**1. Introduction**

While fundamental quantum mechanics describes how isolated quantum systems evolve under unitary evolution, realistic quantum systems are open, as they interact with external environments that are typically too large to exactly model. In order to account for large external environments without directly simulating them, the theory of open quantum systems has developed tools that allow us to study a variety of quantum processes [1,2], including decoherence [3] (the loss of phase information to the environment) and dissipation (the loss of energy to the environment) [4].

The environment, when acting as a heat bath, can lead to the equilibration and thermalisation of quantum systems [5–9]. Meanwhile, in an approach to the quantumto-classical transition called *Quantum Darwinism* [10–14], the environment plays a key role in the process of how quantum systems appear classically objective [13,14]—whereby classical objective systems have properties that are equivalently independently verifiable by independent observers. In the realm of open quantum systems, whether one process or another occurs depends on multiple factors, including details of the system–environment interactions, initial states, time regimes, averaging, etc.

The (classical) second law of thermodynamics generally states that entropy increases over time. Following this strictly, we may imagine that in the far distant future, the entire universe will reach an equilibrium where entropy can no longer increase: this concept is known as "heat death", which can be found in early writings of Bailly, Kelvin, Clausius and von Helmholtz (see references in [15]). An alternative, recent, version of heat death would see a universe composed mostly of vacuum and very far separated particles such that no work is done: this is "cosmological heat death" [16]. There are some caveats to the concept of heat death of the universe: beyond whether or not thermodynamics can be applied at the universal level, it is known that after a sufficiently long time, Poincaré recurrences will return the system/universe to its prior states [17]. Furthermore, the discovery of dark energy and the accelerating rate of expansion of the universe [18] leads to other theories of the universe's ultimate fate such as the "big rip" [19].

These caveats aside, on more familiar temporal and spatial human scales, both classical and quantum objects can thermalise. In fact, thermalisation is quite fundamental: in fairly generic conditions, a local subsystem (of a greater state) will likely be close to thermal [20]. We also see that many everyday physical objects have the same approximate temperature as their environment. This thermality appears to contradict with *objectivity*. In Quantum Darwinism, a system state is considered objective if multiple copies of its information exist, which is mathematically expressed as (classical) correlations between the system and its environment [11,12]. The quintessential example is of the visual information carried in the photon environment. However, information and correlations have an associated *energy* [21,22], and naively, this information should not survive under the process of thermalisation. For example, in the model analysed by Riedel et al. [23], some level of objectivity emerges at finite time, before equilibration sets in; in the model analysed by Mirkin and Wisniacki [24], tuning certain parameters produces either objectivity or thermalisation, but not both.

Furthermore, there is a distance-scale difference. Quantum Darwinism requires strong (classical) correlations between two or indeed many more systems, some of which will invariably be very distant from each other—for example, we can view galaxies billions of light years away. In contrast, thermalisation favours realistic settings that have no or rapidly decaying correlations between distant subsystems of the universe.

In this paper, we investigate this apparent conflict between thermalisation and objectivity and consider whether or not these two can co-exist. To do this, we analyse the overlap between the set of states that are thermal versus the set of states that are objective—if there is no intersection, then there cannot exist any process that produces jointly thermal-objective states. We examine three different sets of thermal states where either: (1) there is system thermalisation, (2) local system and local environment thermalisation, or (3) global system– environment thermalisation. As greater parts of the system-environment become thermal, the overlap between objectivity and thermalisation reduces, often becoming non-existent for many system–environment Hamiltonians. We also find that large environments have better potential to support both thermality and objectivity simultaneously.

This paper is organised as follows. In Section 2, we introduce the mathematical structure of objective states, and in Section 3, we introduce thermalised microcanonical states (for finite systems). Then, in Section 4 we consider the intersection between objective states versus states with a thermal system. In Section 5, we consider states with a locally thermal system and a locally thermal environment. In Section 6, we consider a globally thermal system-environment state. We discuss and conclude in Section 7.

#### **2. Objective States**

In our day-to-day experience, we typically perceive the classical world as being "objective": objects appear to exist regardless of whether we personally look at them, and the properties of these objects can be agreed upon by multiple observers. More formally, we can describe objective states as satisfying the following:

**Definition 1.** *Objectivity [10,11,25]: A system state is* objective *if it is (1) simultaneously accessible to many observers (2) who can all determine the state independently without perturbing it and (3) all arrive at the same result.*

The process of emergen<sup>t</sup> objectivity may be described by Quantum Darwinism [10,26]: as a system interacts and decoheres due to the surrounding environment, information about the system can spread into the environment. The "fittest" information that can be copied tends to record itself in the environment at the expense of other information, thus the name Quantum Darwinism. The paradigmatic example is the photonic environment: multiple photons interact with a physical object and gain information about its physical

features, such as position, colour, size, etc. Multiple independent observers can then sample a small part of this photonic environment to find very similar information about the same system state, thus deeming it objective. We depict this in Figure 1a.

**Figure 1.** (**a**) Objectivity scenario, where a system interacts with multiple sub-environments, such that those sub-environments contain information about the system. (**b**) Thermalisation scenario, where a system interacts with a large heat bath environment and subsequently thermalises to the environment temperature.

There are a number of frameworks to mathematically describe objective states: in order of increasing restriction one has (Zurek's) Quantum Darwinism [10], Strong Quantum Darwinism [12] and Spectrum Broadcast Structure [11] (and invariant spectrum broadcast structure [27]). In this work, we will be focusing primarily on a bipartite systemenvironment, in which case Strong Quantum Darwinism and Spectrum Broadcast Structure coincide. In particular, Spectrum Broadcast Structure gives us a clear geometric state structure which is ideal for state analysis.

Objective states with spectrum broadcast structure can all be written in the following form [11]:

$$\rho\_{S\mathcal{E}} = \sum\_{i} p\_i |i\rangle\langle i| \otimes \bigotimes\_{k=1}^{N} \rho\_{\mathcal{E}\_k|i\prime} \quad \rho\_{\mathcal{E}\_k|i\prime} \rho\_{\mathcal{E}\_k|j\prime} = 0 \,\,\forall i \neq j,\tag{1}$$

where E is the accessible environment and E*k* are the sub-environments. The conditional states "*ρ*E*k* |*i*# can be used to perfectly distinguish index *i*, where {|*i*-} is some diagonal basis of the system and {*pi*} its spectrum. In general, there is no basis dependence in both the system and the environments, and so the overall set of all objective states is non-convex.

#### **3. Thermal States**

Systems can exchange energy and heat through interactions with an external environment that functions as a heat bath. Over time, systems can reach thermal equilibrium. Canonically, the thermal state of a quantum system is the Gibbs state [8]. For a given energy/Hamiltonian expectation value, the thermal Gibbs state maximises the von Neumann entropy [28].

The Gibbs state, which we denote as *γ*, is defined with reference to its Hamiltonian *H* ˆ and inverse temperature *β* = 1/*kBT*:

> *γ*

$$\epsilon = \frac{e^{-\beta \hat{H}}}{Z\_{\beta, \hat{H}}},\tag{2}$$

where *<sup>Z</sup>β*,*H*<sup>ˆ</sup> = tr*e*<sup>−</sup>*βH*<sup>ˆ</sup> is the partition function.

If the Hamiltonian has the spectral decomposition *H* ˆ = ∑*i Ei*|*ii*|, then we can write the canonical thermal state as

$$\gamma = \frac{\sum\_{i} e^{-\beta E\_{i}} |i\rangle\langle i|}{\left(\sum\_{j} e^{-\beta E\_{j}}\right)}.\tag{3}$$

**Remark 1.** *For any state ρ of full rank, there exists a Hamiltonian H*ˆ *ρ and an inverse temperature βρ such that ρ can be considered a thermal state, i.e., we can write ρ* = 1*Z* exp[−*βH*<sup>ˆ</sup> *ρ*]*. To see this, suppose the state ρ has the spectral decomposition ρ* = ∑*i pi*|*ψiψi*| *(pi* > 0*). Then, consider a Hamiltonian with the same eigenvectors, H*ˆ *ρ* = ∑*i Ei*|*ψiψi*|*, with unknown eigenenergies* {*Ei*}*. We want to find* \$*βρ*, *Ei*% *such that*

$$\frac{e^{-\beta\_{\vec{\rho}}E\_{i}}}{\left(\sum\_{j} e^{-\beta\_{\vec{\rho}}E\_{j}}\right)} \stackrel{!}{=} p\_{i\prime} \qquad \forall \, i. \tag{4}$$

*As there is an extra variable in the set* \$*βρ*, *Ei*%*compared to the number of conditions,* |{*pi*}|*, this forms an underdetermined set of equations and there can be infinitely many solutions formed by scaling βρ and Ei inversely.*

Objective states are not globally full-rank, but we could add in a very small (nonobjective) perturbation to make it full-rank. Then, from this perspective, for any full-rank approximately objective state, we can post-select system-environment Hamiltonians and a temperature at which that objective state is also thermal. In a controlled scenario (e.g., with control of the system Hamiltonian and reservoir engineering [29], or in quantum simulators [30]), it is possible to engineer an approximately objective-and-thermal state by choosing system-environment Hamiltonians based on the objective state itself.

In the rest of this paper, we will be considering the reverse scenario, i.e., *given* some system and environment Hamiltonians and inverse temperature *β*, can the subsequent thermal state(s) also support objectivity? By answering this question, we will better understand whether or not objectivity and thermalisation can coexist, and what conditions would allow any coexistence.

In order to answer whether or not there is any overlap between thermalisation and objectivity, we consider the precise state structure. If there is no state overlap, then both properties cannot exist simultaneously, in which case there cannot be any dynamics that produces a non-existent state. More generally, if the two set of states are sufficiently close, then perhaps a compromise is possible.

We will be examining three different types of thermal states:


Examining thermal *states* rather than some time-averaged or instantaneous values of observables means that we are considering thermalisation in a strong sense (or that we have assumed that averaging has already been done). The results are also therefore suitable for more static applications of thermal states, e.g., resource theories.

In order to find the overlap between objective and thermal states, our main method is to start with objective states and successively restrict them to satisfy thermality. As thermal states are full-rank, we will be restricting to objective states where the reduced system and environment states are also full-rank.

Note that if the local system state thermalises, e.g., relative to its energy eigenbasis, then it can also be said to have decohered (relative to that energy eigenbasis). However, whether or not objectivity—an extension of decoherence—arises depends on whether the system thermal information can be encoded in the environment.

#### **4. Objective States with Thermal System**

In this section, we describe the system–environment states that are both objective and have a locally thermal system (and no requirements on the environment thermality or lack thereof).

Consider the situation where a system with self Hamiltonian *H* ˆ S is put in thermal contact with a bath with some temperature *TB*, is left to thermalise, and then de-coupled from the bath. Writing the system Hamiltonian's spectral decomposition as *H* ˆ S = ∑*i Ei*|*ii*| and with fixed inverse temperature *β*, the system thermal state is then

$$\gamma\_S = \frac{\varepsilon^{-\beta H\_S}}{Z\_{\beta, \dot{\Psi}\_S}} = \frac{1}{Z\_{\beta, \dot{\Psi}\_S}} \sum\_i \varepsilon^{-E\_i \emptyset} |i\rangle\langle i|. \tag{5}$$

This implies that objective system-environment states with locally thermal system states must have the following form:

$$\frac{1}{Z\_{\beta,\hat{H}\_S}} \sum\_i \varepsilon^{-E\_i\beta} |i\rangle\langle i| \otimes \rho\_{\mathcal{E}|i\prime} \quad \rho\_{\mathcal{E}|i\prime}\rho\_{\mathcal{E}|j} = 0 \,\forall i \neq j,\tag{6}$$

where the conditional *ρ*E|*i* are perfectly distinguishable.

As we can immediately see, these objective states describe fixed thermal-state information about the system, encoded in the probabilities & *e*<sup>−</sup>*Eiβ <sup>Z</sup>β*,*H*<sup>ˆ</sup> S '*i*. Furthermore, as there are no thermal conditions imposed on the information-carrying environment, the size of set of states satisfying Equation (6) is non-empty, as we have freedom to choose any set of mutually distinguishable environment states "*<sup>ρ</sup>*E|*i*#*i*. Therefore, objectivity and thermalisation overlap: both can occur at the same time.

The set of exact objective states with thermal system in Equation (6) is nowhere dense, as it is a subset of zero-discord states [31]. The set of states in Equation (6) is also nonconvex in general, though convex subsets can be formed by restricting the conditional subspaces on the environment.

Approximate cases would correspond to imperfect information spreading into the environment and/or imperfect system thermalisation before the information spreading stage. As we have a fairly well-defined set of states (Equation (6)), any distance measure to that set can be used to describe approximately objective-with-thermal-system states, e.g.,

$$\left[\left[\mathsf{T}\_{\mathsf{S}}\mathsf{O}\right]\_{\delta} = \left\{\rho\left|\min\_{\rho\_{obj,th}\in\mathsf{T}\_{\mathsf{S}}\mathsf{O}}\left\|\rho-\rho\_{obj,th}\right\|\_{1} \leq \delta\right\}\right.\tag{7}$$

where TSO (thermal-system objective) denotes the set of states satisfying Equation (6), and \$·\$1 is the trace norm. The convex hull of objective-with-thermal-system states are simply zero-discord states with a local thermal system:

$$\left\{ \frac{1}{Z\_{\beta,\hat{H}\_{\mathcal{S}}}} \sum\_{i} e^{-E\_{i}\beta} |i\rangle\langle i| \otimes \rho\_{\mathcal{E}|i} \Big| \rho\_{\mathcal{E}|i} \in \mathcal{H}\_{\mathcal{E}} \right\},\tag{8}$$

i.e., there are no longer any restrictions on the conditional environment states *ρ*E|*<sup>i</sup>*.

#### *Creating Objective States with Thermal Systems*

A two-step process that produces objective-with-thermal-system states is first system thermalisation followed by information broadcasting. Physically, this can occur if the system was first thermalised using one bath, and then we had a fresh environment interact with the system with intent to gain information. As environments in low-entropy state |0- are typically better for quantum Darwinism [32–36], this second 'information-storing' environment could be a very cold bath with states close to the ground state.

The point channel can produce perfectly thermalised states:

$$\Phi\_{\mathcal{S},th}(\cdot) = \text{tr}[\cdot]\gamma\_{\mathcal{S}}.\tag{9}$$

One simple method to broadcast information from system to environment is to start with the information-carrying environment in state |0- (e.g., zero temperature bath). Then, controlled-NOT (CNOT) operations with control system to each individual environment will perfectly broadcast the system information [14,35]:

$$\Phi\_{\text{CNOT}}^{\mathcal{E}\_k}(\rho\_{\mathcal{SE}\_k}) = \mathcal{U}\_{\text{CNOT}}^{\mathcal{SE}\_k} \rho\_{\text{S}\mathcal{E}\_k} \mathcal{U}\_{\text{CNOT}'}^{\text{S}\mathcal{E}\_k\dagger} \tag{10}$$

where *U*SE*<sup>k</sup>* CNOT is the CNOT gate between system S and environment E*k*.

In general, quantum channels that can create the exact objective-with-thermal-system states from Equation (6) are point channels which thermalise the system combined with information broadcasting channels:

$$\Phi\_{\mathsf{T}\mathfrak{S}\mathfrak{O}}(\rho\_{\mathcal{S}\mathcal{E}}) = \frac{1}{Z\_{\beta\_{\ast}\mathfrak{R}\_{\mathcal{S}}}} \sum\_{i} e^{-E\_{i}\beta} |i\rangle\langle i| \otimes \Phi\_{\mathcal{E}|i}(\rho\_{\mathcal{S}\mathcal{E}}) ,\tag{11}$$

where "<sup>Φ</sup>E|*i* : HS ⊗ Hˆ E → Hˆ E #*i* are channels on the environment such that the output states for different *i* are orthogonal.

This process can be performed on a quantum simulator by dividing the available qubits into 'system', 'thermal environment' and 'information-carrying environment', and enacting the suitable gate operations [37].

We can also consider partial thermalisation channels <sup>Λ</sup>*p*−*th*, such that *repeated application* brings the system closer and closer to thermalisation, i.e.,

$$
\Lambda\_{p-th} \diamond \cdots \diamond \Lambda\_{p-th}(\cdot) \to \gamma\_S. \tag{12}
$$

If the system is a qubit, then we can, without loss of generality, consider the system qubit Hamiltonian to be *H* = *<sup>σ</sup>z*/2. One channel which, through repeated application, will lead to the system thermalising is the generalised amplitude damping channel [38]

$$\rho(t) = \Phi\_t^T(\rho\_0) = \sum\_{i=1}^4 E\_i \rho\_0 E\_i^\* \,. \tag{13}$$

with Kraus operators

*E*1 = √*p* 1 0 0 √*η* , *E*2 = √*p* 0 )1 − *η* 0 0 , (14)

$$E\_3 = \sqrt{1-p} \begin{bmatrix} \sqrt{\eta} & 0\\ 0 & 1 \end{bmatrix}, \qquad \qquad E\_4 = \sqrt{1-p} \begin{bmatrix} 0 & 0\\ \sqrt{1-\eta} & 0 \end{bmatrix} \tag{15}$$

where *p* ∈ [0, 1] depends on the temperature of the environment, and *ηt* = 1 − *e*<sup>−</sup>(<sup>1</sup>+2*N*¯ )*t*, where *N* ¯ = 1 *e*1/*<sup>T</sup>* − 1 is the boson occupation number. The equivalent Bloch sphere representation is [38,39]

$$
\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \sqrt{\eta} & & \\ & \sqrt{\eta} & \\ & & \sqrt{\eta} \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ (2p-1)(1-\eta) \end{bmatrix} \tag{16}
$$

with stationary state

$$
\sigma\_{\infty} = \begin{bmatrix} p & 0 \\ 0 & 1-p \end{bmatrix}'\tag{17}
$$

where *x* = tr[*<sup>σ</sup>xρ*0], *y* = tr[*<sup>σ</sup>yρ*0] and *z* = tr[*<sup>σ</sup>zρ*0].

More generally, the following channel, in the Bloch sphere representation, will partially thermalise the system:

$$
\Lambda\_{p-th}(\vec{r}) = A\vec{r} + (1-A)\vec{t}\_{\mathcal{S}\prime} \tag{18}
$$

where*tS* is the Bloch vector of the system thermal state *γ*S, \$*A*\$ < 1 (under matrix norm) and ((*Ar* + (1 − *<sup>A</sup>*)*tS*((2 ≤ 1 for all \$*r*\$2 ≤ 1 (under Euclidean norm). Under repeated application, the state will converge towards the Bloch vector*tS*, i.e., to the thermal state.

Aside from the specific model-dependent methods to produce objective-thermal states, it is possible to produce a quantum circuit that will prepare that state [40,41]. Alternatively, one could also construct a Lindblad generator L (with an unobserved environment) that simulates a chosen quantum channel (in the infinite time limit) [42]. In general, the specific timescales will depend on the situation and also the size of the "unobserved" environment in comparison with the system and observed environment [43–46].

#### **5. Objective States with Thermal System and Thermal Environment**

Thermal environments play a large role in thermodynamics and open quantum systems. In this section, we suppose that both the system and the environment are locally thermal.

As in the previous section, we take the system local Hamiltonian to have some general spectral decomposition *H* ˆ S = ∑*i Ei*|*ii*|. Suppose that the environment's self-Hamiltonian has this spectral decomposition: *H* ˆ E = ∑*k hk*|*ψkψk*|. This leads to the environment thermal state ˆ

$$\gamma\_{\mathcal{E}} = \frac{e^{-\beta H\_{\mathcal{E}}}}{Z\_{\beta,\hat{\mathcal{H}}\_{\mathcal{E}}}}.\tag{19}$$

States that are locally thermal in the system and the environment can be written generally as

$$
\rho\_{\text{S}\mathcal{E}} = \gamma\_{\text{S}} \otimes \gamma\_{\text{E}} + \chi\_{\text{S}\mathcal{E}} \,\prime \tag{20}
$$

where *χ*SE is a correlation matrix where trS *χ*SE = 0 and trE *χ*SE = 0 [47]. Our aim is to determine whether this correlation matrix can hold objective correlations.

If the system and environment have *pure* thermal states, then the combined system– environment thermal state |*<sup>γ</sup>*S *<sup>γ</sup>*S |⊗|*<sup>γ</sup>*E *<sup>γ</sup>*E | is also trivially objective, because there is only one index on the system that the environment needs to distinguish. This can happen if the system and environment only have one energy level, or if the temperature is zero (or very low) and the system and environment both have non-degenerate ground states.

In general though, the system will not have a pure thermal state. With the added restriction of thermal environments, exact co-existence of states that are simultaneously objective and thermal becomes difficult to achieve: the thermality of the environment comes in conflict with the strong condition of classical correlations required by objectivity.

#### *5.1. Equal System and Environment Dimension*

In the scenario where the system and the individual environments have the same dimension, an exact thermal and bipartite-objective state can only exist for highly finetuned system and environment Hamiltonians, i.e., the energy spacing of both must be the same.

**Remark 2.** *If the system and individual environments have the same dimension, there exists a joint state that is both locally-thermal and objective only if they have the same thermal eigenenergies, i.e., the system Hamiltonian eigen-energies* {*Ei*} *differ from the environment Hamiltonian eigen-energies* {*hi*} *by a constant shift, Ei* = *hi* + *c* ∀*i.*

**Proof of Remark 2.** To see this, consider the objective state structure in Equation (1) and enforce the requirement of local thermality. As the environment has the same dimension, the conditional environment states of the objective state must be pure, and orthogonal for *i* = *j*, i.e., have form *ρ*E*k*|*i* = |*φi*|*kφi*|*k*|. This leads to the following state which is objective:

$$\rho\_{S\mathcal{E}} = \sum\_{i} p\_i |i\rangle\langle i| \bigotimes\_{k} |\phi\_{i|k}\rangle\langle\phi\_{i|k}|,\\ \langle\phi\_{i|k}|\phi\_{j|k}\rangle = 0 \,\forall i \neq j, \,\forall k,\tag{21}$$

where "|*φi*|*k*- #are the eigenvectors of the individual environments. This objective structure corresponds to invariant spectrum broadcast structure [27], as the environment states are also objective.

Local thermality of the system and environments means that

$$\rho\_S = \sum\_i p\_i |i\rangle\langle i| \stackrel{!}{=} \gamma\_S \text{ and } \rho\_{\mathcal{E}\_k} = \sum\_i p\_i |\phi\_{i|k}\rangle\langle\phi\_{i|k}| \stackrel{!}{=} \gamma\_{\mathcal{E}\_k}.\tag{22}$$

In order for this to be true, the eigenvalues of both the system thermal state *γ*S *and* the environment thermal states *γ*E*k*must be identical and equalling {*pi*}, i.e.,

$$\frac{e^{-\beta E\_i}}{Z\_{\beta,\hat{\mathcal{H}}\_S}} = \frac{e^{-\beta h\_i}}{Z\_{\beta,\hat{\mathcal{H}}\_{\mathcal{E}\_k}}} \qquad \forall i,\tag{23}$$

with appropriate labelling of "*i*" on the system and the environment.

As the inverse temperature is fixed at some *β*, this means that the Hamiltonian eigenenergies of the system and environment must also be the same, {*Ei*} and {*hi*}, respectively, up to a constant shift. That is, the environment eigenenergies are *hi* = *Ei* + *c*, thus

$$\frac{\mathfrak{e}^{-\left(\tilde{E}\_{i}+\varepsilon\right)\mathfrak{f}}}{\sum\_{j}\mathfrak{e}^{-\left(\tilde{E}\_{j}+\varepsilon\right)\mathfrak{f}}}=\frac{\mathfrak{e}^{-\varepsilon\mathfrak{f}}\mathfrak{e}^{-\left(\tilde{E}\_{i}\mathfrak{f}\right)}}{\sum\_{j}\mathfrak{e}^{-\varepsilon\mathfrak{f}}\mathfrak{e}^{-\left(\tilde{E}\_{j}\mathfrak{f}\right)}}=\frac{\mathfrak{e}^{-\left(\tilde{E}\_{i}\mathfrak{f}\right)}}{Z\_{\mathfrak{f}\_{i}\mathfrak{f}\_{i}\mathfrak{f}}},\tag{24}$$

as required.

Realistically, the scenario of system and environments having identical dimension and equal eigenenergies can occur if both are made out of the same *material*, e.g., they are all photons, all spins, etc. with the same internal and external Hamiltonians up to a constant energy shift.

This shows that randomly independently chosen individual Hamiltonians for the system and the environment, will, in general, *not* support an exact thermal and objective system–environment state. Once a particular system Hamiltonian is chosen, say *H*ˆ S = ∑*i Ei*|*ii*|, an exact thermal-objective system-environment state (with identical system and sub-environment dimensions) can only exist if the environment Hamiltonians have form *H*ˆ E*k* = ∑*i*(*Ei* + *ck*)*Uk*|*ii*|*U*† *k* , with freedom in real value energy *ck* and unitary rotation *Uk* that produces various sets of orthogonal eigenvectors, in order to give rise to the exact thermal-objective state:

$$\rho\_{S\mathcal{E}}^{\text{obj,th}} = \frac{1}{Z\_{\beta,\hat{\mathcal{O}}\_S}} \sum\_i e^{-E\_i \beta} |i\rangle\langle i| \otimes \bigotimes\_{k=1}^N |\phi\_{i|k}\rangle\langle\phi\_{i|k}|,\tag{25}$$

where |*φi*|*k*- = *Uk*|*i*-E*k*.
