Time remains one of the least well-understood concepts in physics, most notably in quantum mechanics. A central goal is to find the fundamental limits of measuring time. One of the main obstacles is the fact that time is not an observable and thus has to be measured indirectly. Here, we explore these questions by introducing a model of time measurements that is complete and autonomous. Specifically, our autonomous quantum clock consists of a system out of thermal equilibrium—a prerequisite for any system to function as a clock—powered by minimal resources, namely, two thermal baths at different temperatures. Through a detailed analysis of this specific clock model, we find that the laws of thermodynamics dictate a trade-off between the amount of dissipated heat and the clock’s performance in terms of its accuracy and resolution. Our results furthermore imply that a fundamental entropy production is associated with the operation of any autonomous quantum clock, assuming that quantum machines cannot achieve perfect efficiency at finite power. More generally, autonomous clocks provide a natural framework for the exploration of fundamental questions about time in quantum theory and beyond.

VL - 7 UR - https://link.aps.org/doi/10.1103/PhysRevX.7.031022 ER - TY - JOUR T1 - Genuine-multipartite entanglement criteria based on positive maps JF - JOURNAL OF MATHEMATICAL PHYSICS Y1 - 2017 A1 - Clivaz, Fabien A1 - Huber, Marcus A1 - Lami, Ludovico A1 - Murta, Glaucia AB - Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition. We thus generalise this bipartite concept to the multipartite setting by introducing non-positive maps that are positive on the subset of biseparable states but can map to a non-positive element if applied to a genuine multipartite entangled state. We explicitly construct examples of multipartite non-positive maps, obtained from positive maps via a lifting procedure, that in this fashion can reveal genuine multipartite entanglement in a robust way. Published by AIP Publishing. VL - 58 ER - TY - JOUR T1 - No-Go Theorem for the Characterization of Work Fluctuations in Coherent Quantum Systems JF - PHYSICAL REVIEW LETTERS Y1 - 2017 A1 - Perarnau-Llobet, Marti A1 - Baumer, Elisa A1 - Hovhannisyan, Karen V. A1 - Huber, Marcus A1 - Acín, Antonio AB -An open question of fundamental importance in thermodynamics is howto describe the fluctuations of work for quantum coherent processes. In the standard approach, based on a projective energy measurement both at the beginning and at the end of the process, the first measurement destroys any initial coherence in the energy basis. Here we seek extensions of this approach which can possibly account for initially coherent states. We consider all measurement schemes to estimate work and require that (i) the difference of average energy corresponds to average work for closed quantum systems and that (ii) the work statistics agree with the standard two-measurement scheme for states with no coherence in the energy basis. We first show that such a scheme cannot exist. Next, we consider the possibility of performing collective measurements on several copies of the state and prove that it is still impossible to simultaneously satisfy requirements (i) and (ii). Nevertheless, improvements do appear, and in particular, we develop a measurement scheme that acts simultaneously on two copies of the state and allows us to describe a whole class of coherent transformations.

VL - 118 ER - TY - JOUR T1 - Quantifying high dimensional entanglement with two mutually unbiased bases JF - Quantum Y1 - 2017 A1 - Erker, Paul A1 - Krenn, Mario A1 - Huber, Marcus AB - Paul Erker, Mario Krenn, and Marcus Huber, Quantum 1, 22 (2017). https://doi.org/10.22331/q-2017-07-28-22 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 whe… VL - 1 UR - https://quantum-journal.org/papers/q-2017-07-28-22/ ER - TY - JOUR T1 - Autonomous {Quantum} {Refrigerator} in a {Circuit}-{QED} {Architecture} {Based} on a {Josephson} {Junction} JF - Physical Review B Y1 - 2016 A1 - Hofer, Patrick P. A1 - Perarnau-Llobet, Marti A1 - Brask, Jonatan Bohr A1 - Silva, Ralph A1 - Huber, Marcus A1 - Brunner, Nicolas KW - Condensed Matter - Mesoscale and Nanoscale Physics KW - Quantum Physics AB - An implementation of a small quantum absorption refrigerator in a circuit QED architecture is proposed. The setup consists of three harmonic oscillators coupled to a Josephson unction. The refrigerator is autonomous in the sense that it does not require any external control for cooling, but only thermal contact between the oscillators and heat baths at different temperatures. In addition, the setup features a built-in switch, which allows the cooling to be turned on and off. If timing control is available, this enables the possibility for coherence-enhanced cooling. Finally, we show that significant cooling can be achieved with experimentally realistic parameters and that our setup should be within reach of current technology. VL - 94 UR - http://arxiv.org/abs/1607.05218 N1 - arXiv: 1607.05218 ER - TY - JOUR T1 - Bipartite depolarizing maps JF - Journal of Mathematical Physics Y1 - 2016 A1 - Lami, Ludovico A1 - Huber, Marcus AB - We introduce a 3-parameter class of maps (1) acting on a bipartite system which are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking, or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, positive partial transposition but not entanglement-breaking maps is found in Theorem 2. A simple example of a positive yet indecomposable map is provided (see the Remark at the end of Section IV). The study of the entanglement-annihilating property is fully addressed by Theorem 7. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture posed in the work of Filippov [J. Russ. Laser Res. 35, 484 (2014)] is affirmatively answered, and the gaps that the imperfect bounds of Filippov and Ziman [Phys. Rev. A 88, 032316 (2013)] left open are closed. To arrive at this result, we furthermore show how the Hadamard product between quantum states can be implemented via local operations. VL - 57 UR - http://aip.scitation.org/doi/10.1063/1.4962339 ER - TY - JOUR T1 - Heisenberg-{Weyl} {Observables}: {Bloch} vectors in phase space JF - Physical Review A Y1 - 2016 A1 - Asadian, Ali A1 - Erker, Paul A1 - Huber, Marcus A1 - Klockl, Claude AB - We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anticommuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first nontrivial example beyond the dichotomic case. VL - 94 UR - https://link.aps.org/doi/10.1103/PhysRevA.94.010301 ER - TY - JOUR T1 - Multi-photon entanglement in high dimensions JF - Nature Photonics Y1 - 2016 A1 - Malik, Mehul A1 - Erhard, Manuel A1 - Huber, Marcus A1 - Krenn, Mario A1 - Robert Fickler A1 - Anton Zeilinger PB - Nature Publishing Group VL - 10 SN - 1749-4885 UR - http://dx.doi.org/10.1038/nphoton.2016.12 IS - 4 ER - TY - JOUR T1 - Quantifying {Entanglement} of {Maximal} {Dimension} in {Bipartite} {Mixed} {States} JF - Physical Review Letters Y1 - 2016 A1 - Sentís, Gael A1 - Eltschka, Christopher A1 - Gühne, Otfried A1 - Huber, Marcus A1 - Siewert, Jens AB - The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the forms of entanglement that can only exist in high-dimensional systems. We derive a framework for lower bounding the appropriate measure of entanglement, the so-called G-concurrence, through few local measurements. Moreover, we show that these bounds have relevant applications also for multipartite states. VL - 117 UR - https://link.aps.org/doi/10.1103/PhysRevLett.117.190502 ER - TY - JOUR T1 - Realising a quantum absorption refrigerator with an atom-cavity system JF - Quantum Science and Technology Y1 - 2016 A1 - Mitchison, Mark T. A1 - Huber, Marcus A1 - Prior, Javier A1 - Woods, Mischa P. A1 - Plenio, Martin B. AB - An autonomous quantum thermal machine comprising a trapped atom or ion placed inside an optical cavity is proposed and analysed. Such a machine can operate as a heat engine whose working medium is the quantised atomic motion or as an absorption refrigerator that cools without any work input. Focusing on the refrigerator mode, we predict that it is possible with state-of-the-art technology to cool a trapped ion almost to its motional ground state using a thermal light source such as sunlight. We nonetheless find that a laser or a similar reference system is necessary to stabilise the cavity frequencies. Furthermore, we establish a direct and heretofore unacknowledged connection between the abstract theory of quantum absorption refrigerators and practical sideband cooling techniques. We also highlight and clarify some assumptions underlying several recent theoretical studies on self-contained quantum engines and refrigerators. Our work indicates that cavity quantum electrodynamics is a promising and versatile experimental platform for the study of autonomous thermal machines in the quantum domain. VL - 1 UR - http://stacks.iop.org/2058-9565/1/i=1/a=015001 ER - TY - JOUR T1 - The role of quantum information in thermodynamics—a topical review JF - Journal of Physics A: Mathematical and Theoretical Y1 - 2016 A1 - Goold, John A1 - Huber, Marcus A1 - Riera, Arnau A1 - Rio, Lídia del A1 - Paul Skrzypczyk VL - 49 UR - http://stacks.iop.org/1751-8121/49/i=14/a=143001?key=crossref.ce80cac2171964cc6d6c51ed12d1e014 ER - TY - JOUR T1 - Should Entanglement Measures be Monogamous or Faithful? JF - Physical Review Letters Y1 - 2016 A1 - Lancien, Cécilia A1 - Di Martino, Sara A1 - Huber, Marcus A1 - Piani, Marco A1 - Adesso, Gerardo A1 - Winter, Andreas VL - 117 IS - 6 JO - Phys. Rev. Lett. ER - TY - JOUR T1 - Steering Maps and Their Application to Dimension-Bounded Steering JF - Physical Review Letters Y1 - 2016 A1 - Moroder, Tobias A1 - Gittsovich, Oleg A1 - Huber, Marcus A1 - Uola, Roope A1 - Gühne, Otfried VL - 116 UR - http://link.aps.org/doi/10.1103/PhysRevLett.116.090403 ER - TY - JOUR T1 - Witnessing entanglement by proxy JF - New Journal of Physics Y1 - 2016 A1 - Bäuml, Stefan A1 - Bruß, Dagmar A1 - Huber, Marcus A1 - Kampermann, Hermann A1 - Winter, Andreas VL - 18 IS - 1 JO - New J. Phys. ER - TY - JOUR T1 - Autonomous quantum thermal machine for generating steady-state entanglement JF - New Journal of Physics Y1 - 2015 A1 - Bohr Brask, Jonatan A1 - Haack, Géraldine A1 - Nicolas Brunner A1 - Huber, Marcus VL - 17 UR - http://stacks.iop.org/1367-2630/17/i=11/a=113029?key=crossref.0c6a75cb628f113c6cbd2a7b0d092f0d ER - TY - JOUR T1 - Coherence-assisted single-shot cooling by quantum absorption refrigerators JF - New Journal of Physics Y1 - 2015 A1 - Mitchison, Mark T A1 - Woods, Mischa P A1 - Prior, Javier A1 - Huber, Marcus VL - 17 UR - http://stacks.iop.org/1367-2630/17/i=11/a=115013?key=crossref.3c2d7f1bc5de4bbea1cba38fd2236293 ER - TY - JOUR T1 - Experimental access to higher-dimensional entangled quantum systems using integrated optics JF - Optica Y1 - 2015 A1 - Christoph Schaeff A1 - Robert Polster A1 - Huber, Marcus A1 - Ramelow, Sven A1 - Anton Zeilinger KW - Integrated optics KW - quantum optics AB - Integrated optics allows for the generation and control of increasingly complex photonic states on chip-based architectures. Here, we implement two entangled qutrits&\#x2014;a nine-dimensional quantum system&\#x2014;and demonstrate an exceptionally high degree of experimental control. The approach, which is conceptually different to common bulk optical implementations, is heavily based on methods of integrated in-fiber and on-chip technologies and further motivated by methods commonly used in today&\#x2019;s telecommunications industry. The system is composed of an in-fiber source creating entangled qutrit states of any amplitude and phase, and an on-chip integrated general Multiport enabling the realization of any desired local unitary transformation within the two qutrit nine-dimensional Hilbert space. The complete design is readily extendible toward higher dimensions with moderate increase in complexity. Ultimately, our scheme allows for complete on-chip integration. We demonstrate the flexibility and generality of our system by realizing a complete characterization of the two-qutrit space of higher-order Einstein&\#x2013;Podolsky&\#x2013;Rosen correlations. PB - OSA VL - 2 UR - http://www.osapublishing.org/optica/abstract.cfm?URI=optica-2-6-523 ER - TY - JOUR T1 - Extractable Work from Correlations JF - Physical Review X Y1 - 2015 A1 - Martí Perarnau-Llobet A1 - Karen V. Hovhannisyan A1 - Huber, Marcus A1 - Paul Skrzypczyk A1 - Nicolas Brunner A1 - Acín, Antonio VL - 5 UR - http://link.aps.org/doi/10.1103/PhysRevX.5.041011 ER - TY - JOUR T1 - Most energetic passive states JF - Physical Review E Y1 - 2015 A1 - Martí Perarnau-Llobet A1 - Karen V. Hovhannisyan A1 - Huber, Marcus A1 - Paul Skrzypczyk A1 - Tura, Jordi A1 - Acín, Antonio VL - 92 UR - http://link.aps.org/doi/10.1103/PhysRevE.92.042147 ER - TY - JOUR T1 - Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes JF - Journal of Physics A: Mathematical and Theoretical Y1 - 2015 A1 - Lancien, Cécilia A1 - Gühne, Otfried A1 - Sengupta, Ritabrata A1 - Huber, Marcus VL - 48 UR - http://stacks.iop.org/1751-8121/48/i=50/a=505302?key=crossref.2e93a28de5494676a0b866fa95ccebfd ER - TY - JOUR T1 - Thermodynamic cost of creating correlations JF - New Journal of Physics Y1 - 2015 A1 - Huber, Marcus A1 - Martí Perarnau-Llobet A1 - Karen V. Hovhannisyan A1 - Paul Skrzypczyk A1 - Klöckl, Claude A1 - Nicolas Brunner A1 - Acín, Antonio VL - 17 UR - http://stacks.iop.org/1367-2630/17/i=6/a=065008?key=crossref.b9c20f4bb628b09efcb912347c01851e ER - TY - JOUR T1 - Thermodynamics of creating correlations: {Limitations} and optimal protocols JF - Physical Review E Y1 - 2015 A1 - Bruschi, David Edward A1 - Martí Perarnau-Llobet A1 - Friis, Nicolai A1 - Karen V. Hovhannisyan A1 - Huber, Marcus VL - 91 UR - http://link.aps.org/doi/10.1103/PhysRevE.91.032118 ER - TY - JOUR T1 - Entanglement enhances cooling in microscopic quantum refrigerators JF - Physical Review E Y1 - 2014 A1 - Nicolas Brunner A1 - Huber, Marcus A1 - Noah Linden A1 - Sandu Popescu A1 - Ralph Silva A1 - Paul Skrzypczyk AB - Small self-contained quantum thermal machines function without external source of work or control, but using only incoherent interactions with thermal baths. Here we investigate the role of entanglement in a small self-contained quantum refrigerator. We first show that entanglement is detrimental as far as efficiency is concerned---fridges operating at efficiencies close to the Carnot limit do not feature any entanglement. Moving away from the Carnot regime, we show that entanglement can enhance cooling and energy transport. Hence a truly quantum refrigerator can outperform a classical one. Furthermore, the amount of entanglement alone quantifies the enhancement in cooling. VL - 89 IS - 3 JO - Phys. Rev. E ER - TY - JOUR T1 - Fifty years of Bell’s theorem JF - Journal of Physics A: Mathematical and Theoretical Y1 - 2014 A1 - Nicolas Brunner A1 - Gühne, Otfried A1 - Huber, Marcus VL - 47 UR - http://stacks.iop.org/1751-8121/47/i=42/a=420301?key=crossref.9d548f14fc7338685392dcd6228de98d ER - TY - JOUR T1 - Inequalities for the ranks of multipartite quantum states JF - Linear Algebra and its Applications Y1 - 2014 A1 - Josh Cadney A1 - Huber, Marcus A1 - Noah Linden A1 - Winter, Andreas AB - We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (\alpha-R\'enyi entropy for \alpha=1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other \alpha-R\'enyi entropies for α∈(0,1)∪(1,∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α∈(0,1) is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of \alpha=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., \alpha=1) and 0-R\'enyi entropy are exceptionally interesting measures of entanglement in the multipartite setting. VL - 452 JO - Linear Algebra and its Applications ER - TY - JOUR T1 - Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information JF - Nature Communications Y1 - 2014 A1 - Robert Fickler A1 - Radek Lapkiewicz A1 - Huber, Marcus A1 - Lavery, Martin P.J. A1 - Padgett, Miles J. A1 - Anton Zeilinger AB - Photonics has become a mature field of quantum information science, where integrated optical circuits offer a way to scale the complexity of the setup as well as the dimensionality of the quantum state. On photonic chips, paths are the natural way to encode information. To distribute those high-dimensional quantum states over large distances, transverse spatial modes, like orbital angular momentum (OAM) possessing Laguerre Gauss modes, are favourable as flying information carriers. Here we demonstrate a quantum interface between these two vibrant photonic fields. We create three-dimensional path entanglement between two photons in a non-linear crystal and use a mode sorter as the quantum interface to transfer the entanglement to the OAM degree of freedom. Thus our results show a novel, flexible way to create high-dimensional spatial mode entanglement. Moreover, they pave the way to implement broad complex quantum networks where high-dimensionally entangled states could be distributed over distant photonic chips. VL - 5 JO - Nat Comms ER - TY - JOUR T1 - Steering Bound Entangled States: A Counterexample to the Stronger Peres Conjecture JF - Physical Review Letters Y1 - 2014 A1 - Moroder, Tobias A1 - Gittsovich, Oleg A1 - Huber, Marcus A1 - Gühne, Otfried AB - Quantum correlations are at the heart of many applications in quantum information science and, at the same time, they form the basis for discussions about genuine quantum effects and their difference to classical physics. On one hand, entanglement theory provides the tools to quantify correlations in information processing and many results have been obtained to discriminate useful entanglement, which can be distilled to a pure form, from bound entanglement, being of limited use in many applications. On the other hand, for discriminating quantum phenomena from their classical counterparts, Schr\"odinger and Bell introduced the notions of steering and local hidden variable models. We provide a method to generate systematically bound entangled quantum states which can still be used for steering and therefore to rule out local hidden state models. This sheds light on the relations between the various views on quantum correlations and disproves a widespread conjecture known as the stronger Peres conjecture. For practical applications, it implies that even the weakest form of entanglement can be certified in a semi-device independent way. VL - 113 IS - 5 JO - Phys. Rev. Lett. ER - TY - JOUR T1 - Unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder inequalities JF - Physical Review A Y1 - 2014 A1 - Wölk, Sabine A1 - Huber, Marcus A1 - Gühne, Otfried AB - We present a unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to use it to derive not only already known entanglement criteria, but also criteria which were unknown so far. We systematically investigate its potential and limitations to detect bipartite and multipartite entanglement. Furthermore, we describe how to apply our generalized entanglement detection scheme to find the measurement directions to verify the entanglement of a given state experimentally. VL - 90 IS - 2 JO - Phys. Rev. A ER - TY - JOUR T1 - Witnessing Genuine Multipartite Entanglement with Positive Maps JF - Physical Review Letters Y1 - 2014 A1 - Huber, Marcus A1 - Sengupta, Ritabrata AB - We derive a general framework that connects every positive map with a corresponding witness for partial separability in multipartite quantum systems. We show that many previous approaches were intimately connected to the witnesses derived from partial transposition and that such criteria can easily be outperformed in higher dimensions by non-decomposable maps. As an exemplary case we present a witness that is capable of detecting genuine multipartite entanglement in bound entangled states. VL - 113 IS - 10 JO - Phys. Rev. Lett. ER - TY - JOUR T1 - Entangled singularity patterns of photons in {Ince}-{Gauss} modes JF - Physical Review A Y1 - 2013 A1 - Krenn, Mario A1 - Robert Fickler A1 - Huber, Marcus A1 - Radek Lapkiewicz A1 - Plick, William A1 - Ramelow, Sven A1 - Anton Zeilinger VL - 87 UR - http://link.aps.org/doi/10.1103/PhysRevA.87.012326 ER - TY - JOUR T1 - Entanglement Generation is Not Necessary for Optimal Work Extraction JF - Physical Review Letters Y1 - 2013 A1 - Hovhannisyan, Karen A1 - Martí Perarnau-Llobet A1 - Huber, Marcus A1 - Acín, Antonio AB - We consider reversible work extraction from identical quantum batteries. From an ensemble of individually passive states, work can be produced only via global unitary (and thus entangling) operations. However, we show here that there always exists a method to extract all possible work without creating any entanglement, at the price of generically requiring more operations (i.e. additional time). We then study faster methods to extract work and provide a quantitative relation between the amount of generated multipartite entanglement and extractable work. Our results suggest a general relation between entanglement generation and the power of work extraction. VL - 111 IS - 24 JO - Phys. Rev. Lett. ER - TY - JOUR T1 - Entropy vector formalism and the structure of multidimensional entanglement in multipartite systems JF - Physical Review A Y1 - 2013 A1 - Huber, Marcus A1 - Martí Perarnau-Llobet A1 - de Vicente, Julio I. VL - 88 UR - http://arxiv.org/abs/1307.3541 IS - 4 JO - Phys. Rev. A ER - TY - JOUR T1 - Examining the dimensionality of genuine multipartite entanglement JF - Quantum Information Processing Y1 - 2013 A1 - Spengler, Christoph A1 - Huber, Marcus A1 - Gabriel, Andreas A1 - Hiesmayr, Beatrix C. VL - 12 UR - http://link.springer.com/10.1007/s11128-012-0369-8 ER - TY - JOUR T1 - Proving the generation of genuine multipartite entanglement in a single-neutron interferometer experiment JF - New Journal of Physics Y1 - 2013 A1 - Erdösi, Daniel A1 - Huber, Marcus A1 - Hiesmayr, Beatrix C. A1 - Hasegawa, Yuji VL - 15 UR - http://stacks.iop.org/1367-2630/15/i=2/a=023033?key=crossref.b714f63bb49eb9e64ce9a70aa4a6ddfe ER - TY - JOUR T1 - Structure of Multidimensional Entanglement in Multipartite Systems JF - Physical Review Letters Y1 - 2013 A1 - Huber, Marcus A1 - de Vicente, Julio I. AB - We explore the structure of multipartite quantum systems which are entangled in multiple degrees of freedom. We find necessary and sufficient conditions for the characterization of tripartite systems and necessary conditions for any number of parties. Furthermore we develop a framework of multilevel witnesses for efficient discrimination and quantification of multidimensional entanglement that is applicable for an arbitrary number of systems and dimensions. VL - 110 IS - 3 JO - Phys. Rev. Lett. ER - TY - JOUR T1 - Weak randomness in device-independent quantum key distribution and the advantage of using high-dimensional entanglement JF - Physical Review A Y1 - 2013 A1 - Huber, Marcus A1 - Pawłowski, Marcin VL - 88 IS - 3 JO - Phys. Rev. A ER -