transmitted over turbulent freespace links. Numerous recent labscale experiments have found significant degradation in the mode quality after transmission through simulated turbulence and consecutive coherent detection. Here, we experimentally analyze the transmission of one prominent class of spatial modesorbitalangular momentum (OAM) modesthrough 3 km of strong turbulence over the city of Vienna. Instead of performing a coherent phasedependent measurement, we employ an incoherent detection scheme, which relies on the unambiguous intensity patterns of the different spatial modes. We use a pattern recognition algorithm (an artificial neural network) to identify the characteristic mode patterns displayed on a screen at the receiver. We were able to distinguish between 16 different OAM mode superpositions with only a similar to 1.7% error rate and to use them to encode and transmit small grayscale images. Moreover, we found that the relative phase of the superposition modes is not affected by the atmosphere, establishing the feasibility for performing longdistance quantum experiments with the OAM of photons. Our detection method works for other classes of spatial modes with unambiguous intensity patterns as well, and can be further improved by modern techniques of pattern recognition.
Generation and confirmation of a (100 x 100)dimensional entangled quantum system
Mario Krenn, Marcus Huber, Robert Fickler, Radek Lapkiewicz, Sven Ramelow, Anton Zeilinger
Proceedings of the National Academy of Sciences of the United States of America
111(17)
62436247
(2014)

Journal
Entangled quantum systems have properties that have fundamentally overthrown the classical worldview. Increasing the complexity of entangled states by expanding their dimensionality allows the implementation of novel fundamental tests of nature, and moreover also enables genuinely newprotocols for quantum information processing. Here we present the creation of a (100 x 100)dimensional entangled quantum system, using spatial modes of photons. For its verification we develop a novel nonlinear criterion which infers entanglement dimensionality of a global state by using only information about its subspace correlations. This allows very practical experimental implementation as well as highly efficient extraction of entanglement dimensionality information. Applications in quantum cryptography and other protocols are very promising.
2013
RealTime Imaging of Quantum Entanglement
Robert Fickler, Mario Krenn, Radek Lapkiewicz, Sven Ramelow, Anton Zeilinger
Quantum Entanglement is widely regarded as one of the most prominent features of quantum mechanics and quantum information science. Although, photonic entanglement is routinely studied in many experiments nowadays, its signature has been out of the grasp for realtime imaging. Here we show that modern technology, namely triggered intensified charge coupled device (ICCD) cameras are fast and sensitive enough to image in realtime the effect of the measurement of one photon on its entangled partner. To quantitatively verify the nonclassicality of the measurements we determine the detected photon number and error margin from the registered intensity image within a certain region. Additionally, the use of the ICCD camera allows us to demonstrate the high flexibility of the setup in creating any desired spatialmode entanglement, which suggests as well that visual imaging in quantum optics not only provides a better intuitive understanding of entanglement but will improve applications of quantum science.
Quantum orbital angular momentum of elliptically symmetric light
William N. Plick, Mario Krenn, Robert Fickler, Sven Ramelow, Anton Zeilinger
We present a quantummechanical analysis of the orbital angular momentum of a class of recently discovered elliptically symmetric stable light fieldsthe socalled InceGauss modes. We study, in a fully quantum formalism, how the orbital angular momentum of these beams varies with their ellipticity, and we discover several compelling features, including nonmonotonic behavior, stable beams with real continuous (noninteger) orbital angular momenta, and orthogonal modes with the same orbital angular momenta. We explore, and explain in detail, the reasons for this behavior. These features may have applications in quantum key distribution, atom trapping, and quantum informatics in generalas the ellipticity opens up an alternative way of navigating the spatial photonic Hilbert space. DOI: 10.1103/PhysRevA.87.033806
Entangled singularity patterns of photons in InceGauss modes
Mario Krenn, Robert Fickler, Marcus Huber, Radek Lapkiewicz, William Plick, Sven Ramelow, Anton Zeilinger
Photons with complex spatial mode structures open up possibilities for new fundamental highdimensional quantum experiments and for novel quantum information tasks. Here we show entanglement of photons with complex vortex and singularity patterns called InceGauss modes. In these modes, the position and number of singularities vary depending on the mode parameters. We verify twodimensional and threedimensional entanglement of InceGauss modes. By measuring one photon and thereby defining its singularity pattern, we nonlocally steer the singularity structure of its entangled partner, while the initial singularity structure of the photons is undefined. In addition we measure an InceGauss specific quantumcorrelation function with possible use in future quantum communication protocols. DOI: 10.1103/PhysRevA.87.012326
2012
Quantum Entanglement of High Angular Momenta
Robert Fickler, Radek Lapkiewicz, William N. Plick, Mario Krenn, Christoph Schaeff, Sven Ramelow, Anton Zeilinger
Single photons with helical phase structures may carry a quantized amount of orbital angular momentum (OAM), and their entanglement is important for quantum information science and fundamental tests of quantum theory. Because there is no theoretical upper limit on how many quanta of OAM a single photon can carry, it is possible to create entanglement between two particles with an arbitrarily high difference in quantum number. By transferring polarization entanglement to OAM with an interferometric scheme, we generate and verify entanglement between two photons differing by 600 in quantum number. The only restrictive factors toward higher numbers are current technical limitations. We also experimentally demonstrate that the entanglement of very high OAM can improve the sensitivity of angular resolution in remote sensing.