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Optimal quantum networks and one-shot entropies. (English) Zbl 1456.81135

Summary: We develop a semidefinite programming method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance measures, defined operationally in terms of interative tests set up by a verifier. We show that the optimal performance is equal to a max relative entropy, which quantifies the informativeness of the test. Building on this result, we extend the notion of conditional min-entropy from quantum states to quantum causal networks. The optimization method is illustrated in a number of applications, including the inversion, charge conjugation, and controlization of an unknown unitary dynamics. In the non-causal setting, we show a proof-of-principle application to the maximization of the winning probability in a non-causal quantum game.

MSC:

81P68 Quantum computation
90C22 Semidefinite programming

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[1] Aspelmeyer M, Zeilinger A, Lindenthal M, Molina-Terriza G, Poppe A, Resch K, Ursin R, Walther P and Jennewein T D 2005 Advanced quantum communications experiments with entangled photons Quantum Communications and Cryptography (Boca Raton, FL: CRC) p 45 · doi:10.1201/9781420026603.ch3
[2] Pirandola S, Eisert J, Weedbrook C, Furusawa A and Braunstein S L 2015 Advances in quantum teleportation Nat. Photon.9 641-52 · doi:10.1038/nphoton.2015.154
[3] Wang X-L, Cai X-D, Su Z-E, Chen M-C, Wu D, Li L, Liu N-L, Lu C-Y and Pan J-W 2015 Quantum teleportation of multiple degrees of freedom of a single photon Nature518 516-9 · doi:10.1038/nature14246
[4] Politi A, Matthews J C F, Thompson M G and O’Brien J L 2009 Integrated quantum photonics IEEE J. Sel. Top. Quantum Electron.15 1673-84 · doi:10.1109/JSTQE.2009.2026060
[5] Crespi A, Ramponi R, Osellame R, Sansoni L, Bongioanni I, Sciarrino F, Vallone G and Mataloni P 2011 Integrated photonic quantum gates for polarization qubits Nat. Commun.2 566 · doi:10.1038/ncomms1570
[6] Barends R et al 2013 Coherent josephson qubit suitable for scalable quantum integrated circuits Phys. Rev. Lett.111 080502 · doi:10.1103/PhysRevLett.111.080502
[7] Devoret M H and Schoelkopf R J 2013 Superconducting circuits for quantum information: an outlook Science339 1169-74 · doi:10.1126/science.1231930
[8] Zwanenburg F A, Dzurak A S, Morello A, Simmons M Y, Hollenberg L C L, Klimeck G, Rogge S, Coppersmith S N and Eriksson M A 2013 Silicon quantum electronics Rev. Mod. Phys.85 961 · doi:10.1103/RevModPhys.85.961
[9] Cirac J I, Zoller P, Kimble H J and Mabuchi H 1997 Quantum state transfer and entanglement distribution among distant nodes in a quantum network Phys. Rev. Lett.78 3221 · doi:10.1103/PhysRevLett.78.3221
[10] Acín A, Cirac J I and Lewenstein M 2007 Entanglement percolation in quantum networks Nat. Phys.3 256-9 · doi:10.1038/nphys549
[11] Kimble H J 2008 The quantum internet Nature453 1023-30 · doi:10.1038/nature07127
[12] Perseguers S, Lewenstein M, Acín A and Cirac J I 2010 Quantum random networks Nat. Phys.6 539-43 · doi:10.1038/nphys1665
[13] Yin J et al 2013 Experimental quasi-single-photon transmission from satellite to Earth Opt. Express21 20032-40 · doi:10.1364/OE.21.020032
[14] Broadbent A, Fitzsimons J and Kashefi E 2009 universal blind quantum computation 50th Annual IEEE Symp. on Foundations of Computer Science, 2009, FOCS’09 (Piscataway, NJ: IEEE) pp 517-26 · Zbl 1292.81023 · doi:10.1109/FOCS.2009.36
[15] Beals R, Brierley S, Gray O, Harrow A W, Kutin S, Linden N, Shepherd D and Stather M 2013 Efficient distributed quantum computing Proc. R. Soc. A 469 20120686 · Zbl 1371.81074 · doi:10.1098/rspa.2012.0686
[16] Barz S, Kashefi E, Broadbent A, Fitzsimons J F, Zeilinger A and Walther P 2012 Demonstration of blind quantum computing Science335 303-8 · Zbl 1355.81057 · doi:10.1126/science.1214707
[17] Gutoski G and Watrous J 2007 Toward a general theory of quantum games Proc. 39th Annual ACM Symp. on Theory of Computing pp 565-574 · Zbl 1232.91112
[18] Chiribella G, D’Ariano G M and Perinotti P 2009 Theoretical framework for quantum networks Phys. Rev. A 80 022339 · Zbl 1255.81045 · doi:10.1103/PhysRevA.80.022339
[19] Chiribella G, D’Ariano G M, Perinotti P, Schlingemann D and Werner R 2013 A short impossibility proof of quantum bit commitment Phys. Lett. A 377 1076-87 · Zbl 1288.81024 · doi:10.1016/j.physleta.2013.02.045
[20] Portmann C, Matt C, Maurer U, Renner R and Tackmann B 2015 Causal boxes: quantum information-processing systems closed under composition (arXiv:1512.02240)
[21] Chiribella G, D’Ariano G M and Perinotti P 2011 Informational derivation of quantum theory Phys. Rev. A 84 012311 · doi:10.1103/PhysRevA.84.012311
[22] Hardy L 2011 Reformulating and reconstructing quantum theory (arXiv:1104.2066)
[23] Hardy L 2012 The operator tensor formulation of quantum theory Phil. Trans. R. Soc. A 370 3385-417 · Zbl 1325.81006 · doi:10.1098/rsta.2011.0326
[24] Tucci R R 1995 Quantum bayesian nets Int. J. Mod. Phys. B 9 295-337 · Zbl 1264.81026 · doi:10.1142/S0217979295000148
[25] Leifer M S and Poulin D 2008 Quantum graphical models and belief propagation Ann. Phys., NY323 1899-946 · Zbl 1146.81017 · doi:10.1016/j.aop.2007.10.001
[26] Leifer M S and Spekkens R W 2013 Towards a formulation of quantum theory as a causally neutral theory of bayesian inference Phys. Rev. A 88 052130 · doi:10.1103/PhysRevA.88.052130
[27] Henson J, Lal R and Pusey M F 2014 Theory-independent limits on correlations from generalized bayesian networks New J. Phys.16 113043 · Zbl 1451.81028 · doi:10.1088/1367-2630/16/11/113043
[28] Ried K, Agnew M, Vermeyden L, Janzing D, Spekkens R W and Resch K J 2015 A quantum advantage for inferring causal structure Nat. Phys.11 414-20 · doi:10.1038/nphys3266
[29] Chaves R, Majenz C and Gross D 2015 Information-theoretic implications of quantum causal structures Nat. Commun.6 5766 · doi:10.1038/ncomms6766
[30] Pienaar J et al 2015 A graph-separation theorem for quantum causal models New J. Phys.17 073020 · Zbl 1452.81006 · doi:10.1088/1367-2630/17/7/073020
[31] Costa F and Shrapnel S 2015 Quantum causal modelling arXiv:1512.07106)
[32] Selinger P 2004 Towards a semantics for higher-order quantum computation Proc. 2nd Int. Workshop on Quantum Programming Languages, TUCS General Publicationvol 33 pp 127-43
[33] Chiribella G, D’Ariano G M, Perinotti P and Valiron B 2009 Beyond quantum computers (arXiv:0912.0195)
[34] Chiribella G, D’Ariano G M, Perinotti P and Valiron B 2013 Quantum computations without definite causal structure Phys. Rev. A 88 022318 · doi:10.1103/PhysRevA.88.022318
[35] Nielsen M and Chuang I 2000 Quantum information and computation Quantum Information and Computation (Cambridge: Cambridge University Press) · Zbl 1049.81015
[36] Kitaev A Y, Shen A and Vyalyi M N 2002 Classical and Quantum Computation vol 47 (Providence, RI: American Mathematical Society) · Zbl 1022.68001
[37] Chiribella G 2012 Perfect discrimination of no-signalling channels via quantum superposition of causal structures Phys. Rev. A 86 040301 · doi:10.1103/PhysRevA.86.040301
[38] Colnaghi T, D’Ariano G M, Facchini S and Perinotti P 2012 Quantum computation with programmable connections between gates Phys. Lett. A 376 2940-3 · Zbl 1266.81046 · doi:10.1016/j.physleta.2012.08.028
[39] Araújo M, Fabio Costa and Brukner Č 2014 Computational advantage from quantum-controlled ordering of gates Phys. Rev. Lett.113 250402 · doi:10.1103/PhysRevLett.113.250402
[40] Hardy L 2009 Quantum gravity computers: on the theory of computation with indefinite causal structure Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle (Berlin: Springer) pp 379-401 · Zbl 1166.81010 · doi:10.1007/978-1-4020-9107-0_21
[41] Oreshkov O, Costa F and Brukner Č 2012 Quantum correlations with no causal order Nat. Commun.3 1092 · doi:10.1038/ncomms2076
[42] Baumeler Ä and Wolf S 2014 Perfect signaling among three parties violating predefined causal order IEEE Int. Symp. on Information Theory pp 526-30
[43] Morimae T 2014 Acausal measurement-based quantum computing Phys. Rev. A 90 010101 · doi:10.1103/PhysRevA.90.010101
[44] Baumeler Ä and Wolf S 2016 Non-causal computation avoiding the grandfather and information antinomies (arXiv:1601.06522)
[45] Akibue S, Owari M, Kato G and Murao M 2016 Entanglement assisted classical communication simulates” classical communication without causal order arXiv:1602.08835
[46] Kitaev A and Watrous J 2000 Parallelization, amplification, and exponential time simulation of quantum interactive proof systems Proc. 32nd Annual ACM Symposium on Theory of Computing pp 608-617 · Zbl 1296.68057
[47] Chiribella G, D’Ariano G M and Perinotti P 2008 Quantum circuit architecture Phys. Rev. Lett.101 060401 · doi:10.1103/PhysRevLett.101.060401
[48] Renner R and Wolf S 2004 Smooth rényi entropy and applications IEEE Int. Symp. on Information Theory p 233
[49] Renner R 2008 Security of quantum key distribution Int. J. Quantum Inf.6 1-27 · Zbl 1151.81007 · doi:10.1142/S0219749908003256
[50] Datta N and Renner R 2009 Smooth entropies and the quantum information spectrum IEEE Trans. Inf. Theory55 2807-15 · Zbl 1367.81022 · doi:10.1109/TIT.2009.2018340
[51] König R, Renner R and Schaffner C 2009 The operational meaning of min-and max-entropy IEEE Trans. Inf. Theory55 4337-47 · Zbl 1367.81028 · doi:10.1109/TIT.2009.2025545
[52] Tomamichel M 2015 Quantum Information Processing with Finite Resources: Mathematical Foundations vol 5 (Berlin: Springer)
[53] Chiribella G, D’Ariano G M and Perinotti P 2008 Transforming quantum operations: quantum supermaps Europhys. Lett.83 30004 · doi:10.1209/0295-5075/83/30004
[54] Brukner Č 2015 Bounding quantum correlations with indefinite causal order New J. Phys.17 083034 · Zbl 1454.81025 · doi:10.1088/1367-2630/17/8/083034
[55] Kraus K, Böhm A, Dollard J D and Wootters W H 1983 States, effects, and operations fundamental notions of quantum theory · Zbl 0545.46049
[56] Choi M-D 1975 Completely positive linear maps on complex matrices Linear Algebr. Appl.10 285-90 · Zbl 0327.15018 · doi:10.1016/0024-3795(75)90075-0
[57] Yurke B and Stoler D 1992 Einstein-podolsky-rosen effects from independent particle sources Phys. Rev. Lett.68 1251 · doi:10.1103/PhysRevLett.68.1251
[58] Zukowski M, Zeilinger A, Horne M A and Ekert A K 1993 event-ready-detectors” bell experiment via entanglement swapping Phys. Rev. Lett.71 4287-90 · doi:10.1103/PhysRevLett.71.4287
[59] Chiribella G, D’Ariano G M and Perinotti P 2010 Probabilistic theories with purification Phys. Rev. A 81 062348 · doi:10.1103/PhysRevA.81.062348
[60] Cormen T H, Leiserson C E, Rivest R L and Stein C 2001 Introduction to Algorithms vol 6 (Cambridge: MIT Press) · Zbl 1047.68161
[61] Chiribella G, D’Ariano G M and Perinotti P 2008 Memory effects in quantum channel discrimination Phys. Rev. Lett.101 180501 · doi:10.1103/PhysRevLett.101.060401
[62] Ziman M 2008 Process positive-operator-valued measure: a mathematical framework for the description of process tomography experiments Phys. Rev. A 77 062112 · doi:10.1103/PhysRevA.77.062112
[63] Bisio A, Chiribella G, D’Ariano G M, Facchini S and Perinotti P 2010 Optimal quantum learning of a unitary transformation Phys. Rev. A 81 032324 · doi:10.1103/PhysRevA.81.032324
[64] Giovannetti V, Lloyd S and Maccone L 2006 Quantum metrology Phys. Rev. Lett.96 010401 · doi:10.1103/PhysRevLett.96.010401
[65] Giovannetti V, Lloyd S and Maccone L 2011 Advances in quantum metrology Nat. Photon.5 222-9 · doi:10.1038/nphoton.2011.35
[66] Gutoski G 2012 On a measure of distance for quantum strategies J. Math. Phys.53 032202 · Zbl 1274.81052 · doi:10.1063/1.3693621
[67] Jenčová A and Plávala M 2016 Conditions for optimal input states for discrimination of quantum channels (arXiv:1603.01437)
[68] Sedlák M, Reitzner D, Chiribella G and Ziman M 2016 Incompatible measurements on quantum causal networks Phys. Rev. A 93 052323 · doi:10.1103/PhysRevA.93.052323
[69] Bisio A, Chiribella G, D’Ariano G M, Facchini S and Perinotti P 2009 Optimal quantum tomography of states, measurements, and transformations Phys. Rev. Lett.102 010404 · doi:10.1103/PhysRevLett.102.010404
[70] Pollock F A, Rodríguez-Rosario C, Frauenheim T, Paternostro M and Modi K 2015 Complete framework for efficient characterisation of non-markovian processes (arXiv:1512.00589)
[71] Modi K et al 2012 Unification of witnessing initial system-environment correlations and witnessing non-markovianity Europhys. Lett.99 20010 · doi:10.1209/0295-5075/99/20010
[72] Deutsch D 1989 Quantum computational networks Proc. R. Soc. A 425 73-90 · Zbl 0691.68054 · doi:10.1098/rspa.1989.0099
[73] Marvian I and Lloyd S 2016 universal quantum emulator arXiv:1606.02734)
[74] Chiribella G, D’Ariano G M and Perinotti P 2008 Optimal cloning of unitary transformation Phys. Rev. Lett.101 180504 · doi:10.1103/PhysRevLett.101.180504
[75] Dür W, Sekatski P and Skotiniotis M 2015 Deterministic superreplication of one-parameter unitary transformations Phys. Rev. Lett.114 120503 · doi:10.1103/PhysRevLett.114.120503
[76] Chiribella G, Yang Y and Huang C 2015 universal superreplication of unitary gates Phys. Rev. Lett.114 120504 · doi:10.1103/PhysRevLett.114.120504
[77] Mičuda M, Stárek R, Straka I, Miková M, Sedlák M, Ježek M and Fiurášek J 2016 Experimental replication of single-qubit quantum phase gates Phys. Rev. A 93 052318 · doi:10.1103/PhysRevA.93.052318
[78] Chiribella G and Yang Y 2016 Quantum superreplication of states and gates Front. Phys.11 1-9 · doi:10.1007/s11467-016-0556-7
[79] Grover L K 1996 A fast quantum mechanical algorithm for database search Proc. Twenty-eighth Annual ACM Symp. on Theory of Computing pp 212-9 · Zbl 0922.68044
[80] Watrous J 2011 Lecture Notes on Semidefinite Programming
[81] Boyd S and Vandenberghe L 2004 Convex Optimization (Cambridge : Cambridge University Press) · Zbl 1058.90049 · doi:10.1017/CBO9780511804441
[82] Datta N 2009 Min-and max-relative entropies and a new entanglement monotone Information Theory, IEEE Transactions on55 2816-26 · Zbl 1367.81021 · doi:10.1109/TIT.2009.2018325
[83] Brandao F G S L and Datta N 2011 One-shot rates for entanglement manipulation under non-entangling maps IEEE Trans. Inf. Theory57 1754-60 · Zbl 1366.81051 · doi:10.1109/TIT.2011.2104531
[84] Horodecki M and Oppenheim J 2013 Fundamental limitations for quantum and nanoscale thermodynamics Nat. Commun.4 2059 · doi:10.1038/ncomms3059
[85] Brandão F, Horodecki M, Ng N, Oppenheim J and Wehner S 2015 The second laws of quantum thermodynamics Proc. Natl Acad. Sci.112 3275-9 · doi:10.1073/pnas.1411728112
[86] Datta N, Mosonyi M, Hsieh M-H and Brandao F G S L 2013 A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels IEEE Trans. Inf. Theory59 8014-26 · Zbl 1364.81057 · doi:10.1109/TIT.2013.2282160
[87] Chiribella G and Xie J 2013 Optimal design and quantum benchmarks for coherent state amplifiers Phys. Rev. Lett.110 213602 · doi:10.1103/PhysRevLett.110.213602
[88] Jenčová A 2014 Base norms and discrimination of generalized quantum channels J. Math. Phys.55 022201 · Zbl 1286.81018 · doi:10.1063/1.4863715
[89] Buscemi F and Datta N 2016 Equivalence between divisibility and monotonic decrease of information in classical and quantum stochastic processes Phys. Rev. A 93 012101 · doi:10.1103/PhysRevA.93.012101
[90] Bae J and Chruscinski D 2016 Operational characterization of divisibility of dynamical maps (arXiv:1601.05522)
[91] Beckman D, Gottesman D, Nielsen M A and Preskill J 2001 Causal and localizable quantum operations Phys. Rev. A 64 052309 · doi:10.1103/PhysRevA.64.052309
[92] Piani M, Horodecki M, Horodecki P and Horodecki R 2006 Properties of quantum nonsignaling boxes Phys. Rev. A 74 012305 · doi:10.1103/PhysRevA.74.012305
[93] D’Ariano G M, Facchini S and Perinotti P 2011 No signaling, entanglement breaking, and localizability in bipartite channels Phys. Rev. Lett.106 010501 · doi:10.1103/PhysRevLett.106.010501
[94] Bennett C, Bernstein H, Popescu S and Schumacher B 1996 Concentrating partial entanglement by local operations Phys. Rev. A 53 2046-52 · doi:10.1103/PhysRevA.53.2046
[95] Popescu S and Rohrlich D 1997 Thermodynamics and the measure of entanglement Phys. Rev. A 56 R3319 · doi:10.1103/PhysRevA.56.R3319
[96] Vedral V, Plenio M B, Rippin M A and Knight P L 1997 Quantifying entanglement Phys. Rev. Lett.78 2275 · Zbl 0944.81011 · doi:10.1103/PhysRevLett.78.2275
[97] Friis N, Dunjko V, Dür W and Briegel H J 2014 Implementing quantum control for unknown subroutines Phys. Rev. A 89 030303 · doi:10.1103/PhysRevA.89.030303
[98] Araújo M, Feix A, Costa F and Brukner Č 2014 Quantum circuits cannot control unknown operations New J. Phys.16 093026 · Zbl 1451.81137 · doi:10.1088/1367-2630/16/9/093026
[99] Nakayama S, Soeda A and Murao M 2015 Quantum algorithm for universal implementation of the projective measurement of energy Phys. Rev. Lett.114 190501 · doi:10.1103/PhysRevLett.114.190501
[100] Bisio A, Dall’Arno M and Perinotti P 2015 The quantum conditional statement arXiv:1509.01062)
[101] Raginsky M 2001 A fidelity measure for quantum channels Phys. Lett. A 290 11-8 · Zbl 1020.81002 · doi:10.1016/S0375-9601(01)00640-5
[102] Nielsen M A 2002 A simple formula for the average gate fidelity of a quantum dynamical operation Phys. Lett. A 303 249-52 · Zbl 0999.81012 · doi:10.1016/S0375-9601(02)01272-0
[103] Horodecki M, Horodecki P and Horodecki R 1999 General teleportation channel, singlet fraction, and quasidistillation Phys. Rev. A 60 1888-98 · Zbl 1005.81505 · doi:10.1103/PhysRevA.60.1888
[104] Nielsen M and Chuang I 1997 Programmable quantum gate arrays Phys. Rev. Lett.79 321 · Zbl 0944.81009 · doi:10.1103/PhysRevLett.79.321
[105] Bužek V, Hillery M and Werner R F 1999 Optimal manipulations with qubits: universal-not gate Phys. Rev. A 60 R2626 · doi:10.1103/PhysRevA.60.R2626
[106] Rungta P, Bužek V, Caves C M, Hillery M and Milburn G J 2001 universal state inversion and concurrence in arbitrary dimensions Phys. Rev. A 64 042315 · doi:10.1103/PhysRevA.64.042315
[107] Chiribella G, D’Ariano G M and Sacchi M F 2005 Optimal estimation of group transformations using entanglement Phys. Rev. A 72 042338 · doi:10.1103/PhysRevA.72.042338
[108] Horodecki P and Ekert A 2002 Method for direct detection of quantum entanglement Phys. Rev. Lett.89 127902 · Zbl 1267.81038 · doi:10.1103/PhysRevLett.89.127902
[109] Horodecki P 2003 From limits of quantum operations to multicopy entanglement witnesses and state-spectrum estimation Phys. Rev. A 68 052101 · doi:10.1103/PhysRevA.68.052101
[110] Buscemi F, D’Ariano G M, Perinotti P and Sacchi M F 2003 Optimal realization of the transposition maps Phys. Lett. A 314 374-9 · Zbl 1052.81527 · doi:10.1016/S0375-9601(03)00954-X
[111] Kalev A and Bae J 2013 Optimal approximate transpose map via quantum designs and its applications to entanglement detection Phys. Rev. A 87 062314 · doi:10.1103/PhysRevA.87.062314
[112] Lim H-T, Kim Y-S, Ra Y-S, Bae J and Kim Y-H 2011 Experimental realization of an approximate partial transpose for photonic two-qubit systems Phys. Rev. Lett.107 160401 · doi:10.1103/PhysRevLett.107.160401
[113] Scott A James 2008 Optimizing quantum process tomography with unitary 2-designs J. Phys. A: Math. Theor.41 055308 · Zbl 1141.81009 · doi:10.1088/1751-8113/41/5/055308
[114] Ringbauer M, Wood C J, Modi K, Gilchrist A, White A G and Fedrizzi A 2015 Characterizing quantum dynamics with initial system-environment correlations Phys. Rev. Lett.114 090402 · doi:10.1103/PhysRevLett.114.090402
[115] Aïmeur E, Brassard G and Gambs S 2006 Machine learning in a quantum world Advances in Artificial Intelligence (Berlin: Springer) pp 431-42 · doi:10.1007/11766247_37
[116] Bisio A, Chiribella G, D’Ariano G M, Facchini S and Perinotti P 2010 Optimal quantum learning of a unitary transformation Phys. Rev. A 81 032324 · doi:10.1103/PhysRevA.81.032324
[117] Paparo G D, Dunjko V, Makmal A, Martin-Delgado M A and Briegel H J 2014 Quantum speedup for active learning agents Phys. Rev. X 4 031002 · doi:10.1103/PhysRevX.4.031002
[118] Chiribella G and Yang Y 2015 arXiv:1502.00259
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