Optimal processing of reversible quantum channels

Article history: We consider the general problem of the optimal transformation of N uses of (possibly different) unitary channels to a single use of another unitary channel in any finite dimension. We show how the optimal transformation can be fully parallelized, consisting in a preprocessing channel followed by a parallel action of all the N unitaries and a final postprocessing channel. Our techniques allow to achieve an exponential reduction in the number of the free parameters of the optimization problem making it amenable to an efficient numerical treatment. Finally, we apply our general results to find the analytical solution for special cases of interest like the cloning of qubit phase gates.

[1]  G. D’Ariano,et al.  Minimal computational-space implementation of multiround quantum protocols , 2010, 1006.1780.

[2]  Robert W. Spekkens,et al.  Foundations of Quantum Mechanics , 2007 .

[3]  Giacomo Mauro D'Ariano,et al.  Quantum learning algorithms for quantum measurements , 2011 .

[4]  G. D’Ariano,et al.  Informational derivation of quantum theory , 2010, 1011.6451.

[5]  William J Munro,et al.  Quantum teleportation of optical quantum gates. , 2003, Physical review letters.

[6]  R. Werner.  OPTIMAL CLONING OF PURE STATES , 1998, quant-ph/9804001.

[7]  Gilles Brassard,et al.  Is information the key? , 2005 .

[8]  Buzek,et al.  Quantum copying: Beyond the no-cloning theorem. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[9]  Hans Halvorson,et al.  Deep beauty : understanding the quantum world through mathematical innovation , 2011 .

[10]  G. D’Ariano,et al.  Optimal quantum tomography of States, measurements, and transformations. , 2008, Physical review letters.

[11]  G M D'Ariano,et al.  Using entanglement improves the precision of quantum measurements. , 2001, Physical review letters.

[12]  G. D’Ariano,et al.  Transforming quantum operations: Quantum supermaps , 2008, 0804.0180.

[13]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[14]  G. D’Ariano,et al.  Theoretical framework for quantum networks , 2009, 0904.4483.

[15]  I. Chuang,et al.  Programmable Quantum Gate Arrays , 1997, quant-ph/9703032.

[16]  Giacomo Mauro D'Ariano,et al.  Memory cost of quantum protocols , 2012 .

[17]  Seth Lloyd,et al.  Continuous Variable Quantum Cryptography using Two-Way Quantum Communication , 2006, ArXiv.

[18]  G. D’Ariano,et al.  Quantum information becomes classical when distributed to many users. , 2006, Physical review letters.

[19]  Minimum error discrimination of Pauli channels , 2005, quant-ph/0506072.

[20]  G. D’Ariano,et al.  Quantum networks: General theory and applications , 2011, 1601.04864.

[21]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[22]  Giulio Chiribella,et al.  Information-disturbance tradeoff in estimating a unitary transformation , 2010 .

[23]  J. Cirac,et al.  Optimal Purification of Single Qubits , 1998, quant-ph/9812075.

[24]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[25]  Maxim Raginsky,et al.  A fidelity measure for quantum channels , 2001, quant-ph/0107108.

[26]  Markus P. Mueller,et al.  A derivation of quantum theory from physical requirements , 2010, 1004.1483.

[27]  G. D’Ariano,et al.  Optimal quantum learning of a unitary transformation , 2009, 0903.0543.

[28]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[29]  Debbie W. Leung,et al.  Adaptive versus nonadaptive strategies for quantum channel discrimination , 2009, 0909.0256.

[30]  London,et al.  Quantum Remote Control: Teleportation of Unitary Operations , 2000, quant-ph/0005061.

[31]  Č. Brukner,et al.  Quantum correlations with no causal order , 2011, Nature Communications.

[32]  T. Ralph,et al.  Quantum process tomography of a controlled-NOT gate. , 2004, Physical review letters.

[33]  G. D’Ariano,et al.  Quantum computation with programmable connections between gates , 2011, 1109.5987.

[34]  Seth Lloyd,et al.  Quantum process tomography of the quantum Fourier transform. , 2004, The Journal of chemical physics.

[35]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .