# Limits to Perception by Quantum Monitoring with Finite Efficiency

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## Abstract

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## 1. Quantum Limits to Perception

## 2. Transition to Complete Descriptions

## 3. Illustrations

#### 3.1. Evolution of the Limits to Perception

#### 3.2. Transition to Complete Descriptions

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Derivation of Bounds to Average Trace Distance

#### Appendix A.2. Derivation of the Average and Variance of the Quantum Relative Entropy

#### Appendix A.3. Bounds to the Difference between Perceptions of Multiple Agents

#### Appendix A.4. Illustration—Evolution of Limits to Perception

**Figure A1.**

**Evolution of the average trace distance and its bounds.**Simulated evolution of the average trace distance $\u2329\mathcal{D}\left({\rho}_{T}^{\mathcal{O}},{\rho}_{T}^{\mathcal{B}}\right)\u232a$ between complete and incomplete descriptions for a spin chain initially in a paramagnetic state on which individual spin components ${\sigma}_{j}^{z}$ are monitored. The simulation corresponds to $N=6$ spins, with couplings $J{\tau}_{m}=h{\tau}_{m}=1/2$. The upper and lower bounds (16) on the average trace distance is depicted by dashed lines, while the shaded area represents the (one standard deviation) confidence region obtained from the upper bound (13) on the standard deviation in the main text, calculated with respect to the mean distance. For $\eta =0$ (

**left**), agent $\mathcal{A}$, without any access to the measurement outcomes, has the most incomplete description of the system. After gaining access to partial measurement results, with $\eta =0.5$ (

**center**) $\mathcal{B}$ gets closer to the complete description of the state of the system. Finally, when $\eta =0.9$ (

**right**), access to enough information provides $\mathcal{B}$ with an almost complete description of the state. Importantly, in all cases the agent can bound how far the description possessed is from the complete one solely in terms solely of the purity $\mathcal{P}\left({\rho}_{T}^{\mathcal{B}}\right)$.

**Figure A2.**

**Evolution of the average relative entropy and its bounds.**Simulated evolution of the average relative entropy $\u2329S\left({\rho}_{T}^{\mathcal{O}}\left|\right|{\rho}_{T}^{\mathcal{B}}\right)\u232a$ between complete and incomplete descriptions for a spin chain on which the z components of individual spins are monitored. The shaded area represents the (one standard deviation) confidence region obtained from the upper bound on the standard deviation of the relative entropy, Equation (14) in the main text. As in the case of the trace distance, access to more information leads to a more accurate state assigned by the agent.

#### Appendix A.5. Illustration—Transition to Complete Descriptions

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**Figure 1.**

**Illustration of the varying degrees of perception by different agents.**The amount of information that an agent possesses of a system can drastically alter its perception, as the expectations of outcomes for measurements performed on the system can differ. (

**a**) The state ${\rho}_{t}^{\mathcal{O}}$ assigned by omniscient agent $\mathcal{O}$, who has full access to the measurement outcomes, corresponds to a complete pure-state description of the system. O thus has the most accurate predictive power. (

**b**) An agent $\mathcal{A}$ completely ignorant of measurement outcomes possesses the most incomplete description of the system. (

**c**) A continuous transition between the two descriptions, corresponding to the worst and most complete perceptions of the system respectively, is obtained by considering an agent $\mathcal{B}$ with partial access to the measurement outcomes of the monitoring process.

**Figure 2.**

**Evolution of the average relative entropy.**Simulated evolution of the average $\u2329S\left({\rho}_{t}^{\mathcal{O}}\left|\right|{\rho}_{t}^{\mathcal{B}}\right)\u232a=\u2329S\left({\rho}_{t}^{\mathcal{B}}\right)\u232a$ of the relative entropy between complete and incomplete descriptions for a spin chain initially in a paramagnetic state on which individual spin components ${\sigma}_{j}^{z}$ are monitored. Here $\langle \xb7\rangle $ denotes an average over all measurement outcomes, and ${\rho}_{t}^{\mathcal{B}}={\langle {\rho}_{t}^{\mathcal{O}}\rangle}_{\mathcal{B}}$ is the state assigned by agent $\mathcal{B}$ after discarding the outcomes unknown to him. The simulation corresponds to $N=6$ spins, with couplings $J{\tau}_{m}=h{\tau}_{m}=1/2$. For $\eta =0$ (black continuous curve), agent $\mathcal{A}$, without any access to the measurement outcomes, has the most incomplete description of the system. For $\eta =0.5$ (red dashed curve), $\mathcal{B}$ gets closer to the complete description of the state of the system, after gaining access to partial measurement results. Finally, when $\eta =0.9$ (blue dotted curve), access to enough information provides $\mathcal{B}$ with an almost complete description of the state. Importantly, in all cases the agent can estimate how far the description possessed is from the complete one solely in terms of the entropy $S\left({\rho}_{t}^{\mathcal{B}}\right)$.

**Figure 3.**

**Transition between levels of perception.**Bounds on average trace distance (

**left**) and average relative entropy (

**right**) as function of measurement efficiency for a harmonic oscillator undergoing monitoring of its position. For such a system the purity of the state ${\rho}_{t}^{\mathcal{B}}$ depends solely on the measurement efficiency with which observer $\mathcal{B}$ monitors the system. This illustrates the transition from complete ignorance of the outcomes of measurements performed ($\eta =0$), to the most complete description as $\eta \to 1$—the situation with the most accurate perception. Efficient use of information happens when a small fraction of the measurement output is incorporated at $\eta \ll 1$, as then both $\mathcal{D}\left({\rho}_{t}^{\mathcal{B}},{\rho}_{t}^{\mathcal{O}}\right)$ and the relative entropy $S\left({\rho}_{t}^{\mathcal{O}}\left|\right|{\rho}_{t}^{\mathcal{B}}\right)$ decay rapidly.

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García-Pintos, L.P.; del Campo, A.
Limits to Perception by Quantum Monitoring with Finite Efficiency. *Entropy* **2021**, *23*, 1527.
https://doi.org/10.3390/e23111527

**AMA Style**

García-Pintos LP, del Campo A.
Limits to Perception by Quantum Monitoring with Finite Efficiency. *Entropy*. 2021; 23(11):1527.
https://doi.org/10.3390/e23111527

**Chicago/Turabian Style**

García-Pintos, Luis Pedro, and Adolfo del Campo.
2021. "Limits to Perception by Quantum Monitoring with Finite Efficiency" *Entropy* 23, no. 11: 1527.
https://doi.org/10.3390/e23111527