**Appendix A.**

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

Using Equations (2) and (4) in the main text and that *ρ*O 0 = *ρ*A 0 , we find

$$\begin{split} \left\langle 1 - \text{Tr} \left( \rho\_{T}^{\mathcal{O}} \rho\_{T}^{\mathcal{A}} \right) \right\rangle &= - \left\langle \int\_{\text{Tr} \left( \rho\_{0}^{\mathcal{O}} \rho\_{0}^{\mathcal{A}} \right)} d \, \text{Tr} \left( \rho\_{t}^{\mathcal{O}} \rho\_{t}^{\mathcal{A}} \right) \right\rangle \\ &= - \int\_{F\_{0}}^{F\_{\text{f}}} d \, \text{Tr} \left( \rho\_{t}^{\mathcal{A}} \rho\_{t}^{\mathcal{A}} \right) \\ &= -2 \int\_{0}^{T} \text{Tr} \left( \rho\_{t}^{\mathcal{A}} A \left[ \rho\_{t}^{\mathcal{A}} \right] \right) dt \\ &= + 2 \sum\_{a} \frac{1}{8 \pi\_{m}^{\mathcal{A}}} \int\_{0}^{T} \text{Tr} \left( \left[ A\_{a'} \left[ A\_{a'} \rho\_{t}^{\mathcal{A}} \right] \right] \rho\_{t}^{\mathcal{A}} \right) dt \\ &= \sum\_{a} \frac{1}{4 \pi\_{m}^{\mathcal{A}}} \int\_{0}^{T} \text{Tr} \left( \left[ \rho\_{t}^{\mathcal{A}}, A\_{a} \right] \left[ A\_{\mathcal{A}}, \rho\_{t}^{\mathcal{A}} \right] \right) dt. \end{split} \tag{14}$$

This identity can be conveniently expressed in terms of the 2-norm of the commutator [*ρ*<sup>A</sup> *t* , *A*] as

$$\left\langle 1 - \text{Tr} \left( \rho\_T^{\mathcal{Q}} \rho\_T^A \right) \right\rangle = \sum\_a \frac{1}{4 \tau\_m^a} \int\_0^T \left\| \left[ \rho\_t^A \, \big|\, A\_a \right] \right\|\_2^2 dt = \sum\_a \frac{T}{4 \tau\_m^a} \overline{\left\| \left[ \rho\_t^A \, \big|\, A\_a \right] \right\|\_{2^\*}^2} \tag{A2}$$

where we denote the time-average of a function *f* by *f* ≡ *T* 0 *f*(*t*)*dt*/*T*. Note that the expression ∑*α* 1 4*τ<sup>α</sup> m* ( ( *ρ*A *t* , *Aα* ( (2 2 plays the role of a time-averaged decoherence time [15,16], generalizing Equation (11) in the main text.

This sets alternative bounds on the average distance between the state *ρ*A*t* assigned by A and the actual state of the system *ρ*O*t* , in terms of the effect of the Lindblad dephasing term acting on the incomplete-knowledge state *ρ*A*t* ,

$$\left\| T \sum\_{a} \frac{1}{4\pi\_{n}^{4}} \overline{\left\| \left[ \rho\_{l}^{A}, A\_{a} \right] \right\|} \right\|\_{2}^{2} \leq \left\langle \mathcal{D} \left( \rho\_{T}^{\mathcal{O}}, \rho\_{T}^{A} \right) \right\rangle \leq \sqrt{T \sum\_{a} \frac{1}{4\pi\_{n}^{4}}} \overline{\left\| \left[ \rho\_{l}^{A}, A\_{a} \right] \right\|}\_{2}^{2} .$$

A short time analysis provides a sense of the evolution of the upper and lower bounds on the trace distance and how they compare to its variance. To leading order in a Taylor series expansion,

$$\mathcal{P}\left(\rho\_{\tau}^{\mathcal{A}}\right) \approx 1 + 2 \operatorname{Tr}\left(\rho\_{0}^{\mathcal{A}} \Lambda \left[\rho\_{0}^{\mathcal{A}}\right]\right) \tau = 1 - \sum\_{a} \frac{1}{4\tau\_{m}^{a}} \operatorname{Tr}\left(\left[\rho\_{0}^{\mathcal{A}}, A\_{a}\right] \left[A\_{a}, \rho\_{0}^{\mathcal{A}}\right]\right) \tau,\tag{A3}$$

and one finds

$$\left|\tau \sum\_{\mathfrak{a}} \frac{1}{4\tau\_{\mathfrak{m}}^{4}}\right| \left|\left[\rho\_{0}^{A}, A\_{\mathfrak{a}}\right]\right|\Big|\_{2}^{2} \leq \left<\mathcal{D}\left(\rho\_{\tau}^{\mathcal{D}}, \rho\_{\tau}^{\mathcal{A}}\right)\right> \leq \sqrt{\tau \sum\_{\mathfrak{a}} \frac{1}{4\tau\_{\mathfrak{m}}^{4}} \left\|\left[\rho\_{0}^{A}, A\_{\mathfrak{a}}\right]\right\|\_{2}^{2}}.\tag{A4}$$

Note that the behavior of the trace distance is determined by the timescale in which decoherence occurs.

Using Equation (9) in the main text and Jensen's inequality, one obtains

$$\left\langle \mathcal{D}^2 \left( \rho\_{\overline{\mathcal{T}}'}^{\mathcal{O}} \rho\_{\overline{\mathcal{T}}}^A \right) \right\rangle \le 1 - \mathcal{P} \left( \rho\_{\overline{\mathcal{T}}}^A \right),\tag{A5}$$

which implies that the variance <sup>Δ</sup>D2*T* ≡ <sup>D</sup><sup>2</sup>*ρ*O*T* , *ρ*A*T* − <sup>D</sup>*ρ*O*T* , *ρ*A*T* 2 satisfies

$$
\Delta \mathcal{D}\_T^2 \le \mathcal{P}\left(\rho\_T^A\right) - \mathcal{P}\left(\rho\_T^A\right)^2. \tag{A6}
$$

In the short time limit this becomes

$$
\Delta \mathcal{D}\_{\tau}^{2} \leq -2 \operatorname{Tr} \left( \rho\_{0}^{4} \Lambda \left[ \rho\_{0}^{4} \right] \right) \tau. \tag{A7}
$$

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

Using that *ρ*O*t* is pure, and that the von Neumann entropy is given by *<sup>S</sup>*(*ρ*) ≡ − Tr(*ρ* log *ρ*), we obtain that the average over the results unknown to agen<sup>t</sup> A satisfy

$$
\begin{split}
\left< \mathcal{S} \left( \rho\_t^{\mathcal{O}} || \rho\_t^A \right) \right> &= \left< \text{Tr} \left( \rho\_t^{\mathcal{O}} \log \rho\_t^{\mathcal{O}} \right) \right> - \left< \text{Tr} \left( \rho\_t^{\mathcal{O}} \log \rho\_t^A \right) \right> \\
&= 0 - \text{Tr} \left( \rho\_t^A \log \rho\_t^A \right) = \mathcal{S} \left( \rho\_t^A \right).
\end{split}
\tag{A8}
$$

This sets a direct connection between the average error induced by assigning state *ρ*A*t* instead of the exact state *ρ*O*t* , as quantified by the relative entropy, in terms of the von Neumann entropy of the state accessible to agen<sup>t</sup> A.

In turn, the variance of the relative entropy satisfies

$$\begin{split} \Delta S^2 \left( \rho\_t^{\mathcal{O}} || \rho\_t^A \right) &= \left< S^2 \left( \rho\_t^{\mathcal{O}} || \rho\_t^A \right) \right> - \left< S \left( \rho\_t^{\mathcal{O}} || \rho\_t^A \right) \right>^2 \\ &= \left< \text{Tr} \left( \rho\_t^{\mathcal{O}} \log \rho\_t^A \right)^2 \right> - S^2 \left( \rho\_t^A \right) \\ &\le \left< \text{Tr} \left( \rho\_t^{\mathcal{O}} \right) \text{Tr} \left( \rho\_t^{\mathcal{O}} \log^2 \rho\_t^A \right) \right> - S^2 \left( \rho\_t^A \right) \\ &= \text{Tr} \left( \rho\_t^{\mathcal{A}} \log^2 \rho\_t^A \right) - S^2 \left( \rho\_t^A \right), \end{split} \tag{4.9}$$

using the Cauchy–Schwarz inequality in the third line. Note that this expression is identical to the variance of the operator − log *ρ*A*t* , which can be thought of as the quantum extension to the notion of the "information content" or "surprisal" (− log *p*) in classical information theory.

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

Consider two agents A and B who simultaneously monitor different observables on a system. Each one has access to the measurement outcomes of their devices, but not to the results obtained by the other agent. The states *ρ*A*T* and *ρ*B*T* that A and B assign to the system differ from the actual pure state *ρ*O*T* that corresponds to the complete description of the system. For simplicity let us consider that A monitors a single observable *A* and B monitors a single observable *B*. The complete-description state of the system assigned by all-knowing agen<sup>t</sup> O evolves according to

$$d\rho\_t^{\mathcal{O}} = L\left[\rho\_t^{\mathcal{O}}\right]dt + I\_{\mathcal{A}}\left[\rho\_t^{\mathcal{O}}\right]dW\_t^{\mathcal{A}} + I\_{\mathcal{B}}\left[\rho\_t^{\mathcal{O}}\right]dW\_t^{\mathcal{B}},\tag{A10}$$

with the Lindbladian *<sup>L</sup>ρ*O*t* ≡ <sup>−</sup>*<sup>i</sup><sup>H</sup>*, *ρ*O*t* + <sup>Λ</sup>A*ρ*O*t* + <sup>Λ</sup>B*ρ*O*t* , with corresponding dephasing terms on observables *A* and *B*. The innovation terms *IA* and *IB* are defined as in Equation (3) in the main text, and *dW*A*t* and *dW*B*t* are independent noise terms.

The states of both observers satisfy

$$d\rho\_t^A = L\left[\rho\_t^A\right]dt + I\_A\left[\rho\_t^A\right]dV\_t^A\tag{A11}$$

$$d\rho\_t^{\mathcal{B}} = L\left[\rho\_t^{\mathcal{B}}\right]dt + I\_B\left[\rho\_t^{\mathcal{B}}\right]dV\_t^{\mathcal{B}}.\tag{A12}$$

Consistency between observers implies that their noises are related to the ones appearing in Equation (A10) by [1,3]:

$$dW\_{l}^{A} = \left(\text{Tr}\left(\rho\_{l}^{A}A\right) - \text{Tr}\left(\rho\_{l}^{\mathcal{O}}A\right)\right)\frac{dt}{\tau\_{\mathcal{M}}} + dV\_{l}^{A}$$

$$dW\_{l}^{B} = \left(\text{Tr}\left(\rho\_{l}^{B}B\right) - \text{Tr}\left(\rho\_{l}^{\mathcal{O}}B\right)\right)\frac{dt}{\tau\_{\mathcal{M}}} + dV\_{l}^{B}.\tag{A13}$$

As the state of each observer satisfies Equation (9), the triangle inequality provides the upper bound

$$\langle \mathcal{D}\left(\rho\_T^A, \rho\_T^B\right)\rangle\_{\mathcal{AB}} \le \sqrt{1 - \text{Tr}\left(\rho\_T^{A^2}\right)} + \sqrt{1 - \text{Tr}\left(\rho\_T^{B^2}\right)}.\tag{A14}$$

and the lower bound

$$\left| \left< \mathcal{D} \left( \rho\_T^{\mathcal{A}}, \rho\_T^{\mathcal{B}} \right) \right> \right|\_{\mathcal{A}\mathcal{B}} \ge \left| \text{Tr} \left( \rho\_T^{\mathcal{A}^2} \right) - \text{Tr} \left( \rho\_T^{\mathcal{B}^2} \right) \right|. \tag{A15}$$

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

We consider the case of observer O monitoring the spin components *σzj* on a 1D transverse field Ising model, with the Hamiltonian defined in Equation (18) of the main text. Figure A1 shows the evolution of the average trace distance <sup>D</sup>*ρ*O*T* , *ρ*B*T* between the complete description and B's partial one, along with the bounds (16), for different values of the monitoring efficiency *η*. Figure A2 shows the evolution of the average relative entropy *<sup>S</sup>ρ*O*T* ||*ρ*B*T* . The dynamics are simulated by implementation of the monitoring process as a sequence of weak measurements modeled by Kraus operators acting on the state of the system. Specifically, the evolution of *ρ*O*t* and corresponding state *ρ*B*t* with partial measurements is numerically obtained from assuming two independent measurement processes, as in [1].

**Figure A1. Evolution of the average trace distance and its bounds.** Simulated evolution of the average trace distance D*ρ*O*T* , *ρ*B*T* between complete and incomplete descriptions for a spin chain initially in a paramagnetic state on which individual spin components *σzj* are monitored. The simulation corresponds to *N* = 6 spins, with couplings *J<sup>τ</sup>m* = *h<sup>τ</sup>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 *η* = 0 (**left**), agen<sup>t</sup> A, without any access to the measurement outcomes, has the most incomplete description of the system. After gaining access to partial measurement results, with *η* = 0.5 (**center**) B gets closer to the complete description of the state of the system. Finally, when *η* = 0.9 (**right**), access to enough information provides B with an almost complete description of the state. Importantly, in all cases the agen<sup>t</sup> can bound how far the description possessed is from the complete one solely in terms solely of the purity <sup>P</sup>*ρ*B*T*.

**Figure A2. Evolution of the average relative entropy and its bounds.** Simulated evolution of the average relative entropy *Sρ*O*T* ||*ρ*B*T* 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*

Consider the case of a one-dimensional harmonic oscillator with position and momentum operators *X* and *P*, respectively. We assume agen<sup>t</sup> B is monitoring the position of the harmonic oscillator, with an efficiency *η*. The dynamics of state *ρ*B*t* is dictated by Equation (15) in the main text for the case of a single monitored observable, with

$$\Lambda\left[\rho\_t^{\mathcal{S}}\right] = \frac{1}{8\pi\_m} \left[X\_\prime \left[X\_\prime \rho\_t^{\mathcal{S}}\right]\right]; \qquad I\_X\left[\rho\_t^{\mathcal{S}}\right] = \frac{1}{\sqrt{4\pi\_m}} \left(\left\{X\_\prime \rho\_t^{\mathcal{S}}\right\} - 2\operatorname{Tr}\left(X \rho\_t^{\mathcal{S}}\right)\rho\_t^{\mathcal{S}}\right). \tag{A16}$$

Such a dynamics preserves the Gaussian property of states. For these, the variances

$$w\_{\mathbf{X}} \equiv \text{Tr}\left(\rho\_l^{\mathcal{B}} \mathbf{X}^2\right) - \text{Tr}\left(\rho\_l^{\mathcal{B}} \mathbf{X}\right)^2\_{\dots} \tag{A17}$$

$$w\_p \equiv \text{Tr}\left(\rho\_t^B P^2\right) - \text{Tr}\left(\rho\_t^B P\right)^2,\tag{A18}$$

and covariance

$$\mathcal{L}\_{xp} \equiv \text{Tr}\left(\rho\_t^{\mathcal{B}} \frac{\{X, P\}}{2}\right) - \text{Tr}\left(\rho\_t^{\mathcal{B}} X\right) \text{Tr}\left(\rho\_t^{\mathcal{B}} P\right),\tag{A19}$$

satisfy the following set of differential equations (in natural units) [1,21]:

$$\frac{d}{dt}\upsilon\_x = 2\omega\upsilon\_{xp} - \frac{\eta}{\tau\_m}\upsilon\_{x'}^2\tag{A20a}$$

$$\frac{d}{dt}\upsilon\_p = -2\omega\upsilon\_{xp} + \frac{1}{4\tau\_m} - \frac{\eta}{\tau\_m}c\_{xp\prime}^2\tag{A20b}$$

$$\frac{d}{dt}\mathbf{c}\_{xp} = \omega \mathbf{v}\_p - \omega \mathbf{v}\_x - \frac{\eta}{\tau\_m} \mathbf{v}\_x \mathbf{c}\_{xp}.\tag{A20c}$$

While the first moments do evolve stochastically, the second moments above satisfy a set of deterministic coupled differential equations. This in turn implies that the purity of the state, which can be obtained from the covariance matrix [22–26]

$$
\sigma(t) \equiv \begin{bmatrix} \upsilon\_x & \upsilon\_{xp} \\ \upsilon\_{xp} & \upsilon\_p \end{bmatrix} \tag{A21}
$$

as

$$\mathcal{P}\left(\rho\_T^{\mathcal{B}}\right) = \frac{1}{2\sqrt{\det\left[\sigma(t)\right]}},\tag{A22}$$

evolves deterministically as well.

> The solution for long times can be derived from Equations (A20), giving

$$
\omega\_{xp}^{\rm sc} = -\frac{\omega \tau\_{\rm m} \pm \sqrt{\omega^2 \tau\_{\rm m}^2 + \eta/4}}{\eta},
\tag{A23a}
$$

$$
\sigma\_x^{ss} = \sqrt{\frac{2\omega \tau\_m}{\eta} \epsilon\_{xp}^{ss}}\tag{A23b}
$$

$$
\sigma\_p^{\rm ss} = \upsilon\_x^{\rm ss} \left( 1 + \frac{\eta}{\omega \tau\_m} \sigma\_{xp}^{\rm ss} \right) \tag{A23c}
$$

which provides the long-time asymptotic value of the purity as a function of the measurement efficiency. The latter turns out to have the following simple expression

$$\begin{split} \mathcal{P}\left(\boldsymbol{\rho}\_{\mathcal{T}}^{\mathcal{B}}\right) &= \frac{1}{2\sqrt{\sigma\_{\boldsymbol{x}}^{\mathcal{S}\boldsymbol{x}}\boldsymbol{v}\_{\mathcal{P}}^{\mathcal{S}\boldsymbol{x}} - (\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{x}})^{2}}} \\ &= \frac{1}{2\sqrt{\frac{2\omega\tau\_{\boldsymbol{m}}}{\eta}\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{s}}\left(1 + \frac{\eta}{\omega\tau\_{\boldsymbol{m}}}\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{s}}\right) - (\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{s}})^{2}}} \\ &= \frac{1}{2\sqrt{\frac{2\omega\tau\_{\boldsymbol{m}}}{\eta}\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{s}} + (\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{s}})^{2}}} \\ &= \frac{1}{2\sqrt{\frac{\tau\_{\boldsymbol{m}}}{\eta}\left(\frac{1}{4\overline{\tau}\_{\boldsymbol{m}}} - \frac{\eta}{\tau\_{\boldsymbol{m}}}(\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{s}})^{2}\right) + (\boldsymbol{c}\_{\boldsymbol{x}\boldsymbol{p}}^{\mathcal{S}\boldsymbol{s}})^{2}} = \frac{1}{2\sqrt{\frac{1}{4\eta}}} = \sqrt{\eta}. \end{split} \tag{A25}$$

Using that

$$1 - \mathcal{P}\left(\rho\_T^{\mathbb{E}}\right) \le \left\langle \mathcal{D}\left(\rho\_T^{\mathcal{O}}, \rho\_T^{\mathbb{E}}\right) \right\rangle\_{\mathcal{B}} \le \sqrt{1 - \mathcal{P}\left(\rho\_T^{\mathbb{E}}\right)},\tag{A26}$$

then implies

$$1 - \sqrt{\eta} \le \left< \mathcal{D} \left( \rho\_{T'}^{\mathcal{O}} \rho\_T^{\mathcal{B}} \right) \right>\_{\mathcal{B}} \le \sqrt{1 - \sqrt{\eta}}.\tag{A27}$$

The entropy of a 1-mode Gaussian state can be expressed in terms of the purity of the state as

$$\begin{split} S\left(\rho\_{\overline{T}}^{\mathbb{S}}\right) &= \left(\frac{1}{2\mathcal{P}\left(\rho\_{\overline{T}}^{\mathbb{S}}\right)} + 1/2\right) \log\left(\frac{1}{2\mathcal{P}\left(\rho\_{\overline{T}}^{\mathbb{S}}\right)} + 1/2\right) \\ &\quad - \left(\frac{1}{2\mathcal{P}\left(\rho\_{\overline{T}}^{\mathbb{S}}\right)} - 1/2\right) \log\left(\frac{1}{2\mathcal{P}\left(\rho\_{\overline{T}}^{\mathbb{S}}\right)} - 1/2\right). \end{split} \tag{A28}$$

Then, using that *<sup>S</sup>ρ*O*t* ||*ρ*B*t* B = *<sup>S</sup>ρ*B*t* and Equation (A25), we obtain that for long times,

$$\begin{split} \left< S\left(\rho\_t^{\mathcal{O}}||\rho\_t^{\mathcal{S}}| \right) \right>\_{\mathcal{S}} &= S\left(\rho\_T^{\mathcal{S}}\right) \\ &= \left(\frac{1}{2\sqrt{\eta}} + \frac{1}{2}\right) \log\left(\frac{1}{2\sqrt{\eta}} + \frac{1}{2}\right) \\ &\quad - \left(\frac{1}{2\sqrt{\eta}} - \frac{1}{2}\right) \log\left(\frac{1}{2\sqrt{\eta}} - \frac{1}{2}\right). \end{split} \tag{A29}$$
