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Quantum Information and Convex Optimization Quantum Information and Convex Optimization Von der Fakult¨atf¨urElektrotechnik, Informationstechnik, Physik der Technischen Universit¨atCarolo-Wilhelmina zu Braunschweig zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Michael Reimpell aus Hildesheim 1. Referent: Prof. Dr. Reinhard F. Werner 2. Referent: PD Dr. Michael Keyl eingereicht am: 22.05.2007 m¨undliche Pr¨ufung(Disputation) am: 25.09.2007 Druckjahr: 2008 Vorver¨offentlichungen der Dissertation Teilergebnisse aus dieser Arbeit wurden mit Genehmigung der Fakult¨atf¨urElek- trotechnik, Informationstechnik, Physik, vertreten durch den Mentor der Arbeit, in folgenden Beitr¨agenvorab ver¨offentlicht: Publikationen [1] M. Reimpell and R. F. Werner. Iterative Optimization of Quantum Error Correcting Codes. Physical Review Letters 94 (8), 080501 (2005); selected for1: Virtual Journal of Quantum Information 5 (3) (2005); arXiv:quant-ph/0307138. [2] M. Reimpell, R. F. Werner and K. Audenaert. Comment on "Optimum Quan- tum Error Recovery using Semidefinite Programming". arXiv:quant-ph/0606059 (2006). [3] M. Reimpell and R. F. Werner. A Meaner King uses Biased Bases. arXiv:quant- ph/0612035 (2006).2 [4] O. G¨uhne,M. Reimpell and R. F. Werner. Estimating Entanglement Measures in Experiments. Physical Review Letters 98 (11), 110502 (2007); arXiv:quant- ph/0607163. Tagungsbeitr¨age 1. M. Reimpell. Optimal codes beyond Knill & Laflamme?. Vortrag. Ringberg Meeting, Rottach-Egern, Mai 2003. 2. M. Reimpell and R. F. Werner. An Iteration to Optimize Quantum Error Correcting Codes. Vortrag, QIT-EQIS, Kyoto, Japan, September 2003. 3. M. Reimpell. Quantum error correction as a semi-definite programme. Vor- trag. 26th A2 meeting, Potsdam, November 2003. 4. D. Kretschmann, M. Reimpell and R. F. Werner. Distributed systems. Poster. 1The Virtual Journal, which is published by the American Physical Society and the American Institute of Physics in cooperation with numerous other societies and publishers, is an edited compi- lation of links to articles from participating publishers, covering a focused area of frontier research. You can access the Virtual Journal at http://www.vjquantuminfo.org. 2Erschienen in: Physical Review A 75 (6), 062334 (2007); Virtual Journal of Quantum Infor- mation 7 (7) (2007) Kolloqium des DFG-Schwerpunktprogramms \Quanten-Informationsverarbei- tung", Bad Honnef, Januar 2004. 5. D. Kretschmann, M. Reimpell and R. F. Werner. Quantum capacity and cod- ing. Poster. Kolloqium des DFG-Schwerpunktprogramms \Quanten-Informations- verarbeitung", Bad Honnef, Januar 2004. 6. M. Reimpell and R. F. Werner. Quantum Error Correcting Codes - Ab Ini- tio. Vortrag. Fr¨uhjahrstagungder Deutschen Physikalischen Gesellschaft, M¨unchen, M¨arz2004. 7. M. Reimpell. Introduction to Quantum Error Correction. Vortrag. A2 - The Next Generation meeting (29th A2 meeting), Potsdam, Juli 2004. 8. M. Reimpell and R. F. Werner. Iterative Optimization of Quantum Error Correcting Codes. Poster. Quantum Information Theory: Present Status and Future Directions, Cambridge, August 2004. 9. M. Reimpell and R. F. Werner. Tough error models. Vortrag. Fr¨uhjahrstagung der Deutschen Physikalischen Gesellschaft, Berlin, M¨arz2005. 10. D. Kretschmann, M. Reimpell, and R. F. Werner. Quantitative analysis of basic concepts in Quantum Information Theory by optimization of elementary operations. Poster. Abschlusskolloquium des DFG-Schwerpunktprogramms \Quanten-Informationsverarbeitung", Bad Honnef, Juni 2005. 11. T. Franz, D. Kretschmann, M. Reimpell, D.Schlingemann and R. F. Werner. Quantum cellular automata. Poster. Abschlusskolloquium des DFG-Schwerpunkt- programms \Quanten-Informationsverarbeitung", Bad Honnef, Juni 2005. 12. M. Reimpell. Postprocessing tomography data via conic programming. Vortrag. Arbeitsgruppe Experimental Quantum Physics LMU M¨unchen, M¨unchen, Februar 2006. 13. M. Reimpell and R. F. Werner. Fitting channels to tomography data. Vortrag. Fr¨uhjahrstagung der Deutschen Physikalischen Gesellschaft, Frankfurt, M¨arz 2006. Abstract This thesis is concerned with convex optimization problems in quantum information theory. It features an iterative algorithm for optimal quantum error correcting codes, a postprocessing method for incomplete tomography data, a method to estimate the amount of entanglement in witness experiments, and it gives necessary and sufficient criteria for the existence of retrodiction strategies for a generalized mean king problem. keywords: quantum information, convex optimization, quantum error correction, channel power iteration, semidefinite programming, tough error models, quantum tomography, entanglement estimation, witness operators, mean king, retrodiction. v Summary This thesis investigates several problems in quantum information theory related to convex optimization. Quantum information science is about the use of quantum mechanical systems for information transmission and processing tasks. It explores the role of quantum mechanical effects, such as uncertainty, superposition and en- tanglement, with respect to information theory. Convexity naturally arises in many places in quantum information theory, as the possible preparations, processes and measurements for quantum systems are convex sets. Convex optimization meth- ods are therefore particularly suitable for the optimization of quantum information tasks. This thesis focuses on optimization problems within quantum error correction, quantum tomography, entanglement estimation and retrodiction. Quantum error correction is used to avoid, detect and correct noisy interactions of the information carrying quantum systems with the environment. In this the- sis, error correction is considered as an optimization problem. The main result is the development of a monotone convergent iterative algorithm, the channel power iteration, which can be used to find optimal coding and decoding operations for arbitrary noise. In contrast to the common approach to quantum error correction, the algorithm does not make any a priori assumptions about the structure of the encoding and decoding operations. In particular, it does not require any error to be corrected completely, and hence can find codes outside the usual quantum error correction setting. More generally, the channel power iteration can be used for the optimization of any linear functional over the set of quantum channels. The thesis also discusses and compares the power iteration to the use of semidefinite program- ming for the computation of optimal codes. Both techniques are applied to different noise models, where they find improved quantum error correcting codes, and suggest that the Knill-Laflamme form of error correction is optimal in a worst case scenario. Furthermore, the algorithms are used to provide bounds on the correction capabili- ties for tough error models, and to disprove the optimality of feedback strategies in higher dimensions. Quantum tomography is used to estimate the parameters of a given quantum state or quantum channel in the laboratory. The tomography yields a sequence of mea- surement results. As the measurement results are subject to errors, a direct interpre- vii tation can suggest negative probabilities. Therefore, parameters are fitted to these tomography data. This fitting is a convex optimization problem. In this thesis, it is shown how to extend the fitting method in the case of incomplete tomography of pure states and unitary gates. The extended method then provides the minimum and maximum fidelity over all fits that are consistent with the tomography data. A common way to qualitatively detect the presence of entanglement in a quantum state produced in an experiment is via the measurement of so-called witness op- erators. In this thesis, convexity theory is used to show how a lower bound on a generic entanglement measure can be derived from the measured expectation values of any finite collection of entanglement witnesses. This is shown, in particular, for the entanglement of formation and the geometric measure of entanglement. Thus, with this method, witness measurements are given a quantitative meaning without the need of further experimental data. More generally, the method can be used to calculate a lower bound on any functional on states, if the Legendre transform of that functional is known. It is proven that the bound is optimal under some conditions on the functional. In the quantum mechanical retrodiction problem, better known as the mean king problem, Alice has to name the outcome of an ideal measurement on a d-dimensional quantum system, made in one of (d + 1) orthonormal bases, unknown to Alice at the time of the measurement. Alice has to make this retrodiction on the basis of the classical outcomes of a suitable control measurement including an entangled copy. In this thesis, the common assumption that the bases are mutually unbiased is relaxed. In this extended setting, it is proven that, under mild assumptions on the bases, the existence of a successful retrodiction strategy for Alice is equivalent to the existence of an overall joint probability distribution for (d+1) random variables, whose marginal pair distributions are fixed as the transition probability matrices of the given bases. This provides a connection between the mean king problem and Bell inequalities. The qubit case is completely analyzed, and it is shown how in the finite dimensional case the existence of a retrodiction strategy can be decided numerically via convex optimization. Contents Summary vii 1 Introduction 1
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