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ON THE BURES METRIC,
$C^*$-NORM AND QUANTUM METRIC
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 27 January 2025, pp. 1-11
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- Article
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Given a unital
$C^*$-algebra and a faithful trace, we prove that the topology on the associated density space induced by the 
$C^*$-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and 
$C^*$-norm when the 
$C^*$-algebra is assumed to be finite dimensional. Finally, we provide an example where the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.
