**5. Conclusions**

We have examined the emergence of quantum Darwinism for a composite system consisting of two qubits interacting with a *N*-partite bath. For an excitation-preserving interaction between the system qubits, we established that the system information is faithfully, redundantly encoded throughout the environment; therefore, we see the emergence of clear Darwinistic signatures. Nevertheless, a decoherence-free subspace permits the system to create and maintain significant non-classical features in the form of quantum discord. Employing the framework of strong quantum Darwinism, which insists that in addition to a mutual information plateau, the discord between the system and an environment fragment must vanish, we have shown that whether or not this state is interpreted as objective and classical depends on how the discord is evaluated. Following the framework of Ref. [6], for measurements on the system, the sizable non-zero coherence present in the decoherence-free subspace implies that this state is definitively not objective. However, as quantum Darwinism posits that classicality and objectivity are dictated by what information can be learned by measuring the environment, and due to the asymmetric nature of the quantum discord when measurements are made on the environment, we find that the discord is vanishing and therefore conclude a classically objective state. To better understand this point, we demonstrated that redundant encoding at the level of the composite system does not imply the same for the individual constituents. Specifically, when non-classical correlations are established between the system qubits, there is still a redundant proliferation of *some* of the system information into the environment; however, the correlations between the two system qubits prevent all of the system's information from being redundantly shared with the environment.

**Author Contributions:** All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** S.C. gratefully acknowledges the Science Foundation Ireland Starting Investigator Research Grant "SpeedDemon" (No. 18/SIRG/5508) for financial support. B.Ç. is supported by the BAGEP Award of the Science Academy and by The Research Fund of Bahçe¸sehir University (BAUBAP) under project No. BAP.2019.02.03. B.V. and acknowledges the UniMi Transition Grant H2020. M.P. is supported by the H2020-FETOPEN-2018-2020 project TEQ (grant No. 766900), the DfE-SFI Investigator Programme (grant 15/IA/2864), the Royal Society Wolfson Research Fellowship (RSWF\R3\183013), the Leverhulme Trust Research Project Grant (grant No. RGP-2018-266), and the U.K. EPSRC (grant No. EP/T028106/1).

**Institutional Review Board Statement:** Not Applicable.

**Informed Consent Statement:** Not Applicable.

**Data Availability Statement:** Not Applicable.

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

#### **Appendix A. Derivations of Equations (6) and (7)**

*Appendix A.1. Direct Approach*

Using the definition of mutual information, one has <sup>I</sup>(*ρS*1*S*2:E*f*) = *<sup>S</sup>*(*ρS*1*S*2 ) + *<sup>S</sup>*(*ρ*E*f*) − *<sup>S</sup>*(*ρS*1*S*2E*f*) and <sup>I</sup>(*ρS*1:E*f*) = *<sup>S</sup>*(*ρS*1 ) + *<sup>S</sup>*(*ρ*E*f*) − *<sup>S</sup>*(*ρS*1E*f*), from which ΔI = <sup>I</sup>(*ρS*1*S*2:E*f*) − <sup>I</sup>(*ρS*1:E*f*) can be calculated as the following:

$$\begin{split} \Delta \mathcal{L} &= \left[ S(\rho\_{S\_1 \mathcal{E}\_f}) - S(\rho\_{S\_1}) \right] - \left[ S(\rho\_{S\_1 S\_2 \mathcal{E}\_f}) - S(\rho\_{S\_1 S\_2}) \right] \\ &= \left[ S(\rho\_{S\_2 \mathcal{E}\_f}) - S(\rho\_{\mathcal{E}\_f}) \right] - \left[ S(\rho\_{S\_2 \mathcal{E}}) - S(\rho\_{\mathcal{E}}) \right]. \end{split} \tag{A1}$$

In passing to the second line, we assume that the total *S*1*S*2E system starts from a pure state and E *f* denotes the part of the environment that is not included in the fraction E*f* , i.e., E *f* is the complement of E*f* . We continue by adding and subtracting *<sup>S</sup>*(*ρS*2 ) to the right hand side of the above equation and rearranging to obtain the following:

$$\begin{split} \Delta \mathcal{L} &= \left[ S(\rho\_{\mathcal{S}\_{2}}) + S(\rho\_{\mathcal{E}}) - S(\rho\_{\mathcal{S}\_{2}\mathcal{E}}) \right] - \left[ S(\rho\_{\mathcal{S}\_{2}}) + S(\rho\_{\mathcal{E}\_{f}}) - S(\rho\_{\mathcal{S}\_{2}\mathcal{E}\_{f}}) \right] \\ \Delta \mathcal{L} &= \mathcal{Z}(\rho\_{\mathcal{S}\_{2}\mathcal{E}}) - \mathcal{Z}(\rho\_{\mathcal{S}\_{2}\mathcal{E}\_{f}}). \end{split} \tag{A2}$$

We note that this result is completely independent of the nature of the dynamics and valid for arbitrary fractions at any given instant. Considering the fraction to be the whole environment, i.e., E *f* is an empty set, the second term in the above equation vanishes and reduces to the following simple form:

$$
\Delta T = \mathcal{T}(\rho\_{\mathbb{S}\_2; \mathcal{E}}),
\tag{A3}
$$

which corresponds to the gap at the end of the curves.

#### *Appendix A.2. Koashi–Winter Relation*

Given an arbitrary tripartite quantum system *ρABC*, the Koashi–Winter (KW)relation [36] states the following inequality:

$$E\_f(\rho\_{AB}) \le \mathcal{S}(\rho\_A) - f^{\leftarrow}(\rho\_{AC}),\tag{A4}$$

with equality attained if *ρABC* is pure and *Ef*(·) denotes the entanglement of formation. Here, we will try to exploit this inequality to present bounds on the mutual information shared between the different fractions of the system and environment.

In addition to our usual assumption of a pure *S*1*S*2E state, we also introduce a pure auxiliary unit A that serves only as a mathematical tool to ensure that our set up fits within the framework of the KW relation. Identifying *A* → *S*1*S*2, *B* → A and *C* → E*f* , and noting that by definition *Ef*(*ρS*1*S*2<sup>A</sup>) = 0, we have the following:

$$J^{\leftarrow}(\rho\_{S\_1S\_2;\mathcal{E}\_f}) \le S(\rho\_{S\_1S\_2}) \tag{A5}$$

$$\mathcal{Z}(\rho\_{S\_1S\_2;\mathcal{E}\_f}) \le S(\rho\_{S\_1S\_2}) + D^{\leftarrow}(\rho\_{S\_1S\_2;\mathcal{E}\_f})$$

which provides an upper bound on the information that we can obtain on the total system by probing a fraction of the environment.

We now use the KW relation to bound the mutual information between a single system qubit and fractions of the environment. To that end, we shift our labeling to *A* → *S*1, *B* → *S*2 and *C* → E*f* again with the assumption that *S*1, *S*2 and the whole environment E is in a pure state. The KW relation gives the following:

$$E\_f(\rho\_{S\_1S\_2}) \le S(\rho\_{S\_1}) - f^{\leftarrow}(\rho\_{S\_1\mathcal{E}\_f}).\tag{A6}$$

Adding <sup>I</sup>(*ρS*1:E*f*) to both sides we ge<sup>t</sup>

$$\mathcal{E}\_f(\rho\_{\mathcal{S}\_1\mathcal{S}\_2}) + \mathcal{Z}(\rho\_{\mathcal{S}\_1\mathcal{E}\_f}) \le \mathcal{S}(\rho\_{\mathcal{S}\_1}) + D^{\leftarrow}(\rho\_{\mathcal{S}\_1\mathcal{E}\_f}) \tag{A7}$$

$$\mathcal{Z}(\rho\_{\mathcal{S}\_1\mathcal{E}\_f}) \le \mathcal{S}(\rho\_{\mathcal{S}\_1}) + D^{\leftarrow}(\rho\_{\mathcal{S}\_1\mathcal{E}\_f}) - \mathcal{E}\_f(\rho\_{\mathcal{S}\_1\mathcal{S}\_2}).$$

Comparing Equations (A5) and (A7) can help us to understand the discrepancy between the mutual information curves presented in Figure 2d. Naturally, we have both <sup>I</sup>(*ρS*1*S*2:E*f*) and <sup>I</sup>(*ρS*1:E*f*) greater than zero, and moreover, we know that discarding a subsystem never increases the mutual information, thus <sup>I</sup>(*ρS*1*S*2:E*f*) ≥ <sup>I</sup>(*ρS*1:E*f*). Together, this allows us to restrict the gap between the curves as the following:

$$0 \le \Delta T \le S(\rho\_{\mathcal{S}\_1 \mathcal{S}\_2}) + D^{\epsilon -}(\rho\_{\mathcal{S}\_1 \mathcal{S}\_2 \mathcal{E}\_f}).\tag{A8}$$

Finally, let us also specifically look at the gap at the end of the curves in Figure 2d, i.e., E*f* = E, where we can use the equality in the KW relations given in Equations (A5) and (A7), and obtain a more precise expression. A pure *S*1*S*2E state implies that <sup>I</sup>(*ρS*1*S*2:<sup>E</sup> ) = <sup>2</sup>*<sup>S</sup>*(*ρS*1*S*2), and we have the following:

$$\begin{split} \Delta \mathcal{Z} &= 2S(\rho\_{S\_1 \mathcal{S}\_2}) - S(\rho\_{S\_1}) - D^{\leftarrow}(\rho\_{S\_1 \mathcal{E}}) + E\_f(\rho\_{S\_1 \mathcal{S}\_2}) \\ &= 2S(\rho\_{S\_1 \mathcal{S}\_2}) - S(\rho\_{S\_1}) - S(\rho\_{\mathcal{E}}) + S(\rho\_{S\_1 \mathcal{E}}) \\ &= S(\rho\_{\mathcal{S}\_2}) + S(\rho\_{\mathcal{E}}) - S(\rho\_{S\_2 \mathcal{E}}) \\ &= \mathcal{Z}(\rho\_{\mathcal{S}\_2 \mathcal{E}}). \end{split} \tag{A9}$$

In passing from the first to the second line we resort to the basic definition of <sup>I</sup>(*ρS*1:<sup>E</sup> ). Note that this is exactly the same result we obtain in Equation (A3) using the direct approach.

#### **Appendix B. Testing the Strong Quantum Darwinism Criteria**

We would like to assess whether the mutual information between the system particles and a single environment state <sup>I</sup>(*ρS*1*S*2:*E*1 ) is purely classical. To that end, we need to check whether the state *ρS*1*S*2*E*1 has a vanishing quantum discord or not, for which we have two options to consider: measurements performed on the system or environment side. The former is the condition of *strong Darwinism* introduced in [6] and the latter is an alternative constraint recently put forward in Ref. [37].

Here, we present the explicit calculation of the nullity condition for quantum discord introduced in [40,41] considering both measurement scenarios mentioned in the paragraph above. An arbitrary state *ρAB* has a vanishing quantum discord with measurements on *A* or *B* if and only if one can find an orthonormal basis {|*n*-} or {|*m*-} in the Hilbert space of *A* or *B* such that the total state can be written in block-diagonal form in this basis. Mathematically, it is possible express this condition as follows:

$$D^{\rightarrow}(\rho\_{A:B}) = 0 \iff \rho\_{AB} = \sum\_{n} p\_n |n\rangle\langle n| \otimes \rho\_{n'}^{B} \tag{A10}$$

$$D^{\leftarrow}(\rho\_{A:B}) = 0 \iff \rho\_{AB} = \sum\_{m} q\_{m} \rho\_{m}^{A} \circledast |m\rangle\langle m|.\tag{A11}$$

Note that in our case, we make the identification *A* → *S*1*S*2 and *B* → *E*1.

Let us start with checking the former condition. We pick an arbitrary orthogonal basis in the Hilbert space of the environmental qubit as {|*ei*-} and express the state at hand as follows:

$$
\rho\_{S\_1 S\_2 E\_1} = \sum\_{i,j} \rho\_{ij}^{S\_1 S\_2} \circledast \left| e\_i^E \right\rangle \left\langle e\_j^E \right|, \tag{A12}
$$

In order for the state in Equation (A12) to be written as the one given in Equation (A11), all *ρS*1*S*<sup>2</sup> *ij* 's must be simultaneously diagonalizable and, if it exists, the basis in which they are diagonal is then {|*n*-}. It was shown in [40,41] that mathematically, this implies the following: *D* <sup>→</sup>(*ρS*1*S*2:*E*1 ) = 0 if and only if one has *ρS*1*S*<sup>2</sup> *ij* , *ρS*1*S*<sup>2</sup> *ij* = 0. Similarly, this condition can be stated as *ρS*1*S*<sup>2</sup> *ij* 's must be normal matrices such that *ρS*1*S*<sup>2</sup> *ij* , *ρS*1*S*<sup>2</sup> *ij* † = 0, and also commute with each other [40,41].

Using our analytics, we can write the general form of *ρS*1*S*2*E*1 at the instant we observe Darwinism, i.e., *JSEt* = *π*/4, as follows:

$$
\rho\_{S\_1S\_2E\_1} = \begin{pmatrix} a & -a & 0 & 0 \\ -a & a & 0 & 0 \\ 0 & 0 & c & c \\ 0 & 0 & c & d \\ \hline \\ 0 & 0 & d^\* & d^\* & e & 0 & 0 \\ \hline \\ 0 & 0 & d^\* & d^\* & e & e & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & f & -f \\ b^\* & -b^\* & 0 & 0 & 0 & 0 & -f & f \\ \end{pmatrix}, \tag{A13}
$$

where *a* = *α*2/2, *b* = ( −<sup>1</sup>)*<sup>N</sup>αδ*/2, *c* = *β*2 cos<sup>2</sup> *Jπ* 2 + *γ*2 sin<sup>2</sup> *Jπ* 2 /2, *d* = [2*βγ* + *i*(*β* − *γ*)*β* + *γ*) sin(*Jπ*)]/4, *e* = *γ*2 cos<sup>2</sup> *Jπ* 2 + *β*2 sin<sup>2</sup> *Jπ* 2 /2 and *f* = *δ*2/2. Recall that parameters *α* = cos *θ*1 cos *θ*2, *β* = cos *θ*1 sin *θ*2, *γ* = sin *θ*1 cos *θ*2, *δ* = sin *θ*1 sin *θ*2 are dependent on the initial states of the system qubits. The dimensions of our the system particles and the environmental qubit are *dS*1*S*2 =4 and *dE*1 =2, respectively, which means that the set {|*ei*-} is composed of two elements and we have 4 *ρS*1*S*<sup>2</sup> *ij* matrices that are 4 × 4 in size. Horizontal and vertical lines dividing the density matrix in Equation (A13) in fact denote these 4 matrices. Explicitly, we have the following:

$$\begin{aligned} \rho\_{S\_1S\_2}^{11} &= \begin{pmatrix} a & -a & 0 & 0 \\ -a & a & 0 & 0 \\ 0 & 0 & c & c \\ 0 & 0 & c & c \end{pmatrix}, & \rho\_{S\_1S\_2}^{12} &= \begin{pmatrix} 0 & 0 & b & -b \\ 0 & 0 & -b & b \\ d & d & 0 & 0 \\ d & d & 0 & 0 \end{pmatrix}, \\\ \rho\_{S\_1S\_2}^{21} &= \begin{pmatrix} 0 & 0 & d^\* & d^\* \\ 0 & 0 & d^\* & d^\* \\ b^\* & -b^\* & 0 & 0 \\ -b^\* & b^\* & 0 & 0 \end{pmatrix}, & \rho\_{S\_1S\_2}^{22} &= \begin{pmatrix} c & e & 0 & 0 \\ e & e & 0 & 0 \\ 0 & 0 & f & -f \\ 0 & 0 & -f & f \end{pmatrix}. \end{aligned}$$

The diagonal matrices are clearly normal matrices, i.e., they satisfy *ρii S*1*S*2 , *ρii S*1*S*2 † = 0. However, the off-diagonal ones are not normal in general. In fact, using the parameters we use in our simulations (setting *S*1-*S*2 interaction *J* = 10), they do not commute independentlyoftheinitialstateofthesystem.Asaresult,itisnotpossibletowrite*ρS*1*S*2*E*1inthe

form given in Equation (A11), and thus *<sup>D</sup>*<sup>→</sup>(*ρS*1*S*2:*E*1 ) > 0 implying that the condition for strong Darwinism, as defined in [6], is not satisfied.

Considering the alternative approach of checking the condition of vanishing discord with measurements on the environment side, *<sup>D</sup>*<sup>←</sup>(*ρS*1*S*2:*E*1 ) > 0, similar to the previous case, we start by expressing our total state in an arbitrary orthogonal basis in the Hilbert space of the system qubits {*sk*} as the following:

$$
\rho\_{S\_1 S\_2 E\_1} = \sum\_{k,l} \left| s\_k^S \right\rangle \left\langle s\_l^S \right| \circledast \rho\_{E\_1}^{kl}.\tag{A14}
$$

Recalling that *dS*1*S*2 = 4, it is possible to identify the set {*sk*} consists of four elements and we have 16 *ρkl E*1 matrices which are 2 × 2 in size, denoted by the horizontal and the vertical lines below:


Following [40,41] again, checking the nullity condition amounts to checking whether the commutators satisfy the condition *ρkl E*1 , *ρkl E*1 = 0. It is possible to show that these commutators indeed vanish, which implies that from the point of view put forward in Ref. [37], all mutual information we have between the system qubits and environment fractions at the instant we observe the plateau is classical, and therefore objective.
