# Quantum Theory, Namely the Pure and Reversible Theory of Information

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## Abstract

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## 1. Introduction

## 2. A Complete Set of Information-Theoretic Principles for Quantum Theory

**Figure 1.**Alice’s laboratory. Alice has at disposal many devices, each of them having an input system and an output system (represented by different wires) and possibly a set of outcomes labelling different processes that can take place. The devices can be connected in series and in parallel to form circuits. A circuit with no input and no output wires represents an experiment starting from the preparation of a state with a given source and ending with some measurement(s). Specifying a theory for Alice’s laboratory means specifying which are the allowed devices and specifying a rule to predict the probability of outcomes in such experiments.

**Principle 1**

**(Causality)**

**Principle 2**

**(Fine-Grained Composition)**

**Principle 3**

**(Perfect Distinguishability)**

**Principle 4**

**(Ideal Compression)**

**Figure 2.**Compressing information. Alice encodes information (here represented by a pile of books) in a suitable system carrying the smallest possible amount of data (here a USB stick). The most advantageous situation is when the compression is lossless (after the encoding Bob is able to perfectly retrieve the information) and maximally efficient (the encoding system contains only the pure states needed to convey the information compatible with ρ).

**Principle 5**

**(Local tomography)**

**Figure 3.**Local Tomography. Alice can reconstruct the state of compound systems using only local measurements on the components. A world where this property did not hold would contain global information that cannot be accessed with local experiments.

**Principle 6**

**(Purity and Reversibility of Physical Processes)**

- physical systems are associated to complex Hilbert spaces;
- the maximum number of perfectly distinguishable states of the system is equal to the dimension of the corresponding Hilbert space;
- the pure states of a system are described by the unit vectors in the corresponding Hilbert space (up to a global phase);
- the reversible processes on a system are described by the unitary operators on the corresponding Hilbert space (up to a global phase);
- the measurements on a system are described by resolutions of the identity in terms of positive operators ${\left\{{P}_{i}\right\}}_{i\in X}$ on the corresponding Hilbert space (aka POVMs, see, e.g., [30] for a didactical presentation);
- the mixed states of a system are described by density matrices on the corresponding Hilbert space;
- the probabilities of outcomes in a measurement are given by the Born rule ${p}_{i}=\mathrm{Tr}\left[{P}_{i}\rho \right]$, where ρ is the density matrix representing the system’s state and $\mathrm{Tr}$ denotes the trace of a matrix;
- the Hilbert space associated to a composite system is the tensor product of the Hilbert spaces associated to the components;
- random processes are described by completely positive trace-preserving maps.

## 3. Conservation of Information and the Purification Principle

## 4. Discussion and Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Chiribella, G.; D’Ariano, G.M.; Perinotti, P.
Quantum Theory, Namely the Pure and Reversible Theory of Information. *Entropy* **2012**, *14*, 1877-1893.
https://doi.org/10.3390/e14101877

**AMA Style**

Chiribella G, D’Ariano GM, Perinotti P.
Quantum Theory, Namely the Pure and Reversible Theory of Information. *Entropy*. 2012; 14(10):1877-1893.
https://doi.org/10.3390/e14101877

**Chicago/Turabian Style**

Chiribella, Giulio, Giacomo Mauro D’Ariano, and Paolo Perinotti.
2012. "Quantum Theory, Namely the Pure and Reversible Theory of Information" *Entropy* 14, no. 10: 1877-1893.
https://doi.org/10.3390/e14101877