# Measurement, Interpretation and Information

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## Abstract

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## 1. Introduction

## 2. The Quantum Measurement Problem

#### 2.1. Three Concepts of Quantum Measurement

- Single measurement: It is a single process, in which the reading of the pointer is registered. A single measurement, considered in isolation, does not supply yet relevant information about the state of the system $S$ prior to the measurement, since the amplitudes are not revealed.
- Frequency measurement: It is a repetition of identical single measurements, whose purpose is to obtain the values ${\left|{\alpha}_{i}\right|}^{2}$ on the basis of the frequencies of the pointer readings in the different single measurements. A frequency measurement supplies relevant information about the state of $S$, but is not yet sufficient to completely identify such a state, since it does not reveal the phases.
- State measurement: It is a collection of frequency measurements, each one of them with its particular experimental arrangement. Each arrangement correlates the pointer $P$ of the apparatus $M$ with an observable ${A}_{i}$ of the system, in such a way that the ${A}_{i}$ are not only different, but also non-commuting to each other. The information obtained by means of such a collection of frequency measurements is sufficient to reconstruct the state of $S$ [3].

#### 2.2. Ideal and Non-Ideal Measurements

- Imperfect measurement (first kind):$$\sum _{i}{\alpha}_{i}|{a}_{i}\rangle \otimes}|{p}_{0}\rangle \text{\hspace{1em}}\to \text{\hspace{1em}}{\displaystyle \sum _{ij}{\mathrm{\beta}}_{ij}|{a}_{i}\rangle \otimes}|{p}_{j}\rangle $$
- Disturbing measurement (second kind):$$\sum _{i}{\alpha}_{i}|{a}_{i}\rangle \otimes}|{p}_{0}\rangle \text{\hspace{1em}}\to \text{\hspace{1em}}{\displaystyle \sum _{i}{\alpha}_{i}|{a}_{i}^{d}\rangle \otimes}|{p}_{i}\rangle $$

## 3. Modal Interpretations of Quantum Mechanics

#### 3.1. The Modal Family

- The interpretation is based on the standard formalism of quantum mechanics.
- The interpretation is realist, that is, it aims at describing how reality would be if quantum mechanics were true.
- Quantum mechanics is a fundamental theory: it describes not only elementary particles but also macroscopic objects; quantum states refer to single systems, not to ensembles.
- The quantum state describes the probabilities of the possible properties of the system. In turn, systems possess actual properties at all times, whether or not a measurement is performed on them. The relationship between the quantum state and the actual properties is probabilistic.
- A quantum measurement is an ordinary physical interaction. There is no collapse: the quantum state always evolves unitarily according to the Schrödinger equation, which gives the time evolution of probabilities, not of actual properties.

#### 3.2. The modal−Hamiltonian Interpretation

Composite systems postulate:A quantum system $S\text{:(}\mathcal{O}\text{,}H\text{)}$, with initial state ${\mathrm{\rho}}_{0}\in \mathcal{O}\text{'}$, is composite when it can be partitioned into two quantum systems ${S}^{1}\text{:(}{\mathcal{O}}^{1}\text{,}{H}^{1}\text{)}$ and ${S}^{2}\text{:(}{\mathcal{O}}^{2}\text{,}{H}^{2}\text{)}$ such that (i) $\mathcal{O}={\mathcal{O}}^{1}\otimes {\mathcal{O}}^{2}$, and (ii) $H={H}^{1}\otimes {I}^{2}+{I}^{1}\otimes {H}^{2}$, (where ${I}^{1}$ and ${I}^{2}$ are the identity operators in the corresponding tensor product spaces). In this case, we say that ${S}^{1}$ and ${S}^{2}$ are subsystems of the composite system, $S={S}^{1}+{S}^{2}$. If the system is not composite, it is elemental.

Actualization rule:Given an elemental quantum system $S\text{:(}\mathcal{O}\text{,}H\text{)}$, the actual-valued observables of $S$ are $H$ and all the observables commuting with $H$ and having, at least, the same symmetries as $H$.

## 4. Modal-Hamiltonian Interpretation in Measurement

#### 4.1. The Ideal Case

#### 4.2. The Non-Ideal Case

- ➣
- If the ${\mathrm{\beta}}_{ni}$, with $n\ne i$, are small in the sense that ${\sum}_{n\ne i}{\left|{\mathrm{\beta}}_{ni}\right|}^{2}}\ll {\left|{\mathrm{\beta}}_{ii}\right|}^{2$, then ${\left|{\mathrm{\beta}}_{ii}\right|}^{2}\simeq {\left|{\alpha}_{i}\right|}^{2}$. This means that, in the frequency measurement performed by repetition of this single measurement, the coefficients ${\left|{\alpha}_{i}\right|}^{2}$ can be approximately obtained and, therefore, the frequency measurement is reliable.
- ➣
- If the ${\mathrm{\beta}}_{ni}$, with $n\ne i$, are not small, then ${\left|{\mathrm{\beta}}_{ii}\right|}^{2}\simeq {\left|{\alpha}_{i}\right|}^{2}$ does not hold. Therefore, the result obtained by means of the frequency measurement will be non-reliable.

#### 4.3. The Source of Non-Reliability

## 5. An Informational Account of Measurement

#### 5.1. Information in Shannon’s Context

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- The source S generates the message to be received at the destination.
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- The channel CH is the medium used to transmit the information from source to destination.
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- The destination D receives the message.

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- $H(S;D)$ is the mutual information: the average amount of information generated at the source S and received at the destination D.
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- $E$ is the equivocity: the average amount of information generated at S but not received at D.
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- $N$ is the noise: the average amount of information received at D but not generated at S.

- If S and D are completely independent, the values of E and N are maximum ( $E=H(S)$ and $N=H(D)$), and the value of $H(S;D)$ is minimum $H(S;D)=0$).
- If the dependence between S and D is maximum, the values of E and N are minimum ( $E=N=0$), and the value of $H(S;D)$ is maximum $H(S;D)=H(S)=H(D)$).

#### 5.2. Measurement as an Informational Process

- ➣
- When the channel is deterministic, then the measurement is ideal. In this case, all and only the information of the measured quantum state is recovered by the pointer. The perfect correlation introduced by the measuring device is embodied in the fact that the conditional probabilities that define the channel are trivial: $\mathrm{Pr}\left({p}_{j}/{a}_{k}\right)={\delta}_{jk}$.
- ➣
- When the channel is equivocal and/or noisy, then the measurement is non-ideal. In this case, there is some loss of relevant information E or/and addition of spurious information N through the process: this is embodied in the fact that the conditional probabilities that define the channel are not trivial. But since those probabilities characterize the measuring device through calibration, they can be used to give a criterion of reliability independent of the particular measurement carried out:
- −
- If the conditional probabilities are approximately trivial, $\mathrm{Pr}\left({p}_{j}/{a}_{k}\right)\simeq {\delta}_{jk}$, then $\mathrm{Pr}\left({p}_{j}\right)\simeq \mathrm{Pr}\left({a}_{j}\right)={\left|{\alpha}_{k}\right|}^{2}$. This means that, in the frequency measurements performed by repetition of the single measurement, the measured frequencies $f{r}_{j}$ approximately supply the value of the coefficients ${\left|{\alpha}_{i}\right|}^{2}$ and, therefore, the frequency measurement is reliable.
- −
- If the conditional probabilities are not approximately trivial, then $\mathrm{Pr}\left({p}_{j}\right)\simeq \mathrm{Pr}\left({a}_{j}\right)={\left|{\alpha}_{k}\right|}^{2}$ does not hold. Therefore, the results obtained by means of frequency measurements will be non-reliable.

## 6. Conclusions

- (a)
- There must be two quantum systems: a system to be measured, $S\text{:(}\mathcal{O}\text{,}{H}_{S}\text{)}$, and a measuring apparatus, $M\text{:(}\mathcal{O}\text{,}{H}_{M}\text{)}$.
- (b)
- The apparatus $M$ must be constructed in such a way to have a pointer observable $P$ such that (i) $\left[{H}_{M},P\right]=0$, (ii) $P$ has, at least, the same degeneracy as ${H}_{M}$, and (iii) the eigenvalues of $P$ are different and macroscopically distinguishable. As argued, these conditions are physically reasonable independently of this interpretation.
- (c)
- During a certain period, $S$ and $M$ interact through an interaction Hamiltonian ${H}_{\text{int}}$ intended to introduce a correlation between an observable $A$ of $S$ and the pointer $P$ of $M$.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Lombardi, O.; Fortin, S.; López, C.
Measurement, Interpretation and Information. *Entropy* **2015**, *17*, 7310-7330.
https://doi.org/10.3390/e17117310

**AMA Style**

Lombardi O, Fortin S, López C.
Measurement, Interpretation and Information. *Entropy*. 2015; 17(11):7310-7330.
https://doi.org/10.3390/e17117310

**Chicago/Turabian Style**

Lombardi, Olimpia, Sebastian Fortin, and Cristian López.
2015. "Measurement, Interpretation and Information" *Entropy* 17, no. 11: 7310-7330.
https://doi.org/10.3390/e17117310