A functorial characterization of von Neumann entropy

Parzygnat, Arthur J.

doi:10.48550/arxiv.2009.07125

Cited by 3 publications

(5 citation statements)

References 18 publications

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“…And since the Hilbert-Schmidt adjoint of the partial trace is a * -homomorphism, the information discrepancy in this case coincides with the "entropy change along a * -homomorphism" defined in Ref. [70].…”

Section: Deterministic Evolutionsupporting

confidence: 67%

“…The third equality follows from Theorem 5.1, while the remaining equalities follow from the previously computed identities as well as basic properties of the logarithm and the von Neumann entropy [70]. ■…”

Section: Extending Von Neumann Entropy To Quasi-density Matricesmentioning

confidence: 85%

“…[4,25,27], and a dynamical axiomatic characterization of von Neumann entropy appears in Ref. [70]. Many other axiomatic characterizations of related entropy functions can be found in Refs.…”

Section: Dynamical Information Measuresmentioning

confidence: 99%

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The Information Loss of a Stochastic Map

Fullwood

Parzygnat

2021

Entropy

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We provide a stochastic extension of the Baez–Fritz–Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure that we call conditional information loss. Although not functorial, these information measures are semi-functorial, a concept we introduce that is definable in any Markov category. We also introduce the notion of an entropic Bayes’ rule for information measures, and we provide a characterization of conditional entropy in terms of this rule.

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Section: Deterministic Evolutionsupporting

confidence: 67%

Section: Extending Von Neumann Entropy To Quasi-density Matricesmentioning

confidence: 85%

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The Information Loss of a Stochastic Map

Fullwood

Parzygnat

2021

Entropy

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“…There have been other categorical approaches to entropy, most notably [BFL11], [BF14], [Lei21], and [Par22]. Our presentation here has almost nothing in common with those.…”

Section: Introductionmentioning

confidence: 85%

Polynomial Functors and Shannon Entropy

Spivak

2023

Electron. Proc. Theor. Comput. Sci.

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Past work shows that one can associate a notion of Shannon entropy to a Dirichlet polynomial, regarded as an empirical distribution. Indeed, entropy can be extracted from any d ∈ Dir by a twostep process, where the first step is a rig homomorphism out of Dir, the set of Dirichlet polynomials, with rig structure given by standard addition and multiplication. In this short note, we show that this rig homomorphism can be upgraded to a rig functor, when we replace the set of Dirichlet polynomials by the category of ordinary (Cartesian) polynomials.In the Cartesian case, the process has three steps. The first step is a rig functor Poly Cart → Poly sending a polynomial p to ṗy, where ṗ is the derivative of p. The second is a rig functor Poly → Set × Set op , sending a polynomial q to the pair (q(1), Γ(q)), where Γ(q) = Poly(q, y) can be interpreted as the global sections of q viewed as a bundle, and q(1) as its base. To make this precise we define what appears to be a new distributive monoidal structure on Set × Set op , which can be understood geometrically in terms of rectangles. The last step, as for Dirichlet polynomials, is simply to extract the entropy as a real number from a pair of sets (A, B); it is given by log A − log A √ B and can be thought of as the log aspect ratio of the rectangle.

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“…perspective and showed that so-called information functions of degree 1 behave "a lot like certain derivations" [EVG15]. A few years prior in 2011, Baez, Fritz, and Leinster gave a category theoretical characterization of entropy in [BFL11], which was recently extended to the quantum setting by Parzygnat in [Par20]. In preparation of that 2011 result, Baez remarked in the informal article [Bae11] that entropy appears to behave like a derivation in a certain operadic context, an observation we verify and make explicit below.…”

Section: Introductionmentioning

confidence: 99%

Entropy as a Topological Operad Derivation

Bradley

2021

Preprint

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We share a small connection between information theory, algebra, and topology-namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian module over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.

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A functorial characterization of von Neumann entropy

Cited by 3 publications

References 18 publications

The Information Loss of a Stochastic Map

The Information Loss of a Stochastic Map

Polynomial Functors and Shannon Entropy

Entropy as a Topological Operad Derivation

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