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"title": "Fast Thermalization from the Eigenstate Thermalization Hypothesis",
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"note": "We thank Charles Xu for early discussions on the topic of this paper. We thank Robert (Hsin-Yuan) Huang for suggesting to check quantum expander properties numerically. We thank Cambyse Rouz\u2000 for discussions on approximate tensorization. CFC is supported by Caltech RA fellowship and the Eddleman Fellowship.",
"abstract": "The Eigenstate Thermalization Hypothesis (ETH) has played a major role in explaining thermodynamic phenomena in quantum systems. However, so far, no connection has been known between ETH and the timescale of thermalization. In this paper, we rigorously show that ETH indeed implies fast thermalization to the global Gibbs state. We show fast convergence for two models of thermalization. In the first, the system is weakly coupled to a bath of (quasi)-free Fermions that we control. We derive a finitely-resolved version of Davies' generator, with explicit error bounds and resource estimates, that describes the joint evolution at finite times. The second is Quantum Metropolis Sampling, a quantum algorithm for preparing Gibbs states on a quantum computer. In both cases, no guarantee for fast convergence was previously known for non-commuting Hamiltonians, partly due to technical issues with a finite energy resolution. The critical feature of ETH we exploit is that the Hamiltonian can be modeled by random matrix theory below a sufficiently small energy scale. We show this gives quantum expander at nearby eigenstates of the Hamiltonian. This then implies fast convergence to the global Gibbs state by mapping the problem to a one-dimensional classical random walk on the spectrum of the Hamiltonian. Our results explain finite-time thermalization in chaotic open quantum systems and suggest an alternative formulation of ETH in terms of quantum expanders, which we confirm numerically for small systems.",
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"title": "Random quantum circuits transform local noise into global white noise",
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"note": "We thank Adam Bouland, Bill Fefferman, Zeph Landau, Yunchao Liu, Oskar Painter, John Preskill, and Thomas Vidick for helpful feedback about this work. AD and FB acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907). This material is also based upon work supported by the NSF Graduate Research Fellowship under Grant No. DGE-1745301. NHJ is supported in part by the Stanford Q-FARM Bloch Fellowship in Quantum Science and Engineering. NHJ would like to thank the Aspen Center for Physics for its hospitality during the completion of part of this work. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.",
"abstract": "We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. We show that, for local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution p_(noisy) of a generic noisy circuit instance and the output distribution pideal of the corresponding noiseless instance shrink exponentially with the expected number of gate-level errors, as F = exp(\u22122s\u03f5 \u00b1 O(s\u03f5\u00b2)), where \u03f5 is the probability of error per circuit location and s is the number of two-qubit gates. Furthermore, if the noise is incoherent, the output distribution approaches the uniform distribution p_(unif) at precisely the same rate and can be approximated as p_(noisy) \u2248 F_(p_(ideal)) + (1\u2212F)p_(unif), that is, local errors are scrambled by the random quantum circuit and contribute only white noise (uniform output). Importantly, we upper bound the total variation error (averaged over random circuit instance) in this approximation as O(F\u03f5\u221as), so the \"white-noise approximation\" is meaningful when \u03f5\u221as \u226a 1, a quadratically weaker condition than the \u03f5s\u226a1 requirement to maintain high fidelity. The bound applies when the circuit size satisfies s \u2265 \u03a9(nlog(n)) and the inverse error rate satisfies \u03f5\u207b\u00b9 \u2265 \u03a9\u0303 (n). The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; it was an underlying assumption in complexity-theoretic arguments that low-fidelity random quantum circuits cannot be efficiently sampled classically. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.",
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"note": "We thank Yuan Su and Mario Berta for helpful discussions and Joel Tropp and Andrew Lucas for related collaborations. CFC is especially thankful for Joel Tropp for introducing to him the subject of matrix concentration inequalities. After this work was completed, we became aware of related work by Qi Zhao, You Zhou, Alexander F. Shaw, Tongyang Li, and Andrew M. Childs that also studies Hamiltonian simulation with random input states. We thank them for letting us know about their work. CFC is supported by Caltech RA fellowship and the Eddleman Fellowship.",
"abstract": "Quantum simulation is expected to be one of the key applications of future quantum computers. Product formulas, or Trotterization, are the oldest and, still today, an appealing method for quantum simulation. For an accurate product formula approximation in the spectral norm, the state-of-the-art gate complexity depends on the number of Hamiltonian terms and a certain 1-norm of its local terms. This work studies the concentration aspects of Trotter error: we prove that, typically, the Trotter error exhibits 2-norm (i.e., incoherent) scaling; the current estimate with 1-norm (i.e., coherent) scaling is for the worst cases. For k-local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from a 1-design ensemble (e.g., computational basis states). Our gate count depends on the number of Hamiltonian terms but replaces the 1-norm quantity by its analog in 2-norm, giving significant speedup for systems with large connectivity. Our results generalize to Hamiltonians with Fermionic terms and when the input state is drawn from a low-particle number subspace. Further, when the Hamiltonian itself has Gaussian coefficients (e.g., the SYK models), we show the stronger result that the 2-norm behavior persists even for the worst input state. Our main technical tool is a family of simple but versatile inequalities from non-commutative martingales called uniform smoothness. We use them to derive Hypercontractivity, namely p-norm estimates for low-degree polynomials, which implies concentration via Markov's inequality. In terms of optimality, we give examples that simultaneously match our p-norm bounds and the spectral norm bounds. Therefore, our improvement is due to asking a qualitatively different question from the spectral norm bounds. Our results give evidence that product formulas in practice may generically work much better than expected.",
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"title": "Emergent Randomness and Benchmarking from Many-Body Quantum Chaos",
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"note": "We acknowledge discussions with Abhinav Deshpande and Alexey Gorshkov as well as funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907), the NSF CAREER award (1753386), the AFOSR YIP (FA9550-19-1-0044), the DARPA ONISQ program (W911NF2010021), the Army Research Office MURI program (W911NF2010136), the NSF QLCI program (2016245), and Fred Blum. JC acknowledges support from the IQIM postdoctoral fellowship. ALS acknowledges support from the Eddleman Quantum graduate fellowship. JPC acknowledges support from the PMA Prize postdoctoral fellowship. HP acknowledges support by the Gordon and Betty Moore Foundation. HH is supported by the J. Yang & Family Foundation. AK acknowledges funding from the Harvard Quantum Initiative (HQI) graduate fellowship. JSC is supported by a Junior Fellowship from the Harvard Society of Fellows and the U.S. Department of Energy under grant Contract Number DE-SC0012567. SC acknowledges support from the Miller Institute for Basic Research in Science. \r\n\r\nJC and ALS contributed equally to this work.",
"abstract": "Chaotic quantum many-body dynamics typically lead to relaxation of local observables. In this process, known as quantum thermalization, a subregion reaches a thermal state due to quantum correlations with the remainder of the system, which acts as an intrinsic bath. While the bath is generally assumed to be unobserved, modern quantum science experiments have the ability to track both subsystem and bath at a microscopic level. Here, by utilizing this ability, we discover that measurement results associated with small subsystems exhibit universal random statistics following chaotic quantum many-body dynamics, a phenomenon beyond the standard paradigm of quantum thermalization. We explain these observations with an ensemble of pure states, defined via correlations with the bath, that dynamically acquires a close to random distribution. Such random ensembles play an important role in quantum information science, associated with quantum supremacy tests and device verification, but typically require highly-engineered, time-dependent control for their preparation. In contrast, our approach uncovers random ensembles naturally emerging from evolution with a time-independent Hamiltonian. As an application of this emergent randomness, we develop a benchmarking protocol which estimates the many-body fidelity during generic chaotic evolution and demonstrate it using our Rydberg quantum simulator. Our work has wide ranging implications for the understanding of quantum many-body chaos and thermalization in terms of emergent randomness and at the same time paves the way for applications of this concept in a much wider context.",
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"title": "Matrix Product Density Operators: when do they have a local parent Hamiltonian?",
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"note": "We thank Jean-Francois Quint for comments on multiplicative ergodic theory. We thank Mario Berta,\r\nMarco Tomamichel, Hao-Chung Cheng for discussions about the DPI for CMI. CFC is thankful for\r\nPhysics TA Relief Fellowship and the Physics TA Fellowship at Caltech. KK acknowledges funding\r\nprovided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF\r\nGrant PHY-1733907) and MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number\r\nJPMXS0120319794. FB acknowledges funding from NSF.",
"abstract": "We study whether one can write a Matrix Product Density Operator (MPDO) as the Gibbs state of a quasi-local parent Hamiltonian. We conjecture this is the case for generic MPDO and give supporting evidences. To investigate the locality of the parent Hamiltonian, we take the approach of checking whether the quantum conditional mutual information decays exponentially. The MPDO we consider are constructed from a chain of 1-input/2-output (`Y-shaped') completely-positive maps, i.e. the MPDO have a local purification. We derive an upper bound on the conditional mutual information for bistochastic channels and strictly positive channels, and show that it decays exponentially if the correctable algebra of the channel is trivial. We also introduce a conjecture on a quantum data processing inequality that implies the exponential decay of the conditional mutual information for every Y-shaped channel with trivial correctable algebra. We additionally investigate a close but nonequivalent cousin: MPDO measured in a local basis. We provide sufficient conditions for the exponential decay of the conditional mutual information of the measured states, and numerically confirmed they are generically true for certain random MPDO.",
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"title": "Fast and robust quantum state tomography from few basis measurements",
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"note": "We thank C. Ferrie, T. Grurl, C. Lancien, R. Konig and J.A.\r\nTropp for valuable input and helpful discussions. F.B. and R.K. acknowledge funding\r\nfrom the US National Science Foundation (PHY1733907). The Institute for Quantum\r\nInformation and Matter is an NSF Physics Frontiers Center. D.S.F. acknowledges financial\r\nsupport from VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059).\r\n\r\nData and code availability:\r\nSource data and code are available for this paper [Fra20].\r\nAll other data that support the plots within this paper and other findings of this study\r\nare available from the corresponding author upon reasonable request.",
"abstract": "Quantum state tomography is a powerful, but resource-intensive, general solution for numerous quantum information processing tasks. This motivates the design of robust tomography procedures that use relevant resources as sparingly as possible. Important cost factors include the number of state copies and measurement settings, as well as classical postprocessing time and memory. In this work, we present and analyze an online tomography algorithm designed to optimize all the aforementioned resources at the cost of a worse dependence on accuracy. The protocol is the first to give provably optimal performance in terms of rank and dimension for state copies, measurement settings and memory. Classical runtime is also reduced substantially and numerical experiments demonstrate a favorable comparison with other state-of-the-art techniques. Further improvements are possible by executing the algorithm on a quantum computer, giving a quantum speedup for quantum state tomography.",
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"title": "Efficient classical simulation of random shallow 2D quantum circuits",
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"note": "We thank Nicole Yunger Halpern, Richard Kueng, Saeed Mehraban, Ramis Movassagh, Anand Natarajan, and Mehdi Soleimanifar for helpful discussions. Numerical simulations were performed using the ITensor Library. This work was funded by NSF grants CCF-1452616, CCF-1729369, PHY-1818914, and DGE-1745301, as well as ARO contract W911NF-17-1-0433, the MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space and the Dominic Orr Fellowship at Caltech. The Institute for Quantum Information and Matter (IQIM) is an NSF Physics Frontiers Center (PHY-1733907).",
"abstract": "Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates. We furthermore conjecture that sufficiently shallow random circuits are efficiently simulable more generally. To this end, we propose and analyze two simulation algorithms. Implementing one of our algorithms numerically, we give strong evidence that it is efficient both asymptotically and, in some cases, in practice. To argue analytically for efficiency, we reduce the simulation of 2D shallow random circuits to the simulation of a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements -- a type of process that has generally been observed to undergo a phase transition from an efficient-to-simulate regime to an inefficient-to-simulate regime as measurement strength is varied. Using a mapping from quantum circuits to statistical mechanical models, we give evidence that a similar computational phase transition occurs for our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied.",
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"title": "Models of quantum complexity growth",
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"note": "The authors would like thank Dorit Aharonov, Thom Bohdanowicz, Elizabeth Crosson, Felix Haehl, Aram Harrow, Tomas Jochym-O'Connor, Hugo Marrochio, Grant Salton, Eugene Tang, and Thomas Vidick for inspiring discussions and valuable feedback. All authors acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907). JP is supported in part by DOE Award DE-SC0018407 and by the Simons Foundation It from Qubit Collaboration. RK is supported in part by the Office of Naval Research (Award N00014-17-1-2146) and the Army Research Office (Award W911NF121054). NHJ would like to thank the IQIM at Caltech, McGill University, and UC Berkeley for their hospitality during the completion of this work. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.",
"abstract": "The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many-body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a short-cut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary k-designs, ensembles which capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind. Moreover, our results apply under a strong definition of quantum complexity based on optimal distinguishing measurements.",
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"note": "We thank the Google AI Quantum team for useful discussion. EF also thanks Soonwon Choi, Misha Lukin, Hannes Pichler, Sheng-Tao Wang and Leo Zhou for many good chats. We acknowledge Jeffrey Goldstone for help with the acknowledgements. The work of EF was partially supported from NSF grant CCF-1729369 and ARO contract W911NF-17-1-0433.",
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"title": "Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning",
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"note": "We thank Scott Aaronson, Joran van Apeldoorn, Andr\u00e1s Gily\u00e9n, Cupjin Huang, and anonymous reviewers for helpful discussions. We are also grateful to Joran van Apeldoorn and Andr\u00e1s Gily\u00e9n for sharing a working draft of [4] with us. FB was supported by NSF. CYL and AK are supported by the Department of Defense. TL is supported by NSF CCF-1526380. XW is supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams program. XW is also supported by NSF grants CCF-1755800 and CCF-1816695.",
"abstract": "We give two quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes access to an oracle to the matrices at unit cost. We show that it has run time O\u0303(s^2(\u221a((m\u03f5)^(\u221210)) + \u221a((n\u03f5)^(\u221212))), with \u03f5 the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms when m \u2248 n. The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is O\u0303 (\u221am + poly(r))\u22c5poly(log m,log n,B,\u03f5^(\u22121)), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in n and polynomially in r. \r\n\r\nWe apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state \u03c1 with rank at most r, we show we can find in time \u221am\u22c5poly(log m,log n,r,\u03f5^(\u22121)) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as \u03c1 on the m measurements, up to error \u03f5. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. \r\n \r\nAs in previous work, we obtain our algorithm by \"quantizing\" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest.",
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"note": "(Submitted on 18 Sep 2016 (v1), last revised 20 Apr 2017 (this version, v4)) \r\n\r\nWe thank Joran van Apeldoorn, Ronald de Wolf, Andras Gilyen, Aram Harrow, Sander Gribling, Matt Hastings, Cedric Yen-Yu Lin, Ojas Parekh, and David Poulin for interesting discussions and useful comments on the paper.",
"abstract": "We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst case running time n^(1/2)m^(1/2)S^2 poly(log(n), log(m), R, r, 1/\u03b4), with n and s the dimension and sparsity of the input matrices, respectively, m the number of constraints, \u03b4 the accuracy of the solution, and R, r upper bounds on the size of the optimal primal and dual solutions. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in n and m. We prove the algorithm cannot be substantially improved giving a \u03a9(n^(1/2) + m^(1/2)) quantum lower bound for solving semidefinite programs with constant s, R, r and \u03b4. \r\n\r\nWe then argue that in some instances the algorithm offer even exponential speed-ups. This is the case whenever the quantum Gibbs states of Hamiltonians given by linear combinations of the input matrices of the SDP can be prepared efficiently on a quantum computer. An example are SDPs in which the input matrices have low-rank: For SDPs with the maximum rank of any input matrix bounded by rank, we show the quantum algorithm runs in time poly(log(n), log(m), rank, r, R, \u03b4)m^(1/2). \r\n\r\nThe quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on an classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need of solving an inner linear program which may be of independent interest.",
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"[1] P.W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a\r\nquantum computer. SIAM review 41.2, 303 (1999).\r\n[2] L.K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev.\r\nLett. 79, 325 (1997).\r\n[3] L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review 38, 49 (1996).\r\n[4] M.X. Goemans. Semidefinite programming in combinatorial optimization. Mathematical Programming\r\n79, 143 (1997).\r\n[5] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004.\r\n[6] M.X. Goemans and D.P. Williamson. Improved approximation algorithms for maximum cut\r\nand satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115\r\n(1995).\r\n[7] Y.T. Lee, A. Sidford, and S.C. Wong. A faster cutting plane method and its implications for\r\ncombinatorial and convex optimization. IEEE 56th Annual Symposium on the Foundations\r\nof Computer Science (FOCS), 2015.\r\n[8] S. Arora, E. Hazan and S. Kale. Fast algorithms for approximate semidefinite programming\r\nusing the multiplicative weights update method. 46th Annual IEEE Symposium on Foundations\r\nof Computer Science, 2005. FOCS 2005.\r\n[9] S. Arora and S. Kale. A combinatorial, primal-dual approach to semidefinite programs. Proceedings\r\nof the thirty-ninth annual ACM symposium on Theory of computing. ACM, 2007.\r\n[10] S. Arora, E. Hazan and S. Kale. The Multiplicative Weights Update Method: a Meta-\r\nAlgorithm and Applications. Theory of Computing 8, 121 (2012).\r\n[11] K. Temme et al. Quantum metropolis sampling. Nature 471, 87 (2011).\r\n[12] M.H. Yung and A. Aspuru-Guzik. A quantum\u2013quantum Metropolis algorithm. Proceedings\r\nof the National Academy of Sciences 109, 754 (2012).\r\n[13] D. Poulin and P. Wocjan. Sampling from the thermal quantum Gibbs state and evaluating\r\npartition functions with a quantum computer. Phys. Rev. Lett. 103, 220502 (2009).\r\n[14] A.N. Chowdhury and R.D. Somma. Quantum algorithms for Gibbs sampling and hittingtime\r\nestimation. arXiv preprint arXiv:1603.02940 (2016).\r\n[15] M. Kastoryano and F.G.S.L. Brandao. Quantum Gibbs Samplers: the commuting case. Comm.\r\nMath. Phys. 344, 915 (2016).\r\n[16] F.G.S.L. Brandao and M. Kastoryano. Finite correlation length implies efficient preparation of\r\nquantum thermal states. In preparation.\r\n[17] G. Brassard et al. Quantum amplitude amplification and estimation. Contemporary Mathematics\r\n305, 53 (2002).\r\n[18] E.T. Jaynes. Information Theory and Statistical Mechanics II. Phys. Rev. 108, 171 (1957).\r\n[19] J.R. Lee, P. Raghavendra, and D. Steurer. Lower bounds on the size of semidefinite programming\r\nrelaxations. Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory\r\nof Computing. ACM, 2015.\r\n[20] R. Ahlswede, A. Winter. Strong Converse for Identification via Quantum Channels\". IEEE\r\nTrans. Information Theory 48, 569 (2003).\r\n[21] N.E. Sherman, T. Devakul, M.B. Hastings, R.R.P. Singh. Phys. Rev. E 93, 022128 (2016).\r\n[22] S. Lloyd, M. Mohseni, P. Rebentrost. Quantum Principal Component Analysis. Nature\r\nPhysics 10, 631 (2014).\r\n[23] A. Childs and R. Kothari. Limitations on the simulation of non-sparse Hamiltonians. arXiv\r\npreprint arXiv:0908.4398 (2009).\r\n[24] W.R. Gilks. Markov chain monte carlo. John Wiley and Sons, Ltd. Chicago (2005).\r\n[25] D.W. Berry, A.M. Childs, R. Kothari. Hamiltonian simulation with nearly optimal dependence\r\non all parameters. Proceedings of the 56th IEEE Symposium on Foundations of Computer\r\nScience (FOCS 2015), 792 (2015).\r\n[26] J. A. Tropp. User-friendly tail bounds for sums of random matrices, 2010, arXiv:1004.4389.\r\n[27] S. Kimmel, C. Yen-Yu Lin, G. Hao Low, M. Ozols, T.J. Yoder. Hamiltonian Simulation with\r\nOptimal Sample Complexity. arXiv:1608.00281.\r\n[28] H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf. Quantum fingerprinting. Phys. Rev. Lett.\r\n87(16):167902, 2001. quant-ph/0102001."
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"note": "Lecture notes for the 5th Summer School on Mathematical Physics at the Universidad de Los Andes, Bogot\u00e1, Colombia. \r\n\r\nWe would like to thank our hosts Alonso Botero, Andres Schlief and Monika Winklmeier from the Universidad de Los Andes for inviting us and putting together the summer school. We would also like to thank the enthusiastic students who attended.",
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"abstract": "The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this work, we relax this assumption, and reconsider the original protocol of Colbeck and Renner, Nature Physics 8, 450-454 (2012), on randomness amplification using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with 0 \u2264 \u03b5 <(2^((1/12))\u22121)/2(2^((1/12))+1) \u2248 0.0144). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source against this attack for the parameter range 0 \u2264 \u03b5 < 0.0132. We leave the case of a yet more general attack on the amplification protocol as an interesting open problem.",
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"[1] R. Arnon-Friedman, A. Ta-Shma, Limits of privacy amplification against non-signalling memory attacks,\r\nPhys. Rev. A 86, 062333 (2012).\r\n[2] S.L. Braunstein & C.M. Caves, Wringing out better Bell inequalities, Annals of Physics 202, 22 (1990).\r\n[3] R. Koenig, R. Renner & C. Schaffner, The operational meaning of min- and max-entropy, IEEE Trans. Inf.\r\nTh., vol. 55, no. 9 (2009).\r\n[4] R. Colbeck & R. Renner, Free randomness can be amplified, Nature Physics 8, 450-454 (2012).\r\n[5] R. Gallego, L. Masanes, G. De La Torre, C. Dhara, L. Aolita, & A. Acin, Full randomness from arbitrarily\r\ndeterministic events, Nature Communications 4, 2654 (2013).\r\n[6] P. Mironowicz, R. Gallego & M. Paw\u0142owski, Amplification of arbitrarily weak randomness, Phys. Rev. A\r\n91, 032317 (2015).\r\n[7] F.G.S.L. Brand\u00e3o, R. Ramanathan, A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, T. Szarek &\r\nH. Wojew\u00f3dka, Robust device-independent randomness amplification with few devices, arXiv:1310.4544v2\r\n[quant-ph] (2015).\r\n[8] K.M. Chung, Y. Shi & X. Wu, Physical randomness extractors: generating random numbers with minimal\r\nassumptions, arXiv:1402.4797 (2014).\r\n[9] R. Ramanathan, F.G.S.L. Brand\u00e3o, K. Horodecki, M. Horodecki, P. Horodecki & H. Wojew\u00f3dka,\r\nRandomness amplification against no-signaling adversaries using two devices, arXiv:1504.06313 [quant-ph]\r\n(2015).\r\n[10] N. Gisin, A.A. M\u00e9thot & V. Scarani, Pseudo-telepathy: input cardinality and Bell-type inequalities, International\r\nJournal of Quantum Information 5: 525-534 (2007).\r\n[11] D.M. Greenberger, M.A. Horne & A. Zeilinger, Bells theorem, quantum theory, and conceptions of the\r\nuniverse, (Kluwer, Dordrecht), 69 (1989).\r\n[12] A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, M. Paw\u0142owski & R. Ramanathan, Free randomness\r\namplification using bipartite chain correlations, Phys. Rev. A 90, 032322 (2014).\r\n[13] O. G\u00fchne, G. T\u00f3th, P. Hyllus & H.J. Briegel, Bell inequalities for graph states, Phys. Rev. Lett. 95, 120405\r\n(2005).\r\n[14] N.S. Jones & L. Masanes, Interconversion of nonlocal correlations, Phys. Rev. A 72, 052312 (2005).\r\n[15] R. Ramanathan, J. Tuziemski, M. Horodecki & P. Horodecki, No quantum realization of extremal nosignaling\r\nboxes, arXiv:1410.0947v2 [quant-ph] (2015).\r\n[16] M. Santha & U.V. Vazirani, Generating quasi-random sequences from slightly-random sources, Proceedings\r\nof the 25th IEEE Symposium on Foundations of Computer Science (FOCS\u201984), 434 (1984)."
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"title": "Estimating operator norms using covering nets",
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"note": "We thank Jop Briet, Pablo Parrilo and Ben Recht for interesting discussions. FGSLB is supported by EPSRC. AWH was funded by NSF grants CCF-1111382 and CCF-1452616, ARO contract W911NF-12-1-0486 and a Leverhulme Trust Visiting Professorship VP2-2013-041. Part of this work was done while A.W. was visiting UCL.",
"abstract": "We present several polynomial- and quasipolynomial-time approximation schemes for a large class of generalized operator norms. Special cases include the 2\u2192q norm of matrices for q>2, the support function of the set of separable quantum states, finding the least noisy output of entanglement-breaking quantum channels, and approximating the injective tensor norm for a map between two Banach spaces whose factorization norm through \u2113^n_1 is bounded. \r\nThese reproduce and in some cases improve upon the performance of previous algorithms by Brand\u00e3o-Christandl-Yard and followup work, which were based on the Sum-of-Squares hierarchy and whose analysis used techniques from quantum information such as the monogamy principle of entanglement. Our algorithms, by contrast, are based on brute force enumeration over carefully chosen covering nets. These have the advantage of using less memory, having much simpler proofs and giving new geometric insights into the problem. Net-based algorithms for similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in each case with a run-time that is exponential in the rank of some matrix. We achieve polynomial or quasipolynomial runtimes by using the much smaller nets that exist in \u2113_1 spaces. This principle has been used in learning theory, where it is known as Maurey's empirical method.",
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"[ABS10] Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games\r\nand related problems. In FOCS, pages 563{572, 2010.\r\n[AIM14] S. Aaronson, R. Impagliazzo, and D. Moshkovitz. AM with multiple Merlins. In Computational\r\nComplexity (CCC), 2014 IEEE 29th Conference on, pages 44{55, June 2014, arXiv:1401.6848.\r\n[ALSV13a] Noga Alon, Troy Lee, Adi Shraibman, and Santosh Vempala. The approximate rank of a matrix\r\nand its algorithmic applications: Approximate rank. In Proceedings of the 45th Annual ACM\r\nSymposium on Symposium on Theory of Computing, STOC '13, pages 675{684, 2013.\r\n[ALSV13b] Noga Alon, Troy Lee, Adi Shraibman, and Santosh Vempala. The approximate rank of a matrix\r\nand its algorithmic applications: Approximate rank. In Proceedings of the Forty-\ufffdfth Annual\r\nACM Symposium on Theory of Computing, STOC '13, pages 675{684. ACM, 2013.\r\n[BaCY11] Fernando G.S.L. Brand~ao, Matthias Christandl, and Jon Yard. A quasipolynomial-time algorithm\r\nfor the quantum separability problem. In Proceedings of the 43rd annual ACM symposium\r\non Theory of computing, STOC '11, pages 343{352, 2011, arXiv:1011.2751.\r\nBBH+12] Boaz Barak, Fernando G.S.L. Brand~ao, Aram W. Harrow, Jonathan Kelner, David Steurer, and\r\nYuan Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In STOC '12,\r\nSTOC '12, pages 307{326, 2012, arXiv:1205.4484.\r\n[BC11] F.G.S.L. Brand~ao and M. Christandl. Detection of multiparticle entanglement: Quantifying the\r\nsearch for symmetric extensions, 2011, arXiv:1105.5720.\r\n[BCL94] Keith Ball, Eric A. Carlen, and Elliott H. Lieb. Sharp uniform convexity and smoothness\r\ninequalities for trace norms. Inventiones mathematicae, 115(1):463{482, 1994.\r\n[BCY11] F. G. S. L. Brand~ao, M. Christandl, and J. Yard. Faithful squashed entanglement. Commun.\r\nMath. Phys., 306(3):805{830, 2011, arXiv:1010.1750.\r\n[BH13] Fernando G. S. L. Brand~ao and Aram W. Harrow. 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Birkh\ufffdauser, 2007.\r\n[Ste05] Daureen Steinberg. Computation of matrix norms with applications to robust optimization.\r\nMaster's thesis, Technion, 2005. Available on A. Nemirovski's website http://www2.isye.gatech.\r\nedu/\ufffdnemirovs/.\r\n[SW12] Yaoyun Shi and Xiaodi Wu. Epsilon-net method for optimizations over separable states. In\r\nProceedings of the 39th International Colloquium Conference on Automata, Languages, and\r\nProgramming - Volume Part I, ICALP'12, pages 798{809, Berlin, Heidelberg, 2012. Springer-\r\nVerlag, arXiv:1112.0808.\r\n[TJ74] Nicole Tomczak-Jaegermann. The moduli of smoothness and convexity and the Rademacher\r\naverages of the trace classes sfpg (1 \ufffd p < 1). Studia Mathematica, 50(2):163{182, 1974.\r\n[Tro10] J. A. Tropp. User-friendly tail bounds for sums of random matrices, 2010, arXiv:1004.4389.\r\n[Yan06] D. Yang. A simple proof of monogamy of entanglement. Phys. Lett., 360(1):249, 2006,\r\narXiv:quant-ph/0604168."
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"note": "The paper is supported by ERC AdG grant QOLAPS, EC grant RAQUEL and by Foundation for Polish Science TEAM project co-financed by the EU European Regional Development Fund. FB acknowledges support from EPSRC and Polish Ministry of Science and Higher Education Grant no. IdP2011 000361. Part of this work was done in National Quantum Information Center of Gd\u00e1nsk. Part of this work was done when F. B., R. R., K. H. and M. H. attended the program \u201cMathematical Challenges in Quantum Information\u201d at the Isaac Newton Institute for Mathematical Sciences in the University of Cambridge.",
"abstract": "Recently the first physically realistic protocol amplifying the randomness of Santha-Vazirani sources using a finite number of no-signaling devices and with a constant rate of noise has been proposed, however there still remained the open question whether this can be accomplished under the minimal conditions necessary for the task. Namely, is it possible to achieve randomness amplification using only two no-signaling devices and in a situation where the violation of a Bell inequality implies only an upper bound for some outcome probability for some setting combination? Here, we solve this problem and present the first device-independent protocol for the task of randomness amplification of Santha-Vazirani sources using a device consisting of only two non-signaling components. We show that the protocol can amplify any such source that is not fully deterministic into a totally random source while tolerating a constant noise rate and prove the security of the protocol against general no-signaling adversaries. The minimum requirement for a device-independent Bell inequality based protocol for obtaining randomness against no-signaling attacks is that every no-signaling box that obtains the observed Bell violation has the conditional probability P(x|u) of at least a single input-output pair (u,x) bounded from above. We show how one can construct protocols for randomness amplification in this minimalistic scenario.",
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"[1] M. Santha and U. V. Vazirani. Generating Quasi-Random\r\nSequences from Slightly-Random Sources. Proceedings of\r\nthe 25th IEEE Symposium on Foundations of Computer\r\nScience (FOCS\u201984), 434 (1984).\r\n[2] B. Chor and O. Goldreich. Unbiased bits from sources of\r\nweak randomness and probabilistic communication complexity.\r\nSIAM Journal on Computing, 17(2): 230 (1988).\r\n[3] J. Barrett and N. Gisin. How Much Measurement Independence\r\nIs Needed to Demonstrate Nonlocality? Phys.\r\nRev. Lett. 106, 100406 (2011).\r\n[4] M. J. W. Hall. Local Deterministic Model of Singlet State\r\nCorrelations Based on Relaxing Measurement Independence.\r\nPhys. Rev. Lett. 105, 250404 (2010).\r\n[5] R. Colbeck and R. Renner. Free randomness can be amplified.\r\nNature Physics 8, 450 (2012).\r\n[6] Xin-Li. Extractors for a constant number of independent\r\nsources with polylogarithmic min-entropy. (to appear in\r\nFOCS 2013).\r\n[7] R. Gallego, L. Masanes, G. de la Torre, C. Dhara, L. Aolita\r\nand A. Acin. Full randomness from arbitrarily deterministic\r\nevents. arXiv:1210.6514 (2012).\r\n[8] F. G. S. L. Brandao, R. Ramanathan, A. Grudka, K.\r\nHorodecki, M. Horodecki, P. Horodecki, T. Szarek and\r\nH.Wojewodka. Robust Device-Independent Randomness\r\nAmplification with Few Devices. arXiv: 1310.4544 (2013).\r\n[9] R. Ramanathan, F. G. S. L. Brandao, A. Grudka, K.\r\nHorodecki, M. Horodecki and P. Horodecki. Robust\r\nDevice-Independent Randomness Amplification. arXiv:\r\n1308.4635 (2013).\r\n[10] O. Guehne, G. Toth, P. Hyllus and H. Briegel. Phys. Rev.\r\nLett. 95, 120405 (2005).\r\n[11] P. Mironowicz, R. Gallego and M. Paw\u0142owski. Robust amplification\r\nof Santha-Vazirani sources with three devices.\r\nPhys. Rev. A 91, 032317 (2015).\r\n[12] A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki,\r\nM. Paw\u0142owski and R. Ramanathan. Free randomness\r\namplification using bipartite chain correlations.\r\narXiv:1303.5591 (2013).\r\n[13] J. E. Pope and A. Kay. Limited FreeWill in Multiple Runs\r\nof a Bell Test. arXiv:1304.4904 (2013).\r\n[14] L. P. Thinh, L. Sheridan and V. Scarani. Bell tests with minentropy\r\nsources. arXiv:1304.3598 (2013).\r\n[15] M. Plesch and M. Pivoluska. Single Min-Entropy Random\r\nSource can be Amplified. arXiv:1305.0990 (2013).\r\n[16] R. Augusiak, M. Demianowicz, M. Pawlowski, J. Tura\r\nand A. Acin. Monogamies of correlations and amplification\r\nof randomness. arXiv: 1307.6390 (2013).\r\n[17] R. Colbeck and A. Kent. Private Randomness Expansion\r\nWith Untrusted Devices. Journal of Physics A: Mathematical\r\nand Theoretical 44(9), 095305 (2011).\r\n[18] B. Barak, G. Kindler, R. Shaltiel, B. Sudakov, and A.\r\nWigderson. Simulating independence: New constructions\r\nof condensers, Ramsey graphs, dispersers, and extractors.\r\nProceedings of the 37th Annual ACM Symposium\r\non Theory of Computing, pp 110 (2005).\r\n[19] A. Rao. 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Universally Composable Privacy Amplification\r\nfrom Causality Constraints. Phys. Rev. Lett. 102,\r\n140501 (2009).\r\n[27] E. H\u00a8anggi, R. Renner and S. Wolf. Efficient Device-\r\nIndependent Quantum Key Distribution. EUROCRYPT\r\n2010, 216 (2010).\r\n[28] U. Vazirani and T. Vidick. Certifiable Quantum Dice\r\n- Or, testable exponential randomness expansion.\r\narXiv:1111.6054 (2011).\r\n[29] U. Vazirani and T. Vidick. Fully device independent quantum\r\nkey distribution. arXiv:1210.1810 (2012).\r\n[30] M. Coudron, T. Vidick, and H. Yuen. Robust Randomness\r\nAmplifiers: Upper and Lower Bounds. arXiv:1305.6626\r\n(2013).\r\n[31] F. G. S. L. Brandao and A.W. Harrow. Quantum de Finetti\r\nTheorems under Local Measurements with Applications.\r\nSTOC 2013: 861-870. arXiv: 1210.6367 (2012).\r\n[32] F. G. S. L. Brandao and A. W. Harrow. Product-state Approximations\r\nto Quantum Groundstates. STOC 2013: 871-\r\n880.\r\n[33] A. Panconesi and A. Srinivasan. Randomized distributed\r\nedge coloring via an extension of the Chernoff-Hoeffding\r\nbounds. SIAM Journal on Computing 26, 350-368 (1997).\r\n[34] R. Impagliazzo and V. Kabanets. APPROX/RANDOM\u201910\r\nProceedings of the 13th international conference on Approximation,\r\nand the International conference on Randomization,\r\nand combinatorial optimization: algorithms\r\nand techniques, 617-631 (2010).\r\n[35] L. Aolita, R. Gallego, A. Ac\u00b4\u0131n, A. Chiuri, G. Vallone, P.\r\nMataloni and A. Cabello, Phys. Rev. A 85, 032107 (2012).\r\n[36] A. Cabello, Phys. Rev. Lett. 101 210401 (2008).\r\n[37] N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990)."
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"note": "FB acknowledges EPSRC for financial support. MC acknowledges the EU Integrated Project SIQS and the Alexander von Humboldt foundation for financial support. Part of this work was done while FB was visiting the Simons Institute for the Theory of Computing in the program Quantum Hamiltonian Complexity.",
"abstract": "We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of N spins arranged on a d-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to O(N^(1/(d+1))) spins. The proof rests on a variant of the Berry-Esseen theorem for quantum lattice systems and ideas from quantum information theory.",
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"items": [
"[1] J.L. Lebowitz and E.H. Lieb, Existence of thermodynamics for real matter with Coulomb forces. Phys.\r\nRev. Lett. 22, 631 (1969).\r\n[2] H.-O. Georgii, The equivalence of ensembles for classical systems of particles. J. Stat. Phys. 80, 1341\r\n(1995).\r\n[3] H. Touchette, The large deviation approach to statistical mechanics. Phys. Rep. 478, 1 (2009).\r\n[4] R. Lima. Equivalence of ensembles in quantum lattice systems. Annales de l\u2019I. H. P. 15 (1), 6168 (1971).\r\n[5] R. Lima. Equivalence of Ensembles in Quantum Lattice Systems: States. Commun. Math. Phys. 24,\r\n180192 (1972).\r\n[6] M.P. M\u00a8 uller, E. Adlam, Ll. Masanes, and Nathan Wiebe. Thermalization and canonical typicality in\r\ntranslation-invariant quantum lattice systems. arXiv:1312.7420.\r\n[7] A. Polkovnikov, K. Sengupta, A. Silva, M. Vengalattore. Non-equilibrium dynamics of closed interacting\r\nquantum systems. Rev. Mod. Phys. 83, 863 (2011).\r\n[8] J. Eisert, M. Friesdorf, C. Gogolin. Quantum many-body systems out of equilibrium. arXiv:1408.5148.\r\n[9] M. Deserno. Microcanonical and canonical two-dimensional Ising model: An example.\r\n[10] H. Araki. Gibbs states of a one dimensional quantum lattice. Comm. Math. Phys. 14, 120 (1969).\r\n[11] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, J. Eisert. Locality of temperature. Phys. Rev. X 4,\r\n031019 (2014).\r\n[12] S. Popescu, A. Short, and W. Winter. Entanglement and the foundations of statistical mechanics. Nature\r\nPhysics 2, 754 (2006).\r\n[13] S. Goldstein, J.L. Lebowitz, R. Tumulka, and N. Zanghi. Canonical Typicality. Phys. Rev. Lett. 96,\r\n050403 (2006).\r\n[14] H. Touchette. Equivalence of statistical-mechanical ensembles: A collection of quotes and notes. 2006.\r\n[15] M. Srednicki. Chaos and Quantum Thermalization. Physical Review E 50, 888 (1994).\r\n[16] B. Simon, The Statistical Mechanics of Lattice Gases, Vol. 1, Princeton University Press, Princeton,\r\n1993.\r\n[17] M. Cramer, F.G.S.L. Brand\u02dcao and M. Guta. A Berry\u2013Essen Theorem for Quantum Lattice Systems. In\r\npreparation (2015).\r\n[18] N. Datta. Min- and Max- Relative Entropies and a New Entanglement Monotone. IEEE Transactions\r\non Information Theory 55, 2816 (2009).\r\n[19] R. Jain, J. Radhakrishnan, P. Sen. A new information-theoretic property about quantum states with\r\nan application to privacy in quantum communication. Journal of the ACM, 56(6), September 2009.\r\nArticle no. 33.\r\n[20] R. Jain and A. Nayak. Short proofs of the Quantum Substate Theorem. arXiv:1103.6067.\r\n[21] N. Datta and R. Renner. Smooth Renyi Entropies and the Quantum Information Spectrum. IEEE Transactions\r\non Information Theory 55, 2807 (2009).\r\n[22] F.G.S.L. Brand\u02dcao and M.B. Plenio. A Generalization of Quantum Stein\u2019s Lemma. Commun. Math.\r\nPhys. 295, 791 (2010).\r\n[23] F.G.S.L. Brand\u02dcao and M. Horodecki. Exponential Decay of Correlations Implies Area Law. To appear\r\nin CMP. arXiv:1206.2947."
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"title": "Quantum Many-Body Phenomena in Coupled Cavity Arrays",
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"keywords": "quantum many-body models, polaritons, cavity QED, photon blockade",
"note": "This work is part of the QIP-IRC supported by EPSRC (GR/S82176/0), the Integrated Project Qubit Applications (QAP) supported by the IST directorate as Contract Number 015848\u2019, the EU STREP project HIP and was supported by the EPSRC grant EP/E058256/1, the Alexander von Humboldt Foundation, the Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq), the Royal Society and the DFG Emmy Noether grant HA 5593/1-1.",
"abstract": "The increasing level of experimental control over atomic and optical systems gained in the past years have paved the way for the exploration of new physical regimes in quantum optics and atomic physics, characterised by the appearance of quantum many-body phenomena, originally encountered only in condensed-matter physics, and the possibility of experimentally accessing them in a more controlled manner. In this review article we survey recent theoretical studies concerning the use of cavity quantum electrodynamics to create quantum many-body systems. Based on recent experimental progress in the fabrication of arrays of interacting micro-cavities and on their coupling to atomic-like structures in several different physical architectures, we review proposals on the realisation of paradigmatic many-body models in such systems, such as the Bose-Hubbard and the anisotropic Heisenberg models. Such arrays of coupled cavities offer interesting properties as simulators of quantum many-body physics, including the full addressability of individual sites and the accessibility of inhomogeneous models.",
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"items": [
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"rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.",
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