5.1.1. Approximate Thermal-Objective States

As noted, an exact thermal-objective state can only emerge when the system and environment Hamiltonians have a very particular relationship. More generally, we can look for the existence of a state that is *approximately* thermal and objective.

Suppose we allow a deviation in the environment Hamiltonian from the ideal Hamiltonian, i.e., where *H*ˆ E = ∑*i*(*Ei* + *c* + *<sup>δ</sup>i*)|*φiφi*|E , where {*<sup>δ</sup>i*}*i* are different for at least two *i*'s (we work with one environment for simplicity). In this situation, while the state in Equation (25) is objective, it no longer has local thermal environments. We can measure the minimum distance between the set of thermal states and the set of objective states with the trace norm as follows:

$$D\_{\text{obj-thm}}(\hat{H}\_{\mathcal{S}}, \hat{H}\_{\mathcal{E}}, \boldsymbol{\beta}) = \min\_{\rho\_{\text{obj-}\mathcal{I}\mathcal{S}\mathcal{E}}} \left\| \rho\_{\text{obj}} - \gamma\_{\text{S}\mathcal{E}} \right\|\_{1'} \tag{26}$$

where *ρobj* are objective states, and *γ*SE = *γ*S ⊗ *γ*E + *χ*SE have locally thermal system and environment and variable correlation matrix *χ*SE .

Taking the ansatz

$$\rho\_{obj}^{\*} = \frac{1}{Z\_{\beta, \mathcal{H}\_S}} \sum\_i e^{-E\_i \beta} |i\rangle\langle i| \otimes |\phi\_i\rangle\langle\phi\_i|\_{\mathcal{E}'} \tag{27}$$

from Equation (25), the distance of this objective state to the set of locally thermal states can be bounded above:

$$\leq \min\_{\boldsymbol{\chi}, \boldsymbol{\chi}, \boldsymbol{\xi}, \text{traceless}} \left\lVert \frac{1}{Z\_{\beta, \Omega\_{\mathcal{S}}}} \sum\_{i} e^{-E\_{i}\boldsymbol{\beta}} |i\rangle\langle i| \otimes |\phi\_{i}\rangle\langle\phi\_{i}| \right. \\ \left. \left. \begin{array}{l} 1 \\ \frac{1}{Z\_{\beta, \Omega\_{\mathcal{S}}}} \sum\_{i} e^{-E\_{i}\boldsymbol{\beta}} |i\rangle\langle i| \otimes |\phi\_{i}\rangle\langle\phi\_{i}| \\\ -\frac{1}{Z\_{\beta, \Omega\_{\mathcal{S}}}} \sum\_{i} e^{-E\_{i}\boldsymbol{\beta}} |i\rangle\langle i| \otimes \frac{1}{Z\_{\beta, \Omega\_{\mathcal{E}}}} \sum\_{j} e^{-\left(E\_{j} + \boldsymbol{c} + \boldsymbol{\delta}\_{j}\right)\boldsymbol{\delta}} |\phi\_{j}\rangle\langle\phi\_{j}| \right. \\ \end{array} \right\vert . \tag{28}$$

By picking a sample matrix,

$$\chi\_{S\mathcal{E}} = \frac{1}{Z\_{\not\beta\hat{\mathcal{H}}\_{S}}} \sum\_{i} e^{-E\_{i}\beta} |i\rangle\langle i| \otimes |\phi\_{i}\rangle\langle\phi\_{i}|\_{\mathcal{E}} - \gamma\_{S} \otimes \frac{1}{Z\_{\not\beta\hat{\mathcal{H}}\_{S}}} \sum\_{i} e^{-E\_{i}\beta} |\phi\_{i}\rangle\langle\phi\_{i}|\_{\mathcal{E}'} \tag{29}$$

the distance is then bounded as

$$\begin{split} & D\_{\text{obj-thm}} \left( \hat{H}\_{\mathcal{S}}, \hat{H}\_{\mathcal{E}\_{\mathcal{E}}}, \boldsymbol{\beta} \right) \\ & \leq \left\| \begin{array}{c} \gamma\_{\mathcal{S}} \odot \frac{1}{Z\_{\mathcal{S}\mathcal{A}\_{\mathcal{E}}}} \sum\_{\boldsymbol{\beta}} \varepsilon^{-\left(\mathcal{E}\_{\mathcal{I}} + \varepsilon + \boldsymbol{\delta}\_{\hat{\boldsymbol{\beta}}}\right) \boldsymbol{\beta}} |\boldsymbol{\phi}\_{\hat{\boldsymbol{\beta}}}\rangle \langle \boldsymbol{\phi}\_{\hat{\boldsymbol{\beta}}}|\_{\mathcal{E}} - \gamma\_{\mathcal{S}} \odot \frac{1}{Z\_{\mathcal{S}\mathcal{A}\_{\mathcal{E}}}} \sum\_{\boldsymbol{\delta}} \varepsilon^{-\operatorname{E}\_{\mathcal{E}} \boldsymbol{\beta}} |\boldsymbol{\phi}\_{\hat{\boldsymbol{\beta}}}\rangle \langle \boldsymbol{\phi}\_{\hat{\boldsymbol{\beta}}}|\_{\mathcal{E}} \end{array} \right\|\_{1} \end{split} \tag{30}$$

$$\overset{\cdots}{\varepsilon} = \left\| \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \end{array} \begin{array}{c} \cdots \\ \hline \end{array} \begin{array}{c} \cdots \\ \hline \\ \end{array} \begin{array}{c} \cdots \\ \hline \end{$$

$$\hat{\rho} = \sum\_{i} \left| \frac{\varepsilon^{-(E\_i + \varepsilon + \delta\_i)\beta}}{Z\_{\beta, \hat{\Omega}\_{\mathcal{E}}}} - \frac{\varepsilon^{-E\_i \beta}}{Z\_{\beta, \hat{\Omega}\_{\mathcal{S}}}} \right|. \tag{32}$$

The distance is bounded by the difference between the thermal-state eigenenergies, which here is a nonlinear function of the deviations {*<sup>δ</sup>i*}.

In Figure 2, we consider if this error is Normal-distributed *δi* ∼ N (0, *σ*) with mean zero and standard deviation *σ*. We see that, on average, increasing the spread *σ* linearly increases the upper bound on the distance measure of Equation (32) in the domain considered.

**Figure 2.** Averaged upper bound to the distance (Equation (32)) between the set of objective states vs. the set of thermal states (locally thermal system and environment) versus standard deviation *σ* of the deviations *δi*. That is, the environment Hamiltonian is less-than-optimal: for a system Hamiltonian energy distribution {*Ei*}, the environment Hamiltonian energies are {*Ei* + *<sup>δ</sup>i*}, where the deviations are *δi* ∼ N (0, *σ*) (normal distribution). The inverse temperature is *β* = 1, with qubit system and qubit environment. Averaged across 1000 random instances.

#### 5.1.2. Employing Macrofractions

A known technique for improving distinguishability of environments is the use of macrofractions, i.e., grouping multiple subenvironments into a greater environment fragment [48–50]. By doing this, even if the deviation of the environment Hamiltonian energies from the system Hamiltonian energies is large, we may be able to construct an approximate objective-thermal state.

Consider the distance between the set of objective states and the set of states with locally thermal subsystems similarly as above:

$$D\_{\text{obj-thm}}\left(\hat{H}\_{\text{S}}, \{\hat{H}\_{\mathcal{E}\_k}\}\_{k=1'}^N \beta\right) = \min\_{\rho\_{\text{obj}}, \gamma\_{\text{S}\mathcal{E}}} \left\|\rho\_{\text{obj}} - \gamma\_{\text{S}\mathcal{E}}\right\|\_{1'} \tag{33}$$

where the following state consists of locally thermal system and environments: *γ*SE = *γ*S ⊗ *γ*E1 ⊗···⊗ *γ*E*N* + *χ*SE , with correlation matrix *χ*SE such that trS [*<sup>χ</sup>*SE ] = 0 and trE*k* [*<sup>χ</sup>*SE ] = 0 for all *k*.

Using *<sup>ρ</sup>*<sup>∗</sup>*obj* = 1 *<sup>Z</sup>β*,*H*<sup>ˆ</sup> S ∑*i e*<sup>−</sup>*Eiβ*|*ii*| ⊗ *Nk*=<sup>1</sup> |*φiφi*|E*k* as an example close-by objective state, and with matrix

$$\chi\_{\mathcal{SE}}^{\*} = \rho\_{obj}^{\*} - \gamma\_{\mathcal{S}} \otimes \bigotimes\_{k=1}^{N} \left( \frac{1}{Z\_{\beta\_{\ast}\hat{\mathcal{H}}\_{\mathcal{S}}}} \sum\_{i} e^{-E\_{i}\beta} |\phi\_{i}\rangle\langle\phi\_{i}|\_{\mathcal{E}\_{k}} \right), \tag{34}$$

the distance is then bounded as


*<sup>D</sup>*obj-thm*H*<sup>ˆ</sup> S, *H*ˆ E1 ,..., *H*ˆ E*N* , *β*≤ ((((((((((( *Nk*=1⎛⎝ 1 *<sup>Z</sup>β*,*H*<sup>ˆ</sup> E*k* ∑*i e*<sup>−</sup>(*Ei*+*c*+*δi*|*<sup>k</sup>* )*β*|*φiφi*|E*k*⎞⎠ − *Nk*=<sup>1</sup> <sup>1</sup> *<sup>Z</sup>β*,*H*<sup>ˆ</sup> S ∑*i <sup>e</sup>*<sup>−</sup>*Eiβ*|*φiφi*|E*k*! (((((((((((1 (35) = ∑ *i*1,...,*iN* -----*e*<sup>−</sup>*Ei*1+*c*+*δi*1|1*<sup>β</sup> <sup>Z</sup>β*,*H*<sup>ˆ</sup> E1+ ··· + *e*<sup>−</sup>*EiN* <sup>+</sup>*c*+*δiN*|*N<sup>β</sup> <sup>Z</sup>β*,*H*<sup>ˆ</sup> <sup>E</sup>*N*− *e*<sup>−</sup>*Ei*1 *β* ···*e*<sup>−</sup>*EiN β <sup>Z</sup>β*,*H*<sup>ˆ</sup>*N* ------. (36)

 S -

We plot the behaviour of the bound Equation (36) in Figure 3. As expected, the figure shows that increasing the number of environments included into a macrofraction leads to a decreasing distance between the set of thermal states versus the set of objective states. This is essentially as though we considered increasingly larger environments, which is the focus of the next subsection.

NE macrofraction size

**Figure 3.** Upper bound to the distance (Equation (36)) between the set of objective states and the set of thermal states (locally thermal system and environment) versus macrofractions of size *NE*. A macrofraction is collection of environments. Here, the environment Hamiltonians are less-than-optimal, i.e., for a system Hamiltonian energy distribution {*Ei*}, the environment Hamiltonians energies are "*Ei* + *<sup>δ</sup>i*|*k*#, *k* = 1, ... , *NE*, where the "error" is *δi* ∼ N (0, *σ* = 0.05) (Normal distribution). The inverse temperature is *β* = 1, with qubit system and qubit environments. Averaged across 500 random instances.

#### *5.2. Environment Dimension Larger than System Dimension*

In common situations, environments are much larger than the system. Intuitively, larger environment dimensions should give greater flexibility to form approximately objective-thermal states. In this section, we find that the existence of exact thermalobjective states requires very fine tuned system and environment Hamiltonians. However, we will also find that as the dimension of the environment goes up (e.g., towards the classical/thermodynamic limit), there will exist states that are close to both objectivity and local thermality.

**Theorem 1.** *The distance between the set of objective states and the set of states with locally thermal system and environment goes to zero as the dimension of the environment goes to infinity.*

**Proof of Theorem 1.** Consider the distance between these two sets for some given system thermal state *γ*S and environment thermal state *γ*E :

$$D\_{\text{obj-thm}}(\gamma\_{\mathcal{S}\prime}\gamma\_{\mathcal{E}}) = \min\_{\substack{\rho\_{\text{obj}}\\ \chi\_{\mathcal{S}\mathcal{E}}}} \left\| \rho\_{\text{obj}} - (\gamma\_{\mathcal{S}} \otimes \gamma\_{\mathcal{E}} + \chi\_{\mathcal{S}\mathcal{E}}) \right\|\_{1'} \tag{37}$$

where *ρobj* are objective states and *χ*SE are correlation matrices. Decomposing the system thermal state as *γ*S = ∑*i pi*|*ii*|, where *pi* = *e* −*Eiβ <sup>Z</sup>β*,*H*<sup>ˆ</sup> S , we can bound Equation (37) by fixing the local state on the system in the objective states *ρobj* as

$$D\_{\text{obj-thm}}(\gamma\_{\mathcal{S}\prime}\gamma\_{\mathcal{E}}) \leq \min\_{\substack{\rho\_{\mathcal{E}|i} \vdash \rho\_{\mathcal{E}|i} \\ \chi\_{\mathcal{S}\mathcal{E}}}} \left\| \sum\_{i} p\_{i} |i\rangle\langle i| \otimes \rho\_{\mathcal{E}|i} - (\gamma\_{\mathcal{S}} \otimes \gamma\_{\mathcal{E}} + \chi\_{\mathcal{S}\mathcal{E}}) \right\|\_{1} \tag{38}$$

where *ρ*E|*i* ⊥ *ρ*E|*i* denotes that the conditional environment states should be perfectly distinguishable as per objectivity.

By picking a sample matrix *χ*SE = ∑*i pi*|*ii*| ⊗ *ρ*E|*i* − *γ*S ⊗ ∑*i piρ*E|*<sup>i</sup>*, the distance Equation (38) is then bounded as

$$D\_{\text{obj-thm}}(\gamma\_{\mathcal{S}\prime}\gamma\_{\mathcal{E}}) \leq \min\_{\rho\_{\mathcal{E}|i^\perp}, \boldsymbol{\rho}\_{\mathcal{E}|i^\parallel}} \left\| \sum\_{i} p\_i \rho\_{\mathcal{E}|i} - \gamma\_{\mathcal{E}} \right\|\_{1} \tag{39}$$

Write the environment thermal state as *γ*E = ∑*j e* −*hjβ <sup>Z</sup>β*,*H*<sup>ˆ</sup> E |*ψjψj*|. Suppose that the states *ρ*E|*i* are diagonal in the same eigenstates as \$|*ψj*-%, i.e., take *ρ*E|*i* = ∑*j cj*|*i*|*ψjψj*|, where ∑*jcj*|*i*=1,and*cj*|*icj*|*i*=0for*i*=*i* fororthogonality.Then,

$$D\_{\text{obj-thm}}(\gamma\_{\mathcal{S}}, \gamma\_{\mathcal{E}}) \leq \min\_{c\_{j|i} \text{ orthogonal}} \sum\_{j} \left| \sum\_{i} p\_{i} c\_{j|i} - \frac{\varepsilon^{-h\_{j}\p}}{Z\_{\beta, H\_{\mathcal{E}}}} \right|. \tag{40}$$

As *cj*|*icj*|*i* = 0 (i.e., for orthogonality), we can define disjoint sets *Ci* where *j* ∈ *Ci* means *cj*|*i* = 0 and *cj*|*i* = 0 if *i* = *i*. We are essentially partitioning the environment eigenvectors |*ψj*-Einto groups labelled by the *system* eigenvectors |*i*-S.

$$D\_{\text{obj-thm}}(\gamma\_{\mathcal{S}}, \gamma\_{\mathcal{E}}) \leq \min\_{\{\mathcal{C}\_{i}\}: \text{disjpoint}} \sum\_{k=1}^{d\_{\mathcal{S}}} \sum\_{j \in \mathcal{C}\_{k}} \left| p\_{k} c\_{j|k} - \frac{e^{-h\_{j}\beta}}{Z\_{\beta, \mathcal{H}\_{\mathcal{E}}}} \right|. \tag{41}$$

Naively, the minimum would occur if *c*∗*j*|*i* = *<sup>e</sup>*<sup>−</sup>*hj<sup>β</sup> <sup>Z</sup>β*,*H*<sup>ˆ</sup> E !/*pi*. However, such *cj*|*i* may not lead to a real state, due to lack of normalisation. Instead, we can upperbound this with

the candidate *c*˜*j*|*i* = *c*<sup>∗</sup> *j*|*i* ∑*<sup>k</sup>*∈*Ci c*<sup>∗</sup> *k*|*i* , which *is* normalised. In the optimal case, *c*˜*j*|*i* = *c*<sup>∗</sup> *j*|*i* and the distance would go to zero. Simplifying, then our candidates are

$$\mathcal{E}\_{j|i} = \frac{\mathcal{e}^{-h\_j \beta}}{\sum\_{k \in \mathcal{C}\_i} \mathcal{e}^{-h\_k \beta}} \, \prime \tag{42}$$

.

leading to

$$D\_{\text{obj-thm.}}(\gamma\_{\mathcal{S}}, \gamma\_{\mathcal{E}}) \leq \min\_{\{\mathcal{C}\_{\mathcal{i}}\}: \text{disjoint}} \sum\_{k=1}^{d\_{\mathcal{S}}} \left| p\_k - \frac{\left(\sum\_{j \in \mathcal{C}\_k} e^{-h\_j \mathcal{S}}\right)}{Z\_{\mathcal{\beta}, \mathcal{R}\_{\mathcal{E}}}} \right|. \tag{43}$$

Without loss of generality, we can consider the smallest *hj* to be zero, and therefore *e* −*hjβ* = 1 .

$$\max\_{\square} \overline{Z\_{\emptyset, \mathcal{H}\_{\mathcal{E}\_{\square}}}} = \overline{Z\_{\emptyset, \mathcal{H}\_{\mathcal{E}\_{\square}}}}$$

Consider the following algorithm for picking *j* indices to include in *Ck*. Every time we include in another index ˜*j* into *Ck*, the value of ∑*j*∈*Ck e* −*hjβ <sup>Z</sup>β*,*H*<sup>ˆ</sup> E increases by at most 1 *<sup>Z</sup>β*,*H*<sup>ˆ</sup> E Therefore, a basic procedure is to start with *Ck* = {·} (empty) and randomly add in *j*1, *j*2, ... until we are close to the value of *pk*. We stop adding more *j s* when ∑*j*∈*Ck e* −*hjβ <sup>Z</sup>β*,*H*<sup>ˆ</sup> E exceeds the value of *pk*, and can choose to either keep or remove the last *j* depending on whether its inclusion or exclusion leads to a value closer to *pk*.

Because the maximum step-change is 1 *<sup>Z</sup>β*,*H*<sup>ˆ</sup> E , this means that the maximum difference isbounded:







$$\left| p\_k - \frac{\left(\sum\_{j \in \mathcal{C}\_k} e^{-h\_j \beta} \right)}{Z\_{\beta, \mathcal{H}\_{\mathcal{E}}}} \right| \le \frac{1}{2} \frac{1}{Z\_{\beta, \mathcal{H}\_{\mathcal{E}}}}. \tag{44}$$

We depict this in Figure 4.

*j*3 term as the sum is closer to *pk* without it.

$$\begin{array}{c|c} \text{(a)} & \text{(b)} \\\\ p\_{k} & \text{(c)} \\ \hline \\ \begin{array}{c} \text{(a)} \\ \text{(b)} \\ \text{(c)} \\ \hline \\ \text{(c)} \end{array} & \begin{array}{c} \text{(b)} \\ \begin{array}{c} \text{(c)} \\ \text{(c)} \\ \text{(c)} \\ \text{(c)} \\ \text{(d)} \\ \text{(e)} \\ \text{(e)} \\ \end{array} & \begin{array}{c} \text{(d)} \\ \text{(e)} \\ \text{(e)} \\ \text{(e)} \\ \text{(f)} \\ \text{(f)} \\ \end{array} & \begin{array}{c} \text{(d)} \\ \text{(e)} \\ \text{(e)} \\ \text{(f)} \\ \text{(f)} \\ \text{(e)} \\ \text{(f)} \\ \end{array} & \begin{array}{c} \text{(d)} \\ \text{(e)} \\ \text{(e)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \end{array} & \begin{array}{c} \text{(d)} \\ \text{(e)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \end{array} & \begin{array}{c} \text{(d)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{(f)} \\ \text{($$

**Figure 4.** Illustration of part of the proof for Theorem 1, following after Equation (43). The aim is to assign environment indices *j* to groups labelled by system indices *k*. As we add in more indices *j* into *Ck*, the sum ∑*j*∈*Ck e*<sup>−</sup>*hjβ <sup>Z</sup>β*,*H*<sup>ˆ</sup> E increases. In this example, in (**a**), we stop adding more indices after *j*3, as *j*3 leads to overshooting the value of *pk*. We either keep the last *j*3 if the sum with *j*3 is closer to *pk* or we do not include it if the sum is closer to *pk* without it. In (**b**), we have decided not to keep the last

Repeat this for all *pk*. In a random procedure, it may happen that some *Ck* have been overassigned, leading to many sums which are large <sup>∑</sup>*j*∈*Ck e*<sup>−</sup>*hjβ <sup>Z</sup>β*,*H*<sup>ˆ</sup> E > *pk*, thus leading to a shortage of indices *j* left for the remaining *pk*. Therefore, we may have to suboptimally remove earlier indices ˜ *j*, leading to a greater maximum difference:

$$\left| p\_k - \frac{\left(\sum\_{j \in \mathcal{C}\_k} e^{-h\_j \theta} \right)}{Z\_{\beta, \hat{\mathcal{H}}\_{\mathcal{E}}}} \right| \le \frac{1}{Z\_{\beta, \hat{\mathcal{H}}\_{\mathcal{E}}}}. \tag{45}$$

Therefore,

$$D\_{\text{obj\text{-}thm}}(\gamma\_{\mathcal{S}\prime}\gamma\_{\mathcal{E}}) \le \sum\_{k=1}^{d\_{\mathcal{S}}} \frac{1}{Z\_{\beta,\hat{\mathcal{H}}\_{\mathcal{E}}}} = \frac{d\_{\mathcal{S}}}{Z\_{\beta,\hat{\mathcal{H}}\_{\mathcal{E}}}}.\tag{46}$$

As the dimension of the environment, *dE*, increases, this leads to more environment Hamiltonian eigenvalues \$*hj*%. This in turn increases the value of the partition function *<sup>Z</sup>β*,*H*<sup>ˆ</sup> E = <sup>∑</sup>*dEj*=<sup>1</sup> *e*<sup>−</sup>*hjβ* → ∞ as *dE* → ∞. Thus, the distance between the set of thermal states and the set of objective states goes to zero: *<sup>D</sup>*obj-thm(*<sup>γ</sup>*S, *γ*E ) → 0 (provided the system dimension remains fixed).

#### *5.3. Low Temperature and High Temperature Limits*

Provided that the Hamiltonian of the *system* has a non-degenerate ground state, then in the low temperature limit, the thermal state of the system will be (approximately) pure. At *T* = 0, we will have the trivial objective and thermal state |*ψ<sup>S</sup>*groun<sup>d</sup>*ψ*<sup>S</sup>groun<sup>d</sup>| ⊗ *γ*E ground (trivially objective in the sense that there is only one index/single piece of information available).

In contrast, in the high temperature limit, the thermal states of the system and environment will approach maximally mixed states. If the dimension of the environment, *dE* is the same as the system *d* = *dS* = *dE*, then at the infinite temperature limit, the following state satisfies both local thermality and objectivity:

$$\rho\_{T \to \infty} = \frac{1}{d} \sum\_{i} |i\rangle\langle i|\_{\mathcal{S}} \otimes |\psi\_{i}\rangle\langle\psi\_{i}|\_{\mathcal{E}'} \tag{47}$$

along with any other local permutation of indices. This leads to *d*! different objectivethermal states.

If the dimension of the environment is a *multiple* of the system dimension, then it is also possible to have an exact locally-thermal and objective state: Suppose *dE* = *MdS* where *M* ∈ N is a positive integer. Note that the system thermal state at this infinite temperature is *γ*S = <sup>∑</sup>*dSi*=<sup>1</sup> 1*dS*|*ii*|. The environment thermal state can be written as

$$\gamma\_{\mathcal{E}} = \sum\_{i=1}^{d\_{\mathcal{E}}} \frac{1}{d\_{\mathcal{E}}} |\psi\_i\rangle\langle\psi\_i| = \sum\_{i=1}^{Md\_{\mathcal{S}}} \frac{1}{Md\_{\mathcal{S}}} |\psi\_i\rangle\langle\psi\_i| = \sum\_{i=1}^{d\_{\mathcal{S}}} \frac{1}{d\_{\mathcal{S}}} \rho\_{\mathcal{E}|i\prime} \tag{48}$$

$$\rho\_{\mathcal{E}|i} := \sum\_{k=M(i-1)+1}^{Mi} \frac{1}{M} |\psi\_k\rangle\langle\psi\_k|. \tag{49}$$

Therefore, the joint state

$$\rho\_{S\mathcal{E}} = \sum\_{i=1}^{d\_S} \frac{1}{d\_S} |i\rangle\langle i| \otimes \rho\_{\mathcal{E}|i} \tag{50}$$

is both objective *and* satisfies local thermality. We can also see that this state is not unique, i.e., permutations of |*ψk*- in each *ρ*E|*i* are possible, thus there is more than one state that is

both objective and satisfies local thermality. To be precise, there are (*Md*S )! (*M*!)*<sup>d</sup>*S such exactly

objective-thermal states.

However, in general, the environment dimension is not an exact multiple of the system dimension. Then, in the high temperature limit, there does not exist an *exact* objectivethermal state. We can apply Theorem 1 to bound the distance between the set of thermal states (at *T* → ∞) and the set of objective states:

$$D\_{\text{obj\text{-}thm}}(\gamma\_{S\text{-}}\gamma\_{\mathcal{E}})|\_{T\to\infty} \le \frac{d\_S}{Z\_{\beta\_{\mathcal{E}}\mathcal{H}\_{\mathcal{E}}}} = \frac{d\_S}{d\_E}.\tag{51}$$

That is, the higher the environment dimension relative to the system dimension, the more likely it is to have a state that is both closely thermal and closely objective.

#### **6. Objective States That Are Globally Thermal**

When the system–environment interaction is strong and/or non-commuting with the local Hamiltonians, the thermal state cannot be described by just the local Hamiltonian. Instead, the joint-system environment thermal state is given by the total Hamiltonian, *H*ˆtotal = *H*ˆ S + *H*ˆ E + *H*ˆint, where *H*ˆint is the interaction Hamiltonian:

$$\gamma\_{\mathcal{SE}} = \frac{e^{-\beta \hat{H}\_{\text{total}}}}{Z\_{\beta, \hat{H}\_{\text{total}}}}. \tag{52}$$

This type of scenario assumes that the system and environment continue to interact for all time, in all the relevant time frames. As there is only one such thermal state for finite systems, we do not have the extra degrees of freedom for forming objective states as we did in the previous two sections. As such, it is highly unlikely that this one global thermal state is also exactly objective. Furthermore, thermal states are full-rank, but exact objective states are not globally full-rank. So at best, there could only an approximately objective-thermal state.

The global thermal state *γ*SE will only be approximately objective if the relevant total Hamiltonian structure itself fits a very particular form such that its thermal state is also objective at the appropriate energy scale. The eigenstates of the total Hamiltonian become the eigenstates of the thermal state. Therefore, the Hamiltonians must have a particular *system-environment correlated* eigenstate structure. We give two examples:

**Example 1.** *Consider the Hamiltonian*

$$
\hat{H}\_{\text{total}} = \sum\_{i} E\_{i} |i\rangle\langle i| \otimes |\phi\_{i}\rangle\langle\phi\_{i}| + \hat{H}\_{\text{high-energy}y},\tag{53}
$$

*where H*ˆ *high-energy is an orthogonal addition with eigenenergies much higher than the energy scale given by the temperature T and with eigenstates such that <sup>H</sup>*<sup>ˆ</sup>*total is full-rank. This produces the following global thermal state that is also approximately objective:*

$$\gamma\_{S\mathcal{E}} = \frac{1}{Z\_{\beta\_\*\beta\_{\text{total}}}} \sum\_i e^{-\beta E\_i} |i\rangle\langle i| \otimes |\phi\_i\rangle\langle\phi\_i| + \delta\_{\text{high-energy}}.\tag{54}$$

*where <sup>δ</sup>high-energy is a perturbative term corresponding to high-energy states.* **Example 2.** *Consider Hamiltonians of the following form:*

$$
\hat{H}\_{\text{total}} = \sum\_{i} E\_{i} |i\rangle\langle i| \otimes \sum\_{a} q\_{a|i} |\phi\_{a}\rangle\langle\phi\_{a}| + \hat{H}\_{\text{high-energy}}.\tag{55}
$$

*where qa*|*iqa*|*j* = 0 ∀*i* = *j, and where H*ˆ *high-energy is an orthogonal addition with eigenenergies much higher than the energy scale given by the temperature T and with eigenstates such that <sup>H</sup>*<sup>ˆ</sup>*total is full-rank. These give rise to a Gibbs thermal state that is also approximately objective:*

$$\gamma\_{S\mathcal{E}} = \frac{1}{Z\_{\not\mathfrak{F}, \not\mathcal{H}\_{\text{total}}}} \sum\_{i,a} e^{-\not\mathfrak{F}E \cdot q\_{a[i]}} |i\rangle\langle i| \otimes |\phi\_{a}\rangle\langle\phi\_{a}| + \delta\_{\text{high\text{-}energy}}\tag{56}$$

$$\Phi\_i = \sum\_i p\_i |i\rangle\langle i| \otimes \sum\_a c\_{a|i} |\phi\_a\rangle\langle\phi\_{a'}| + \delta\_{\text{high-energy}}\tag{57}$$

$$p\_i := \frac{\sum\_b e^{-\beta E\_i q\_{|b|i}}}{Z\_{\beta, \mathcal{H}\_{\text{total}}}}, \quad \mathcal{c}\_{a|i} = \frac{e^{-\beta E\_i q\_{|a|i}}}{\left(\sum\_b e^{-\beta E\_i q\_{|b|i}}\right)'}, \mathcal{c}\_{a|i} \mathcal{c}\_{a|j} = 0 \; \forall i \neq j,\tag{58}$$

*where <sup>δ</sup>high-energy is a perturbative term corresponding to high-energy states.*

**Remark 3.** *Recall Remark 1 where, for any given state of full rank, a Hamiltonian and temperature can be found such that it can be considered thermal. As such, for any full rank approximately objective state, a Hamiltonian and temperature can be found such that it can be considered also thermal. However, the objective states form a set of measure zero (as discord-free states have zero measure [31]). Thus, the set of sub-component Hamiltonians directly corresponding to those objective states (up to a mutiplicative coefficient, and not including high-energy terms) is also zero measure.*

Since objective states are not globally full-rank, there are no objective states that are also exactly globally thermal, and most Hamiltonians will not produce an approximately objective state either. The Hamiltonians that do give rise to (approximately) objective thermal states such as those given in Equations (53) and (55) consist of strong, *constant*, interactions between the system and the environments, which is unrealistic.

### **7. Conclusions**

In our everyday experience, there are a number of phenomena which appear natural to us. One of them is *thermalisation*, in which physical objects eventually reach thermal equilibrium with the surrounding environment, e.g., an ice cream melting in hot weather. We also typically take for granted that physical objects are *objective*, i.e., their existence and properties can be agreed upon by many people. On the quantum mechanical level, thermalisation and objectivisation of quantum systems can arise through their interaction with external environments.

Thermalisation itself is thought to be a generic process and will occur approximately in general scenarios [20], more so than objectivity [50,51]. In contrast, objectivity requires classical correlations that are more sensitive to the situation, though components of objectivity can occur generically [50–53].

In general, the set of objective states does not have a preferred basis. Imposing (approximate) thermality can help select a preferred basis on the system and environment, which also leads to a preferred arrangemen<sup>t</sup> of classical correlations. If the system local Hamiltonian commutes with the interaction Hamiltonian (among the more straight forward scenarios in which quantum Darwinism has been explored [48,54–57]), then the preferred basis of objectivity would coincide with the "thermal" basis. The joint analysis of objectivity and thermalisation is further motivated by the fact that we observe everyday classical objects that are both objective and thermal.

In this paper, we examined the intersection of thermalisation and objectivity, especially when a single environment is required to fulfil both roles. In particular, we examined whether they can exist simultaneously by exploring whether a system-environment state can be both thermal (having the microcanonical Gibbs form) and objective (having state structure that satisfies spectrum broadcast structure).

By sequentially considering whether only the local system is thermal, or the local system and local environment, or the joint system-environment is thermal, we are able to characterise how rare it is for thermality and objectivity to coincide. This is summarised in Table 1. As we increased the thermalisation requirement from local system to global system and environment, the likelihood of an overlapping objectivity-thermal state existing decreases. This shows that in general, thermality and objectivity *are* at odds.

**Table 1.** Summary table. *H*ˆ S is the system Hamiltonian, *H*ˆ E is the environment Hamiltonian and *β* is the inverse temperature.


By studying the intersection of the sets of thermal and objective states, we can therefore also give a statement about the dynamics that have either objective states or thermal states as their fixed points or as their asymptotic state(s): due to the fine-tuned structure of thermal-objective states, only finely tuned dynamics would produce those states.

Quantum Darwinism can be hindered by numerous factors, such as non-Markovianity [32,58–64], non-ideal environments [33,34], initial system–environment correlations [32,35], environment–environment interactions [23,24,32,62,65], etc. It was shown that environment-environment interactions can lead to thermalisation at the detriment of objectivity in [23,24], but it is still open whether the other factors would lead to similar behaviour.

Based on these results, we conclude that if the hypothetical entropic death of the universe is characterised by the global thermalisation of the entire (observable) universe, then it is extremely unlikely for objectivity to remain. This is consistent with our intuition that, at thermalisation (heat death), there should be no work left to be done. In contrast, objectivity implies information about one system in another, which usually contains extractable work [22].

That said, there are (very) rare situations where a global thermal state can still support objective correlations, at least theoretically. *If* objectivity and information does remain, then this implies that there are highly nonlocal, strong interactions, as such giving rise to Hamiltonians like in Equation (55), which are required to maintain correlations in the global thermal state. While this is unrealistic that the entire universe can have such strong interactions, it may be possible for smaller parts of the universe to maintain interactions and thus have subcomponents that are objective.

Another possibility is that the system alone thermalises on the short time scale, while on more intermediate timescales the system and (information-carrying) environment locally thermalises. Meanwhile, perhaps only at long time scales does the global systemenvironment thermalise, achieving an ultimate "heat death". We found that objectivity is more likely to be able to coexist with thermality in the first two situations. This suggests that objectivity can survive in the short and intermediate timescales, before fading away at the long timescale.

The following narrative feels intuitive: e.g., decoherence occurs first as a loss of phase information, followed by the classical information spread that characterises objectivity; the classical information fades, followed by thermalisation in which all information is lost (aside from select information such as temperature) [23]. Whether this is 'common' remains an open question.

**Author Contributions:** Conceptualisation, T.P.L.; methodology, T.P.L.; formal analysis, T.P.L., A.W. and G.A.; writing—original draft preparation, T.P.L.; writing—review and editing, T.P.L., A.W. and G.A.; visualisation, T.P.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** TPL acknowledges financial support from the UKRI Engineering and Physical Sciences Research Council (EPSRC) under the Doctoral Prize Award (Grant No. EP/T517902/1) hosted by the University of Nottingham. GA acknowledges financial support from the Foundational Questions Institute (FQXi) under the Intelligence in the Physical World Programme (Grant No. RFP-IPW-1907). AW was supported by the Spanish MINECO (projects FIS2016-86681-P and PID2019-107609GB-I00/AEI/10.13039/501100011033), both with the support of FEDER funds, and by the Generalitat de Catalunya (project 2017-SGR-1127).

**Conflicts of Interest:** The authors declare no conflict of interest.
