# Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

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## Abstract

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## 1. Introduction

## 2. Classical Probabilities

## 3. Quantum Probabilities

#### 3.1. Elementary Tests in Quantum Mechanics

#### 3.2. von Neumann’s Axioms

#### 3.3. Quantum Correlations

## 4. Generalized Setting

#### 4.1. Non-Kolmogorovian Information Theory and Contextuality

- R.T. Cox [72,73] showed that if a rational agent is confronted with a Boolean algebra representing empirical events, then any function measuring his degree of belief of the occurrence of any event must be equivalent to a classical probability calculus. In [112], it is shown that if a rational agent is confronted with a non-distributive algebra of physical events, then the consistent probabilities must be those of Equation (9).
- In a similar way, in the Cox approach (see [59] and the references therein), it is shown that Shannon’s entropy:$$H\left({p}_{i}\right)=-\sum _{i}{p}_{i}ln\left({p}_{i}\right)$$$$S\left(\rho \right)=-\mathrm{tr}(\rho ln(\rho \left)\right)$$

#### 4.2. Communication and Correlations in the Generalized Setting

**Definition 1.**A state $\nu \in {\mathcal{C}}_{A}\otimes {\mathcal{C}}_{B}$ will be called separable if there exist ${p}_{i}\in {\mathbb{R}}_{\ge 0}$, ${\nu}_{A}^{i}\in {\mathcal{C}}_{A}$ and ${\nu}_{B}^{i}\in {\mathcal{C}}_{B}$, such that:

## 5. Generalized Entropies and Majorization

#### 5.1. Entropies and Majorization in Classical and Quantum Theories

**Definition 2.**For an N-dimensional probability vector $p=\left\{{p}_{i}\right\}$ with ${p}_{i}\ge 0$ and ${\sum}_{i=1}^{N}{p}_{i}=1$, the classical $(h,\varphi )$-entropies are defined as:

**Definition 3.**Let us consider a quantum system described by a density operator ρ acting on an N-dimensional Hilbert space $\mathcal{H}$. The quantum $(h,\varphi )$-entropies (under the same assumptions for h and ϕ in Definition 2) are defined as follows:

**Definition 4.**Under the same assumptions in Definition 3, the quantum $(h,\varphi )$-entropies are also defined as:

#### 5.2. Entropies and Majorization in General Probabilistic Theories

**Definition 5.**Let us consider a state $\nu \in \mathcal{C}$. The general $(h,\varphi )$-entropies (under the same assumptions for h and ϕ in Definition 2) are defined as follows:

**Definition 6.**Given a state ν, if the majorant of the set ${M}_{\nu}$ (partially ordered by majorization) exists, it is called the spectrum of ν, and it is denoted by $\overline{p}\left(\nu \right)$.

**Definition 7.**Given two states μ and ν, one has that μ is majorized by ν, denoted by $\mu \prec \nu $, if and only if:

**Figure 1.**The generalized spectral decomposition, Equation (29), can be computed in a variety of probabilistic theories. (

**a**) When the convex set is a simplex, the decomposition in terms of pure states is unique, and so, it determines the spectrum of ν. In the triangle above, ν can be written in a unique way as a mixture of ${\nu}_{1}$, ${\nu}_{2}$ and ${\nu}_{3}$. (

**b**) For the state ν of a qubit, the spectrum is given by the eigendecomposition of its density matrix in terms of the orthogonal pure states ${\nu}_{1}$ and ${\nu}_{2}$. The same happens for any other quantum mechanical model. (

**c**) For a general probabilistic theory, there are, a priori, many decompositions of a state in terms of pure ones, and we have to look for the majorant one. For example, for the non-regular polygon with four vertices, the state in the barycenter is $\nu =\frac{1}{2}{\nu}_{1}+\frac{1}{2}{\nu}_{2}=x{\nu}_{1}^{\prime}+(1-x){\nu}_{2}^{\prime}$, with $x>\frac{1}{2}$. The second set of coefficients majorize the first one, so $\overline{p}\left(\nu \right)=\{x,1-x\}$ constitute the spectrum of ν. Note, however, that in both decompositions, the pure states are perfectly distinguishable.

**Definition 8.**Under the same assumptions as in Definition 5, we define the $(h,\varphi )$-entropies:

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Holik, F.; Bosyk, G.M.; Bellomo, G.
Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory. *Entropy* **2015**, *17*, 7349-7373.
https://doi.org/10.3390/e17117349

**AMA Style**

Holik F, Bosyk GM, Bellomo G.
Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory. *Entropy*. 2015; 17(11):7349-7373.
https://doi.org/10.3390/e17117349

**Chicago/Turabian Style**

Holik, Federico, Gustavo M. Bosyk, and Guido Bellomo.
2015. "Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory" *Entropy* 17, no. 11: 7349-7373.
https://doi.org/10.3390/e17117349