A Search for Classical Subsystems in Quantum Worlds

TL;DR

The paper tackles how to derive quasi-classical subsystems from a fixed quantum Hamiltonian by exploring tensor-product structures (TPS) that support pointer states via einselection. It introduces a variational algorithm that optimizes over both the TPS (via a unitary $B$) and initial states to minimize a purity-based cost, demonstrating that every Hamiltonian admits a pointer basis in a separable energy-eigenstate TPS and uncovering multiple coexisting real-mtls or “realms” with distinct dynamical behavior. The results categorize destination Hamiltonians into five classes (Decoupled, Block Diagonal with DFS, Approximate Eigenstates, Furnace, Extended coherence time), showing that even random spectra can yield classical-like subsystems. The findings have foundational implications for quantum interpretations and practical relevance for decoherence-free subspaces and passive error-avoidance protocols, while suggesting avenues to extend to larger, multipartite systems and to connect with broader classicality formalisms.

Abstract

Decoherence and einselection have been effective in explaining several features of an emergent classical world from an underlying quantum theory. However, the theory assumes a particular factorization of the global Hilbert space into constituent system and environment subsystems, as well as specially constructed Hamiltonians. In this work, we take a systematic approach to discover, given a fixed Hamiltonian, (potentially) several factorizations (or tensor product structures) of a global Hilbert space that admit a quasi-classical description of subsystems in the sense that certain states (the "pointer states") are robust to entanglement. We show that every Hamiltonian admits a pointer basis in the factorization where the energy eigenvectors are separable. Furthermore, we implement an algorithm that allows us to discover a multitude of factorizations that admit pointer states and use it to explore these quasi-classical "realms" for both random and structured Hamiltonians. We also derive several analytical forms that the Hamiltonian may take in such factorizations, each with its unique set of features. Our approach has several implications: it enables us to derive the division into quasi-classical subsystems, demonstrates that decohering subsystems do not necessarily align with our classical notion of locality, and challenges ideas expressed by some authors that the propensity of a system to exhibit classical dynamics relies on minimizing the interaction between subsystems. From a quantum foundations perspective, these results lead to interesting ramifications for relative-state interpretations. From a quantum engineering perspective, these results may be useful in characterizing decoherence free subspaces and other passive error avoidance protocols.

Figures (8)

• Figure 1: Best late time linear entropy achieved by the different optimization methods for various Hamiltonians. The columns correspond to the Hamiltonians described in Sec. \ref{['subsec:SR-Hamiltonians-considered']}. The rows correspond to the different initial state optimizations from Sec. \ref{['subsec:SR-initial-conds']}; the unitary $B$ is optimized in each case.
• Figure 2: Time evolution of the purity of the reduced system density matrix for each of the categories identified by our algorithm, as discussed in Sec. \ref{['sec:analysis']}. When the purity is close to one, the training state is separable, i.e. $|\psi\rangle_w = |\chi\rangle_s|\phi_e\rangle$ in the destination factorization. Thus, we also show the effect on the purity resulting from substituting the system state for the orthogonal state, $|\chi\rangle_s \rightarrow |\chi'\rangle_s$, and of randomizing the environment state $|\phi\rangle_e$. While in the decoupled case all initial product states remain unentangled for arbitrary times, the other destinations only allow very particular states to stay unentangled beyond the decoherence time. The furnace case is interesting in that it admits a pointer state only for particular choices of the environment state, as elaborated on in Fig. \ref{['fig:furnace_details']}.
• Figure 3: The decomposition of $|\psi\rangle_w$ in the energy eigenbasis for the case of the 'approximate eigenstate" solutions discussed in Sec. \ref{['sec:analysis']} (see Eq. \ref{['eq:approx-eig-psi']}) is shown in blue along with their corresponding linear entropy of each energy eigenstate (defined in Eq. \ref{['eq:lin-entropy']}). These solutions are dominated by a single low entropy energy eigenstate, although high entropy eigenstates contribute to a small degree. The plot is arranged in order of decreasing $|\alpha_i|$.
• Figure 4: In the furnace Hamiltonian, a pointer state for the system exists only if the environment is in a particular subspace of $\mathcal{H}_e$ (unlike the Block diagonal case, c.f. Fig. \ref{['fig:Destination-Hams']}b, where the pointer state is indifferent to the choice of $|\phi\rangle_e$). Here we depict this phenomenon by perturbing the trained state $|\psi\rangle_w$ (turquoise) such that the new state $|\psi'\rangle_w$ (purple) has support from the same energy eigenvectors as those that contribute to the trained state (top panel). The bottom panel verifies that such a state indeed remains a product state at late times. Note that $|\psi\rangle$ and $|\psi'\rangle$ share the same system state in $\mathcal{H}_s$ (up to an overall phase). We show, for comparison, another state where $|\phi\rangle_e$ is randomized leading to a large increase in the linear entropy in the typical decoherence time (the furnace "incinerates" in the analogy of the text in Section \ref{['sec:analysis']}(d)).
• Figure 5: Here we depict the effect of varying the environment state for a particular pointer state in a 1-qubit system and 2-qubit environment world. The figure shows the amplitudes of the coefficients of a product state composed of the pointer state and a random environment state in the energy eigenbasis. As discussed in the text, varying the environment state varies only the components in the subspace of eigenvectors that is projected by the particular choice of pointer state.