Abstract
We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product of locally compact Abelian group G with its dual \({\hat{G}}\). The method is based upon the Pontryagin duality allowing to establish an isometrical isomorphism between the space of Hilbert–Schmidt operators in \(L^2(G)\) and the Hilbert space \(L^2({\hat{G}}\times G)\). Any such a measure determines a pair of hybrid (containing classical and quantum parts) quantum channels consisting of the measurement channel and the channel transmitting an initial quantum state to the ensemble of quantum states on the group. It is shown that the second channel can be called a complementary channel to the measurement channel.
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References
Holevo, A.: Probabilistic and statistical aspects of quantum theory. Edizioni della Normale, Pisa (2011)
Carmeli, C., Heinosaari, T., Pellonpaa, J.-P., Toigo, A.: Extremal covariant positive operator valued measures: the case of a compact symmetry group. J. Math. Phys. 49, 063504 (2008)
Decker, T., Janzing, D., Roetteler, M.: Implementation of group-covariant POVMs by orthogonal measurements. J. Math. Phys. 46, 012104 (2005)
Holevo, A.S., Yashin, V.I.: Maximum information gain of approximate quantum position measurement. Quantum Inf. Process. 20, 97 (2021)
Holevo, A.: On the classical capacity of general quantum gaussian measurement. Entropy 23(3), 377 (2021)
Amosov, G.G.: On general properties of non-commutative operator graphs. Lobachevskii J. Math. 39(3), 304–308 (2018)
Amosov, G.G., Mokeev, A.S., Pechen, A.N.: Noncommutative graphs based on finite-infinite system couplings: quantum error correction for a qubit coupled to a coherent field. Phys. Rev. A 103(4), 042407 (2021)
Amosov, G.G., Mokeev, A.S., Pechen, A.N.: Non-commutative graphs and quantum error correction for a two-mode quantum oscillator. Quantum Inf. Process. 19(3), 95 (2020)
Amosov, G.G., Mokeev, A.S.: Non-commutative graphs in the Fock space over one-particle Hilbert space. Lobachevskii J. Math. 41(4), 592–596 (2020)
Pechen, A., Trushechkin, A.: Measurement-assisted Landau-Zener transitions. Phys. Rev. A 91(5), 052316 (2015)
Il’in, N.B., Pechen, A.N.: Critical point in the problem of maximizing the transition probability using measurements in an \(n\)-level quantum system. Theor. Math. Phys. 194(3), 384–389 (2018)
Amosov, G.G.: On quantum tomography on locally compact groups. Phys. Lett. A 431, 128002 (2022)
Holevo, A.S.: Accessible information of a general quantum Gaussian ensemble. J. Math. Phys. 62(9), 092201 (2021)
Holevo, A.S.: Complementary channels and the additivity problem. Theory Probab. Appl. 51(1), 92–100 (2007)
Amosov, G.G., Mokeev, A.S., Pechen, A.N.: On the construction of a quantum channel corresponding to non-commutative graph for a qubit interacting with quantum oscillator. Lobachevskii J. Math. 42(10), 2280–2284 (2021)
Shirokov, M.E.: Entropy reduction of quantum measurements. J. Math. Phys. 52(5), 052202 (2011)
Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1962)
Pontrjagin, L.S.: The theorie of topological commutative groups. Ann. Math. 35(2), 361–388 (1934)
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This work is supported by Russian Science Foundation under the grant No 19-11-00086.
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Amosov, G.G. On quantum channels generated by covariant positive operator-valued measures on a locally compact group. Quantum Inf Process 21, 312 (2022). https://doi.org/10.1007/s11128-022-03655-x
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DOI: https://doi.org/10.1007/s11128-022-03655-x