Gauge invariant information concerning quantum channels

Łukasz Rudnicki1,2, Zbigniew Puchała3,4, and Karol Zyczkowski2,4

1Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77, D-50937, Cologne, Germany
2Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
3Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ulica Bałtycka 5, 44-100 Gliwice, Poland
4Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland

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Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.


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[1] R. Blume-Kohout, et al., Robust, self-consistent,closed-form tomography of quantum logic gates on a trapped ion qubit, arXiv:1310.4492 (2013).

[2] D. Kim, et al., Microwave-driven coherent operation of a semiconductor quantum dot charge qubit, Nat. Nanotechnology 10, 243 (2015).

[3] D. Greenbaum, Introduction to Quantum Gate Set Tomography, arXiv:1509.02921 (2015).

[4] J. F. Poyatos, J. I. Cirac, and P. Zoller, Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate, Phys. Rev. Lett. 78, 390 (1997).

[5] S. T. Merkel, et al., Self-consistent quantum process tomography, Phys. Rev. A 87, 062119 (2013).

[6] T. Baumgratz, D. Gross, M. Cramer, and M. B. Plenio, Scalable Reconstruction of Density Matrices, Phys. Rev. Lett. 111, 020401 (2013).

[7] R. Blume-Kohout, et al., Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography, Nat. Commun. 8, 14485 (2017).

[8] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.

[9] A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and quantum computation, Vol. 47 (American Mathematical Society, 2002).

[10] J. Emerson, R. Alicki, K. Życzkowski, Scalable Noise Estimation with Random Unitary Operators, J. Opt. B: Quantum Semiclass. Opt. 7, S347 (2005).

[11] J. Emerson, et al., Symmetrized Characterization of Noisy Quantum Processes, Science 317, 1893 (2007).

[12] E. Knill, et al., Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).

[13] J. J. Wallman and S. T. Flammia, Randomized benchmarking with confidence, New J. Phys. 16, 103032 (2014).

[14] K. Życzkowski and I. Bengtsson, On duality between quantum maps and quantum states, Open Sys. & Information Dyn. 11, 3 (2004).

[15] W. Bruzda, V. Cappellini, H.-J. Sommers, and K. Życzkowski, Random Quantum Operations, Phys. Lett. A 373, 320 (2009).

[16] D. Evans, R. Hoegh-Krohn, Spectral properties of positive maps on $C_*$-algebras, J. London Math. Soc. 17, 345 (1978).

[17] A. Jamiołkowski, On applications of PI-algebras in the analysis of quantum channels, Int. J. Quantum Inf. 10, 1241007 (2012).

[18] D. R. Farenick, Irreducible positive linear maps on operator algebras, Proc. AMS 124, 3381, (1996).

[19] U. Groh, The peripheral point spectrum of Schwarz operators on $C_*$-algebras, Math. Z. 176, 311 (1981).

[20] M. M. Wolf and J. I. Cirac, Dividing Quantum Channels, Commun. Math. Phys. 279, 147 (2008).

[21] M. M. Wolf and D. Perez-Garcia, The inverse eigenvalue problems for quantum channels, preprint arXiv:1005.4545, (2010).

[22] M. Rahaman, Multiplicative Properties of Quantum Channels, J. Phys. A: Math. Theor. 50, 345302 (2017).

[23] M. Sanz, D. Perez-Garcia, M. M. Wolf, and J. I. Cirac, A quantum version of Wielandt's inequality, IEEE Trans. Inf. Theory 56, 4668 (2010).

[24] W. Bruzda, M. Smaczyński, V. Cappellini, H.-J. Sommers, K. Życzkowski, Universality of spectra for interacting quantum chaotic systems, Phys. Rev. E 81, 066209 (2010).

[25] I. Bengtsson and K. Życzkowski, Geometry of Quantum States, II edition, Cambridge University Press, Cambridge, 2017.

[26] A. Fujiwara and P. Algoet, One-to-one parametrization of quantum channels, Phys. Rev. A 59, 3290 (1999).

[27] M. B. Ruskai, S. Szarek, and E. Werner, An analysis of completely positive trace-preserving maps on $\mathscr{M}_{2}$, Linear Algebra Appl. 347, 159 (2002).

[28] D. Braun, O. Giraud, I. Nechita, C. Pellegrini, and M. Znidaric, A universal set of qubit quantum channels, J. Phys. A 47, 135302 (2014).

[29] J. Watrous, Simpler semidefinite programs for completely bounded norms, Chicago J. of Th. Comp. Sci. 2013, 8 (2013).

[30] I. Nechita, Z. Puchała, Ł. Pawela, and K. Życzkowski Almost all quantum channels are equidistant, arXiv:1612.00401 (2016).

[31] M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002).

[32] M. D. Bowdrey, D. K. L. Oi, A. J. Short, K. Banaszek, and J. A. Jones, Fidelity of single qubit maps, Phys. Lett. A. 294, 258 (2002).

[33] M. Horodecki, P. Horodecki, and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Phys. Rev. A 60, 1888 (1999).

[34] E. Magesan, R. Blume-Kohout, and J. Emerson, Gate fidelity fluctuations and quantum process invariants, Phys. Rev. A 84, 012309 (2011).

[35] L. H. Pedersen, N. M. Møller, and K. Mølmer, The distribution of quantum fidelities, Phys. Lett. A 372, 7028 (2011).

[36] N. Johnston and D. W. Kribs, Quantum gate fidelity in terms of Choi matrices, J. Phys. A 44, 495303 (2011).

[37] J. Wallman, C. Granade, R. Harper, and S. T. Flammia, Estimating the Coherence of Noise, New J. Phys. 17, 113020 (2015).

[38] R. Kueng, D. M. Long, A. C. Doherty, and S. T. Flammia, Comparing Experiments to the Fault-Tolerance Threshold, Phys. Rev. Lett. 117, 170502 (2016).

[39] J. Wallman, Bounding experimental quantum error rates relative to fault-tolerant thresholds, arXiv:1511.00727v2 (2016).

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