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Entanglement Breaking Rank Via Complementary Channels and Multiplicative Domains

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Operators, Semigroups, Algebras and Function Theory (IWOTA 2021)

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 292))

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Abstract

Quantum entanglement can be studied through the theory of completely positive maps in a number of ways, including by making use of the Choi-Jamiolkowski isomorphism, which identifies separable states with entanglement breaking quantum channels, and optimal ensemble length with entanglement breaking rank. The multiplicative domain is an important operator structure in the theory of completely positive maps. We introduce a new technique to determine if a channel is entanglement breaking and to evaluate entanglement breaking rank, based on an analysis of multiplicative domains determined by complementary quantum channels. We give a full description of the class of entanglement breaking channels that have a projection as their Choi matrix, and we show the entanglement breaking rank and Choi rank of such channels are equal.

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References

  1. J. Ahiable, D.W. Kribs, J. Levick, R. Pereira, M. Rahaman, Entanglement breaking channels, stochastic matrices, and primitivity. Linear Algebra Appl. 629, 219–231 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras. Ill. J. Math. 18(4), 565–574 (1974)

    MathSciNet  MATH  Google Scholar 

  3. M.-D. Choi, N. Johnston, D.W. Kribs, The multiplicative domain in quantum error correction. J. Phys. A Math. Theor. 42(24), 245303 (2009)

    Google Scholar 

  4. M. Girard, D. Leung, J. Levick, C.-K. Li, V. Paulsen, Y.T. Poon, J. Watrous, On the mixed-unitary rank of quantum channels. Commun. Math. Phys. 394, 919–951 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  5. S.J. Harris, R.H. Levene, V.I. Paulsen, S. Plosker, M. Rahaman, Schur multipliers and mixed unitary maps. J. Math. Phys. 59(11), 112201 (2018)

    Google Scholar 

  6. A.S. Holevo, Coding theorems for quantum channels. Russ. Math. Surv. 53, 1295–1331 (1999)

    Article  MATH  Google Scholar 

  7. A.S. Holevo, Complementary channels and the additivity problem. Theory Probab. Appl. 51(1), 92–100 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. A.S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, vol. 16 (Walter de Gruyter, Berlin, 2012)

    Book  MATH  Google Scholar 

  9. M. Horodecki, P.W. Shor, M.B. Ruskai, Entanglement breaking channels. Rev. Math. Phys. 15(06), 629–641 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Johnston, D.W. Kribs, Generalized multiplicative domains and quantum error correction. Proc. Am. Math. Soc. 139(2), 627–639 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. D.W Kribs, J. Levick, M.I. Nelson, R. Pereira, M. Rahaman, Quantum complementarity and operator structures. Quantum Inf. Comput. 19, 67–83 (2019)

    Google Scholar 

  12. D.W. Kribs, J. Levick, K. Olfert, R. Pereira, M. Rahaman, Nullspaces of entanglement breaking channels and applications. J. Phys. A Math. Theor. 54(10), 105303 (2021)

    Google Scholar 

  13. J. Levick, D.W. Kribs, R. Pereira, Quantum privacy and Schur product channels. Rep. Math. Phys. 80(3), 333–347 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. M.A. Nielson, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  15. S.K. Pandey, V.I. Paulsen, J. Prakash, M. Rahaman, Entanglement breaking rank and the existence of SIC POVMs. J. Math. Phys. 61(4), 042203 (2020)

    Google Scholar 

  16. V. Paulsen, Completely Bounded Maps and Operator Algebras. Number 78 (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  17. M. Rahaman, A new bound on quantum Wielandt inequality. IEEE Trans. Inf. Theory 66(1), 147–154 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Rahaman, S. Jaques, V.I. Paulsen, Eventually entanglement breaking maps. J. Math. Phys. 59(6), 062201 (2018)

    Google Scholar 

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Acknowledgements

We are grateful to the referee for helpful comments. D.W.K. was supported by NSERC Discovery Grant 400160. R.P. was supported by NSERC Discovery Grant 400550. M.R. is supported by the European Research Council (ERC Grant Agreement No. 851716). Part of this work was completed during IWOTA Lancaster UK 2021, and as such, the authors would like to express thanks to the conference organizers and acknowledge their funding via EPSRC grant EP/T007524/1.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada

    David W. Kribs, Jeremy Levick & Rajesh Pereira

  2. Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, Lyon, France

    Mizanur Rahaman

Authors
  1. David W. Kribs
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  2. Jeremy Levick
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  3. Rajesh Pereira
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  4. Mizanur Rahaman
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Corresponding author

Correspondence to David W. Kribs .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Lancaster University, Lancaster, UK

    Yemon Choi

  2. University of Central Lancashire, Preston, UK

    Matthew Daws

  3. Department of Mathematics and Statistics, Lancaster University, Lancaster, UK

    Gordon Blower

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Kribs, D.W., Levick, J., Pereira, R., Rahaman, M. (2023). Entanglement Breaking Rank Via Complementary Channels and Multiplicative Domains. In: Choi, Y., Daws, M., Blower, G. (eds) Operators, Semigroups, Algebras and Function Theory. IWOTA 2021. Operator Theory: Advances and Applications, vol 292. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-38020-4_8

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