Pointer States and Quantum Darwinism with Two-Body Interactions

Quantum Darwinism explains the emergence of classical objectivity within a quantum universe. However, to date, most research on quantum Darwinism has focused on specific models and their stationary properties. To further our understanding of the quantum-to-classical transition, it appears desirable to identify the general criteria a Hamiltonian has to fulfill to support classical reality. To this end, we categorize all N-qubit models with two-body interactions, and show that only those with separable interaction of the system and environment can support a pointer basis. We further demonstrate that “perfect” quantum Darwinism can only emerge if there are no intra-environmental interactions. Our analysis is complemented by solving the ensuing dynamics. We find that in systems exhibiting information scrambling, the dynamical emergence of classical objectivity directly competes with the non-local spread of quantum correlations. Our rigorous findings are illustrated through the numerical analysis of four representative models.


I. INTRODUCTION
We live in a quantum universe, yet our everyday reality is well-described by classical physics.Hence, the obvious question to ask is where all the quantum information and correlations hide.The quantum nature of our universe is captured by its ability to be in a superposition of classically allowed states.The transition from quantum to classical is a two-step process.The first, necessary but not sufficient, step is the destruction of quantum superpositions, i.e., destruction of all interference phenomena.The theory of decoherence teaches us that it is the interaction between the quantum system and its environment that is the cause of this phenomenon [1].The destruction of quantum superpositions presupposes a privileged and unique quantum basis.The elements of this basis are called pointer states [2][3][4].Any quantum superposition written in this basis decomposes into a classical mixture under the effect of environmental interaction.Thus, pointer states are precisely the only quantum states that are stable under this interaction.
Quantum Darwinism [5][6][7][8][9][10][11][12][13] builds on decoherence theory and goes a step further, approaching the problem from the point of view of quantum information theory.An outside observer has no direct access to a system of interest S, but rather the environment E acts as a communication channel.Since any real environment is tremendously large, "observing" S actually means that an observer intercepts only a small, possibly even tiny fragment F of E, and then reconstructs the state of S from the information carried by F.
If the constituents of the environment do not interact, such as, for instance, photons [14,15], then the infor-mation about S is accessible by local measurements on E. However, reality is a little more complicated, and in general, the constituents of E do interact.Such intraenvironmental interactions lead to the build-up of nonlocal correlations, which is the root-cause of information scrambling [16][17][18][19][20][21].Thus, an observer has to access a macroscopic fraction of E to reconstruct unambiguous information about S.
More directly relevant to our present work is Ref. [32], which analyzed a specific model where intraenvironmental interactions scramble the information encoded in different fragments F. This scenario, scrambling only in E, but not in S, makes it easier to highlight the competition between the local transfer of information from S to each degree of freedom of E, and the scrambling of information between the different F of E due to their interactions.
Current research in quantum Darwinism [13] is driven by the analysis of increasingly complex model systems.However, the focus has remained on particular qubitmodels [32][33][34][35][36][37], since their dynamics is tractable.Despite, or rather because of continued progress in our understanding it appears desirable to elucidate the general properties of Hamiltonians that support the emergence of quantum Darwinism.More precisely, it is instrumental to sort all possible interacting many-body Hamiltonians into classes that support a pointer basis for S, and which sub-classes of these will further exhibit the emergence of classical objectivity.Such a classification will also unveil if and under what circumstances, quantum Darwinism can emerge in the presence of scrambling dynamics.
In the present work, we consider a qubit of interest S, that interacts with an environment E also comprised of qubits.Hence, scrambling of information may only occur in E, but not in S. Further, for the sake of simplicity we restrict ourselves to arbitrary two-body interactions.
In a first part of our analysis, we show that the existence of a pointer basis for S imposes a specific structure for the total Hamiltonian describing the evolution of the universe S ⊗ E. In fact, we will see that a pointer basis for S exists for any interactions within E, yet S may only interact with all fragments of E identically, cf.Fig. 1.The second part of the analysis is then focused on the dynamics induced by such Hamiltonians that support a pointer basis.We find that the efficiency of the information transfer between S and E is governed by the statistics of the interaction terms.The average information transfer is irreversible if and only if the support of the coupling coefficients is continuous, and the "speed of communication" is determined by the shape of the distribution of the interaction coefficients.Our general findings are illustrated with four models that correspond to a variety of situations, including scrambling or no scrambling, pointer basis or no pointer basis, quantum Dar-winism or no quantum Darwinism.

II. STRUCTURE OF THE HAMILTONIAN
We start by defining the problem in mathematically rigorous terms.Consider a set of (N + 1) qubits, where the 0th qubit is the system S. Hence, the environment E is comprised of N qubits.For the sake of simplicity, we further restrict ourselves to 2-body interaction models.The most general Hamiltonian corresponding to this scenario then reads where α and β take the values x, y and z, corresponding to the Pauli matrices σ x , σ y and σ z .Indices i and j count the qubits, with i, j = 0 for S and i, j ≥ 1 for E. Further, J ij and ⃗ B i are real coefficients, which in the following we will choose to be random variables.

A. Existence of a pointer basis
The natural question now is, what conditions J ij and ⃗ B i have to fulfill, such that H in Eq. ( 1) supports a pointer basis for S. Pointer states are the particular states of S that are stable under the interaction with E [2-4].Formally, these states can be identified in the following way: |ψ S ⟩ ∈ S is a pointer state of S if and only if for any |ψ E ⟩ ∈ E, an initial product state |ψ S ⟩ ⊗ |ψ E ⟩ evolves under H (1) to remain within an epsilon ball around the product state |ψ S ⟩ ⊗ |ψ E (t)⟩.In other words, the reduced state |ψ S ⟩ remains pure under the evolution of the total Hamiltonian.
It will prove convenient to separate the total Hamiltonian into terms corresponding to S, E, and their interaction.Hence, we write Comparing with Eq. ( 1) we identify the system Hamiltonian as whereas we have for the environment Notice that the first term in Eq. ( 4) describes the intraenvironmental interactions.The interaction between S and E is given by From this separation of terms it becomes immediately obvious that the pointer basis for S can only exist if certain necessary and sufficient conditions for the interaction term H SE are fulfilled.These conditions become particularly intuitive by considering the original motivation for pointer states.These states are not only immune to the dynamics induced by the interaction with the environment, but can be also thought of as states that correspond to the pointer of a measurement apparatus.Mathematically, such an apparatus is described by the pointer observable By construction, the pointer observable A commutes with the total Hamiltonian (1), and hence A and H share an eigenbasis.Due to form of A the corresponding eigenstates can be written in tensor-product form Correspondingly, we can factorize the time-evolution operator as where the H i act only on E. Now, choosing any (reduced) eigenstate of A as initial state of S, In Appendix A we show that by enforcing the commutation relation [A, H] = 0 we have that any Hamiltonian (4) supporting a pointer basis for S has to be of the form where H S is the system Hamiltonian (3), and the h i are arbitrary traceless Hermitian operators acting on the ith qubit of E.
In conclusion, we have shown that any model of interacting qubits that supports a pointer basis for a system qubit S may have at most a separable interaction term H SE .Moreover, this interaction terms has to be factorizable into the system Hamiltonian H S and traceless terms acting on the environmental qubits E. It is important to emphasize that no additional conditions are required pertaining to, for instance, the intra-environmental interactions.Schematically, our findings are illustrated in Fig. 1.

B. Further conditions for quantum Darwinism
It was shown in Ref. [11] that only a special structure of states is compatible with the emergence of quantum Darwinism.These states are of the singly-branching form [12,38], which are the only states to support epsilon quantum correlations as measured by quantum discord [39].
Singly branching states are pointer states of S correlated with the environment states in the special form, It is easy to see that such a singly branching form can emerge if and only if there are no intra-environmental interactions.
Thus, we conclude that quantum Darwinism can only be supported by Hamiltonians with separable interaction between S and E, cf.Eq. ( 8), and no intra-environmental interactions, i.e., J αβ ij = 0 in Eq. ( 4).The remaining question now is whether all such Hamiltonians provide socalled good decoherence, which makes their corresponding E good channels for information transfer.

III. COEFFICIENTS OF THE HAMILTONIAN
To analyze the dynamical emergence of quantum Darwinism, we now solve for the average dynamics under an arbitrary, random Hamiltonian for which the system S has a pointer basis of the single branching form (9). We will find that the efficiency of information transfer within E is governed by the randomness of the interaction coefficients.

A. Solving the dynamics
To this end, consider an arbitrary Hamiltonian of the form (8), where we further enforce vanishing intraenvironmental interactions J αβ ij = 0. Note that in Eq. ( 8) the h i are hermitian, and traceless.Hence, we can write equivalently (and without loss of generality) where the B i are real random variables.
We are now interested in the dynamics induced by Eq. ( 10), and we choose an arbitrary separable initial condition.Therefore, we write where the α i and β i are arbitrary, complex coefficients.Evolving this |ψ 0 ⟩ under the corresponding Schrödinger equation, i∂ t |ψ⟩ = H |ψ⟩ we obtain the time-dependent solution, where we introduced As usual, the reduced density matrix of S is given by tracing out E, ρ S (t) = tr E {|ψ(t)⟩ ⟨ψ(t)|}.The corresponding decoherence factor [32] is given by the amplitude of the off-diagonal coefficients of the reduced density matrix ρ S in the basis {|0⟩ , |1⟩}.We have and, since the B i are stochastically independent, we can write It is easy to see that we have, For random B i it is, however, more instructive to compute the decoherence factors averaged over all possible values for B i .Further denoting the probability density function of B i as P (B i = x) = f i (x), we show in Appendix B that we obtain where In conclusion, we have derived an analytic expression for the average decoherence function, which governs the rate with which information about S is communicated through E.

B. Rate and irreversibility of information transfer
Equation (17) demonstrates the relationship between the emergence of classicality and the randomness of system-environment interactions.Indeed, the probability distributions f i of the couplings B i between S and E play a central role in the rate of information transfer.Observe that the decoherence factors decrease rapidly if f i decrease rapidly.Since f i is the Fourier transform of the probability distribution f i , the order of differentiability of f i gives us the order of decay of f i , while the smallest characteristic length in the distribution f i gives us the inverse of the characteristic time of decay of f i .
Furthermore, if the support of B i is discrete and finite, then the characteristic function f i (k) is a periodic (or quasi-periodic) function and therefore does not converge to 0. Hence, having continuous support for the f i is essential for the emergence of truly classical behavior.In fact, if the f i distribution is continuous, then the information is transferred irreversibly.In this case, the f i are integrable and thus, by virtue of the Riemann-Lebesgue Lemma, (20) where ϵ i depends on the initial state.
Finally, we note that a perfect record of the information about S in the ith qubit corresponds to Γ i = 0.This is typically not the case.However, as the become strictly less than one, Eq. 15 shows that |0⟩ i∈F |0 i (t)⟩ and |1⟩ i∈F |1 i ⟩ become orthogonal on average for a sufficiently large fragment F. Thus, a small set of qubits of the environment is enough to obtain an almost complete record of the state of S.
To this end, consider again a fragment F of E. If any F carries the same information about S, then any two observers accessing different F learn exactly the same information about S. The amount of information that a fragment F of E contains about the system S can be quantified with the mutual information I(S : F) defined as where S(ρ) = −tr {ρ log(ρ)}.The maximal classical information that can be accessed by any observer is upperbounded by the Holevo quantity [40,41] χ(S : F) = S S − S S| F (22) where S S| F is the conditional von Neumann entropy defined as the minimal von Neumann entropy of S obtained after a measurement on F.
The difference of the mutual information, I(S : F), and the Holevo quantity, χ(S : F) has been called quantum discord [39], Quantum discord measures the genuinely quantum information encoded in F.
For each F and its complement, F = E \ F we define the corresponding decoherence factors With these definitions, one can then show that for small enough decoherence factors [32] we have where 2 ), which is the maximal value that the von Neumann entropy of S. Correspondingly, we obtain for the Holevo quantity (see Appendix C) which is fully consistent with earlier findings [42].
Finally, in the limit of long times, t ≫ 1, and for smooth enough f i (20) we obtain the following asymptotic expression for the mutual information and the Holevo quantity Further, averaging over every possible separable initial states, we have and where n is the size of F and ϵ = 2/3 (see Appendix C).These results are depicted in Fig. 2 for N = 50 environmental qubits.Both, the mutual information and the Holevo quantity exhibit a steep initial rise with increasing fragment size n, as larger fragments provide more data about S. This initial rise is followed by the classical plateau.

IV. REPRESENTATIVE EXAMPLES
We conclude the analysis with the numerical solution of four representative examples.To support quantum Darwinism a Hamiltonian must obey the following conditions: existence of a pointer basis, continuous support, and no intra-environment interactions.Our first example exhibits these three conditions, and for each following example we successively remove one of these conditions, cf.Tab.I.

A. Continuous Parallel Decoherent Interaction
The first model has a pointer basis, random coupling coefficients with a continuous spectrum, and does not exhibit scrambling in E. The corresponding Hamiltonian reads where B i are independent random variables, drawn uniformly from For specificity, we call this model Continuous Parallel Decoherent Interaction (CPDI).
In Fig. 3a we plot the resulting mutual information (21), as a function of the fragment size, which rapidly converges towards the asymptotic expression (27).Note the distinct classical plateau indicative of quantum Darwinism.Moreover, we observe relaxation of S into its stationary pointer states over a typical time τ ≃ 1/4, at which point the information transfer becomes irreversible.

B. Discrete Parallel Decoherent Interaction
Our second example is called, Discrete Parallel Decoherent Interaction (DPDI).The corresponding Hamilto- nian is where B i are again independent random variables.However, in contrast to the continuous case in Eq. ( 31), the B i are now drawn uniformly from the discrete set B i ∈ {−1, −0.5, 0.5, 1}.
In Fig. 3b we depict the resulting mutual information.As expected, we observe that the classical plateau appears and disappears periodically, and hence the information transfer is no longer irreversible.At instants at which the plateau completely vanishes, I(S : F) is linear in the fragment size.This indicates that the information about S encoded in E is distributed throughout the entire environment (no redunancy).

C. Continuous Orthogonal Decoherent Interaction
As a next example, we consider Continuous Orthogonal Decoherent Interaction (CODI), which refers to our first model (31) but with an added external field.This additional term is chosen to be not parallel to the interaction between S and E, and hence CODI does not support a pointer basis.The Hamlitonian reads where, as before, B i are independent random variables, drawn uniformly from In this model, information about the observable σ z 0 can be registered in E, but the eigenstates of this observable are not stable and hence not classically objective.This observation is further supported by Fig. 3c, which does not exhibit any form of classical plateau.Moreover, at all instants I(S : F) is a linear function of the size of F, which is a consequence of the complete absence of any redundancy.

D. Continuous Parallel Decoherent Interaction with Scrambling
As a final example, we again consider Eq. ( 31) but now design E to exhibit scrambling dynamics.Accordingly, this model is called Continuous Parallel Decoherent Interaction with Scrambling (CPDI-S), and the Hamiltonian becomes where again B i are independent random variables, drawn uniformly from B i ∈ [−1, 1], and J ij are independent random variables, drawn uniformly from J ij ∈ [−0.03, 0.03].As Fig. 3d shows, a classical plateau rapidly emerges over a time scale of τ SE ≃ 1/4.However, this plateau quickly "disperses" as the quantum information becomes non-local due to scrambling in E. We refer to Ref. [32] for a more detailed analysis of this particular model.Furthermore, it is interesting to note that Eq. ( 34) is another example that demonstrates the competition of decoherence and scrambling as a "sink for quantum information" as analyzed by (some of) us in Ref. [25].

V. CONCLUDING REMARKS
In the present work we determined the set of qubit models, which support the emergence of classicality.In particular, we established a classification of 2-body interaction models based on the structure of the Hamiltonian and on the nature of its coefficients.
The existence of an "exact" pointer basis for the qubit S requires the interaction Hamiltonian to be separable between S and its environment E such that the part acting on S is proportionnal to the self Hamiltonian of S. We call that type of structure a parallel decoherent interaction.Furthermore, without any intra-environment interactions, this Hamiltonian structure leads to a branch-ing state structure, the only one compatible with quantum Darwinism [11].
Furthermore, intra-environment interactions can lead to information scrambling in E, which deteriorates the branching structure.In such situations, the state of S is still a classical mixture, but this classicality is hidden to an outside observer who must take a measurement on a non-local part of E in order to recover almost all the information about S. This indicates a clear competition of the emergence of classical objectivity and scrambling dynamics.
The conceptual notions and the gained insight of our work may open the door for further inquiry, such as the study of k−body interactions.In particular, there is every reason to believe that this type of interaction leads to a non-local information transfer.Indeed, for such interactions, the information about S is directly encoded in E by the entanglement of S with fragments F of size k.This results in a lower redundancy of information.However, the analytical analysis of the dynamics is much more involved than the present 2-interaction case, which is why we leave k−body interactions for future work.by considering the reduced density operator of fragment F, which is given by ρ F = tr SF {ρ}, and we have
Since we are working with qubits, it is then a simple exercise to show that and (C8) Note that we obtain an equivalent expression for the complement F. Thus, the mutual information (21) becomes (C9) Following similar steps as detailed in Ref. [42] the corresponding Holevo quantity can be written as (C10) which for weak decoherence in leading order simply is Finally, we note that for continuous distributions and t ≫ 1 we have (with Eq. ( 20))

FIG. 2 :
FIG. 2: Asymptotic mutual information (27) and Holevo quantity (28) as functions of the fragment size.The classical plateau corresponds to the value S max = 1.