Differentiating the State Evaluation Map from Matrices to Functions on Projective Space
The pure state evaluation map from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>n</mi></msub><mrow><mo>(</mo><mi mathvariant=...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-02-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/2/474 |
| Summary: | The pure state evaluation map from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>n</mi></msub><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi mathvariant="double-struck">C</mi><msup><mi mathvariant="double-struck">P</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> is a completely positive map of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mo>*</mo></msup></semantics></math></inline-formula>-algebras intertwining the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mi>n</mi></msub></semantics></math></inline-formula> symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>n</mi></msub><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to the holomorphic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mo>∂</mo><mo>¯</mo></mover></semantics></math></inline-formula> calculus on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><msup><mi mathvariant="double-struck">P</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>. The method uses connections on Hilbert <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mo>*</mo></msup></semantics></math></inline-formula>-bimodules. |
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| ISSN: | 2073-8994 |