# Quantum Dynamical Entropies and Gács Algorithmic Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Qubits: Fundamentals

#### Quantum Information Sources

## 3. Classical Algorithmic Complexity and Entropy

#### 3.1. Algorithmic Complexity

#### 3.2. Universal Probability

## 4. Alicki–Fannes Entropy

**Remark 1.**The lim sup in (23) has to be used for the sequence of density matrices $\rho \left[{\mathcal{X}}^{\left(n\right)}\right]$ is not a stationary one [3,18]. In fact, while consistency holds as tracing $\rho \left[{\mathcal{X}}^{\left(n\right)}\right]$ over the n-th factor yields the density matrix corresponding to the first $n-1$ factors, ${\mathrm{Tr}}_{n}\rho \left[{\mathcal{X}}^{\left(n\right)}\right]=\rho \left[{\mathcal{X}}^{(n-1)}\right]$, stationarity does not. Indeed, in general, ${\mathrm{Tr}}_{1}\rho \left[{\mathcal{X}}^{\left(n\right)}\right]\ne \rho \left[{\mathcal{X}}^{(n-1)}\right]$. Thence, the density matrix for the local algebra ${M}_{p,q}$ corresponding to sites from p to q in line of principle depends on the starting factor ${M}_{2}\left(\mathbb{C}\right)$ at site p.

**Remark 2.**With respect to the AF-quantum dynamical entropy, the quantum dynamical entropy introduced by Connes, Narnhofer and Thirring [2] has a more complicated construction essentially due to the fact that it is based on a classical symbolic modeling with superimposed quantum corrections [17,21]. Consider a quantum spin chain endowed with a translation invariant state with sufficiently fast decaying correlations, the simplest case being the tensor product of a same density matrix ρ, that is

#### AF-Entropy: Operational Interpretation

## 5. Quantum Dynamical Entropies and Gács Complexity

#### 5.1. Gács Complexity and CNT-Entropy for Quantum Spin Chains

**Remark 3.**A quantum Brudno’s relation was proved in [16] for an ergodic quantum spin chain and the quantum complexity of a quantum state ψ, ${\mathbf{QC}}^{\delta}\left(\psi \right)$, introduced by [13,16] and defined by

- σ is any density matrix acting on the Hilbert space $\mathcal{H}}_{{[0,1]}^{*}}=\underset{n\to +\infty}{lim}\underset{k=0}{\overset{n}{\u2a01}}{\mathcal{H}}_{k$, where ${\mathcal{H}}_{k}={\mathbb{C}}^{\otimes k}$ is the Hilbert space of k qubits: ${\mathcal{H}}_{{[0,1]}^{*}}$ accommodates classical binary strings of arbitrary length as quantum states and makes meaningful referring to generic density matrices acting on it as quantum programs. Furthermore, these density matrices constitute a Banach space ${\mathcal{T}}_{1}^{+}\left({\mathcal{H}}_{{[0,1]}^{*}}\right)$ under the so-called trace-distance ${\parallel X\parallel}_{tr}=\frac{1}{2}\mathrm{Tr}\left(\sqrt{{X}^{\u2020}X}\right)$.
- $\ell \left(\sigma \right)$ is the length of quantum programs and is defined by$$\ell \left(\sigma \right)=min\left\{n\in \mathbb{N}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\sigma \in {\mathcal{T}}_{1}^{+}\left({\mathcal{H}}_{\le n}\right)\right\}\phantom{\rule{4pt}{0ex}}$$
- M is a quantum operation, that is a completely positive map from ${\mathcal{T}}_{1}^{+}\left({\mathcal{H}}_{{[0,1]}^{*}}\right)$ into itself: these maps are interpreted as Quantum Turing Machines (see [22] for a detailed mathematical characterization and a discussion of the halting time in the context of the quantum superposition principle).

#### 5.2. Gács Complexity and AF-Entropy for Quantum Spin Chains

## 6. Conclusions

## Acknowledgement

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Benatti, F.
Quantum Dynamical Entropies and Gács Algorithmic Entropy. *Entropy* **2012**, *14*, 1259-1273.
https://doi.org/10.3390/e14071259

**AMA Style**

Benatti F.
Quantum Dynamical Entropies and Gács Algorithmic Entropy. *Entropy*. 2012; 14(7):1259-1273.
https://doi.org/10.3390/e14071259

**Chicago/Turabian Style**

Benatti, Fabio.
2012. "Quantum Dynamical Entropies and Gács Algorithmic Entropy" *Entropy* 14, no. 7: 1259-1273.
https://doi.org/10.3390/e14071259