On Linear Structure of Non-commutative Operator Graphs

Abstract

We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the circle group has the system of unitary generators fulfilling permutations of basis vectors. For the graph generated by the Heisenberg-Weyl group the explicit formula for a dimension is given. Thus, we found a new description of the linear structure for the operator graphs introduced in our previous works.

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References

  1. 1.

    G. G. Amosov and A. S. Mokeev, “On non-commutative operator graphs generated by covariant resolutions of identity,” Quantum Inform. Process. 17(12), 325 (2018).

    MathSciNet  Article  Google Scholar 

  2. 2.

    G. G. Amosov and A. S. Mokeev, “On non-commutative operator graphs generated by reducible unitary representation of the Heisenberg-Weyl group,” Int. J. Theor. Phys. doi https://doi.org/10.1007/s10773-018-3963-4

  3. 3.

    G. G. Amosov and A. S. Mokeev, “On construction of anticliques for noncommutative operator graphs,” Zap. Nauch. Sem. SPb. Otd. Mat. Inst. Steklov(POMI) 456, 5–15 (2017)

    MATH  Google Scholar 

  4. 3a.

    G. G. Amosov and A. S. Mokeev, J. Math. Sci. 234, 269–275 (2018).

    MathSciNet  Article  Google Scholar 

  5. 4.

    R. Duan, S. Severini, and A. Winter, “Zero-error communication via quantum channels, noncommutative graphs and a quantum Lovasz theta function,” IEEE Trans. Inf. Theory 59, 1164–1174 (2013).

    Article  Google Scholar 

  6. 5.

    E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Phys. Rev. A 55, 900 (1997).

    MathSciNet  Article  Google Scholar 

  7. 6.

    G. G. Amosov, “On general properties of non-commutative operator graphs,” Lobachevskii J. Math. 39(3), 304–308 (2018).

    MathSciNet  Article  Google Scholar 

  8. 7.

    N. Weaver, “A quantum Ramsey theorem for operator systems,” Proc. Am. Math. Soc. 145, 4595–4605 (2017).

    MathSciNet  Article  Google Scholar 

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Funding

The work is supported by Russian Science Foundation under the grant no. 19-11-00086.

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Correspondence to G. G. Amosov or A. S. Mokeev.

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Submitted by S. A. Grigoryan

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Amosov, G.G., Mokeev, A.S. On Linear Structure of Non-commutative Operator Graphs. Lobachevskii J Math 40, 1440–1443 (2019). https://doi.org/10.1134/S1995080219100032

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Keywords and phrases

  • non-commutative operator graphs
  • covariant resolution of identity
  • quantum anticliques