Functional calculus of quantum channels for the holomorphic discrete series of SU(1,1)
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| Title: | Functional calculus of quantum channels for the holomorphic discrete series of SU(1,1) |
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| Authors: | van Haastrecht, Robin, 2000 |
| Source: | Journal of Functional Analysis. 289(6) |
| Subject Terms: | Wehrl inequality, Quantum channels, Reproducing kernels, Hermitian symmetric spaces |
| Description: | The tensor product of two holomorphic discrete series representations of SU(1,1) can be decomposed as a direct multiplicity-free sum of infinitely many holomorphic discrete series representations. I shall introduce equivariant quantum channels for each component of the direct sum by mapping the tensor product of an operator and the identity onto the projection onto one of the irreducible components, generalizing the construction of pure equivariant quantum channels for compact groups. Then I calculate the functional calculus of this operator for polynomials and prove a limit formula for the trace of the functional calculus for any differentiable function. The methods I used are the theory of reproducing kernel Hilbert spaces and a Plancherel theorem for the disk D=SU(1,1)/U(1), together with exact constants for the eigenvalues of the Berezin transform. I prove that the limit of the trace of the functional calculus can be expressed using generalized Husimi functions or using Berezin transforms. |
| File Description: | electronic |
| Access URL: | https://research.chalmers.se/publication/546322 https://research.chalmers.se/publication/546322/file/546322_Fulltext.pdf |
| Database: | SwePub |
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Nájsť tento článok vo Web of Science | Abstract: | The tensor product of two holomorphic discrete series representations of SU(1,1) can be decomposed as a direct multiplicity-free sum of infinitely many holomorphic discrete series representations. I shall introduce equivariant quantum channels for each component of the direct sum by mapping the tensor product of an operator and the identity onto the projection onto one of the irreducible components, generalizing the construction of pure equivariant quantum channels for compact groups. Then I calculate the functional calculus of this operator for polynomials and prove a limit formula for the trace of the functional calculus for any differentiable function. The methods I used are the theory of reproducing kernel Hilbert spaces and a Plancherel theorem for the disk D=SU(1,1)/U(1), together with exact constants for the eigenvalues of the Berezin transform. I prove that the limit of the trace of the functional calculus can be expressed using generalized Husimi functions or using Berezin transforms. |
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| ISSN: | 00221236 10960783 |
| DOI: | 10.1016/j.jfa.2025.111036 |