Asymptotic Equipartition Theorems in von Neumann algebras
- URL: http://arxiv.org/abs/2212.14700v2
- Date: Tue, 16 May 2023 12:59:10 GMT
- Title: Asymptotic Equipartition Theorems in von Neumann algebras
- Authors: Omar Fawzi, Li Gao, and Mizanur Rahaman
- Abstract summary: We show that the smooth max entropy of i.i.d. states on a von Neumann algebra has an rate given by the quantum relative entropy.
Our AEP not only applies to states, but also to quantum channels with appropriate restrictions.
- Score: 24.1712628013996
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The Asymptotic Equipartition Property (AEP) in information theory establishes
that independent and identically distributed (i.i.d.) states behave in a way
that is similar to uniform states. In particular, with appropriate smoothing,
for such states both the min and the max relative entropy asymptotically
coincide with the relative entropy. In this paper, we generalize several such
equipartition properties to states on general von Neumann algebras.
First, we show that the smooth max relative entropy of i.i.d. states on a von
Neumann algebra has an asymptotic rate given by the quantum relative entropy.
In fact, our AEP not only applies to states, but also to quantum channels with
appropriate restrictions. In addition, going beyond the i.i.d. assumption, we
show that for states that are produced by a sequential process of quantum
channels, the smooth max relative entropy can be upper bounded by the sum of
appropriate channel relative entropies.
Our main technical contributions are to extend to the context of general von
Neumann algebras a chain rule for quantum channels, as well as an additivity
result for the channel relative entropy with a replacer channel.
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