#### Document Type

Article

#### Publication Date

2010

#### Abstract

We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality (IC) recently proposed by Pawlowski et al (2009 arXiv:0905.2292). We consider two entropic quantities, which we term *measurement* and *mixing* entropy. In the context of classical and quantum theory, these coincide, being given by the Shannon and von Neumann entropies, respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entropy is not concave whenever the state space is a non-simplicial polytope. Thus, the condition that measurement and mixing entropies coincide is a strong constraint on possible theories. We call theories with this property *monoentropic*.

Measurement entropy is subadditive, but not in general strongly subadditive. Equivalently, if we define the mutual information between two systems A and B by the usual formula *I (A : B) = H(A) + H(B)− H(AB)*, where *H* denotes the measurement entropy and *AB* is a non-signaling composite of *A* and *B*, then it can happen that *I (A : BC) < I (A : B)*. This is relevant to IC in the sense of Pawlowski *et al*: we show that any monoentropic non-signaling theory in which measurement entropy is strongly subadditive, and also satisfies a version of the Holevo bound, is informationally causal, and on the other hand we observe that Popescu–Rohrlich boxes, which violate IC, also violate strong subadditivity. We also explore the interplay between measurement and mixing entropy and various natural conditions on theories that arise in quantum axiomatics.

#### Recommended Citation

Howard Barnum et al 2010 *New J. Phys.* 12 033024. doi:10.1088/1367-2630/12/3/033024

#### Peer Reviewed

1

#### Copyright

IOP Publishing

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

## Comments

This article was originally published in

New Journal of Physics, volume 12, in 2010. DOI: 10.1088/1367-2630/12/3/033024