Steady-state Bell nonlocality in an autonomous quantum thermal machine

Technical Summary: Steady-state Bell nonlocality in an autonomous quantum thermal machine

Problem Statement
Quantum correlations, specifically Bell nonlocality, are essential for secure information processing tasks like device-independent quantum key distribution. While environment-assisted entanglement generation is a known technique, generating steady-state Bell nonlocality in an autonomous setting (requiring no external control or coherence sources) has proven difficult. Previous two-qubit thermal machines could generate steady-state entanglement, but it was typically too weak to violate Bell inequalities. Achieving Bell nonlocality in these systems previously required additional resources, such as local filters or continuous feedback protocols. Furthermore, existing theoretical models often assumed the presence of a static "free-evolution" term in the system Hamiltonian (fixed energy gaps), which may not be present in certain physical realizations like Bose-Hubbard dimers or Su-Schrieffer-Heeger models.

Methodology
The author presents a simple two-qubit autonomous thermal machine consisting of interacting qubits weakly coupled to two separate thermal baths.

• System Hamiltonian: The model utilizes a Hamiltonian $\hat{H}_S$ that generates a flip-flop interaction ($g$) between qubits but explicitly excludes free-evolution terms (static energy gaps). This is realized, for example, by coupling position quadratures of hard-core bosons to baths.
• Derivation: A Lindblad master equation is derived for the reduced state of the two-qubit system under the secular approximation and weak coupling assumptions. The environment is modeled as either bosonic or spin baths.
• Symmetry Analysis: The Liouvillian is analyzed for symmetries. The author identifies a "strong symmetry" related to the two-qubit spin-x parity operator ($\hat{X} = \hat{\sigma}_x \otimes \hat{\sigma}_x$) in the noiseless case.
• Noise Modeling: To address experimental reality, stochastic perturbations are introduced to the system Hamiltonian, modeling local and collective dephasing noise. This breaks the strong symmetry but preserves a weak symmetry.
• Analysis: The steady states are calculated using the right and left eigenoperators of the Liouvillian. The degree of Bell nonlocality is quantified using the Horodecki criterion to determine the maximum CHSH value ($B$).

Key Contributions and Results

1. Generation of Bell Nonlocality without Additional Resources:
   The paper demonstrates that a simple two-qubit autonomous thermal machine, lacking a free-evolution term, can generate steady states that violate the CHSH inequality ($B > 2$) without external filters or feedback. This circumvents previous limitations where such states were deemed impossible without extra resources.

2. Role of Strong Symmetry and High Temperatures:
   In the absence of noise, the system exhibits a strong symmetry that decouples the Liouvillian into even and odd parity sectors. The steady state is a convex combination of Bell states determined by the initial state's projection onto these sectors.

   • Crucially, the system can generate Bell-nonlocal steady states at arbitrarily high temperatures (for certain initial conditions and bath types).
   • For spin baths, which saturate at high temperatures, Bell nonlocality persists regardless of temperature.
   • For bosonic baths, nonlocality persists up to a temperature bound defined by the weak-coupling condition ($g \gg \gamma$).
   • The non-equilibrium steady states (NESS) of this machine can violate Bell inequalities more strongly than the corresponding thermal equilibrium state at the same temperature.

3. Impact of Experimental Imperfections (Noise):
   The study introduces local and collective dephasing noise to model experimental imperfections.

   • Local Noise: If the sole source of noise is local dephasing, the strong symmetry is broken, and the resulting unique steady state is always Bell local ($B \leq 2$).
   • Collective Noise: The co-presence of collective dephasing noise is shown to be advantageous. It restores the ability to generate Bell-nonlocal steady states.
   • Noise as a Resource: Remarkably, increasing the intensity of collective noise (while maintaining local noise) can increase the degree of Bell-inequality violation. The machine can transform a Bell-local steady state (which would occur with only local noise) into a Bell-nonlocal one by introducing collective noise. The Bell parameter in the pure collective noise limit is independent of the noise intensity.

Significance and Claims
The paper claims that the algebraic properties of thermal machines, specifically the presence of strong symmetries in the absence of free-evolution terms, are significant for preserving coherence and stabilizing long-range quantum correlations.

• Theoretical Insight: The work highlights that the absence of a static free-evolution term, often assumed in previous models, is not a hindrance but a feature that enables strong symmetries and high-temperature Bell nonlocality.
• Practical Implication: The findings suggest that experimental imperfections, specifically collective dephasing noise, should not be viewed solely as detrimental. Instead, they can be harnessed to enhance quantum correlations in autonomous thermal machines, potentially improving the performance of previously proposed devices.
• Future Directions: The author modestly notes that while the current steady states are diagonal in the Bell basis (preventing improvement via single-copy local filtering), multi-copy filtering or extension to multi-qubit systems for multipartite nonlocality remain open questions.

The paper concludes that steady-state Bell nonlocality is achievable in simple, autonomous settings and that the interplay between symmetry and noise offers a pathway to optimizing quantum thermal machines.