Quantum Energy Teleportation under Equilibrium and Nonequilibrium Environments

Quantum energy teleportation (QET), implemented via local operations and classical communication, enables carrier-free energy transfer by exploiting quantum resources. While QET has been extensively studied theoretically and validated experimentally in various quantum platforms, enhancing energy output for mixed initial states, as the system inevitably interacts with environments, remains a significant challenge. In this work, we study QET performance in a two-qubit system coupled to equilibrium or nonequilibrium reservoirs. We derive an analytical expression for the energy output in terms of the system Hamiltonian eigenstates, enabling analysis of energy output for mixed states. Using the Redfield master equation, we systematically examine the effects of qubit detuning, nonequilibrium temperature difference, and nonequilibrium chemical potential difference on the energy output. We find that the energy output for mixed states often follows that of the eigenstate with the highest population, and that nonequilibrium environments can enhance the energy output in certain parameter regimes.

💡 Research Summary

The paper investigates quantum energy teleportation (QET) in a realistic setting where a two‑qubit system interacts with external reservoirs. The authors adopt the minimal Hotta model consisting of qubits A (Alice) and B (Bob) with Hamiltonian
(H_{AB}= \varepsilon_A \sigma_z^A + \varepsilon_B \sigma_z^B + 2\kappa \sigma_x^A\sigma_x^B).
The eigen‑spectrum consists of four energies (E_{1,2,3,4}) and corresponding eigenstates that depend on the sum (\Omega=\varepsilon_A+\varepsilon_B), the detuning (\Delta=\varepsilon_A-\varepsilon_B), and the interaction strength (\kappa). By allowing (\varepsilon_A\neq\varepsilon_B) the authors explore the effect of detuning on QET performance.

The QET protocol proceeds in three steps: (i) Alice performs a projective measurement of (\sigma_x^A) obtaining outcome (u=\pm1); (ii) she sends (u) to Bob via a classical channel; (iii) Bob applies a local unitary (U_B(u)=\cos\theta,\mathbb{I} - i u \sin\theta,\sigma_y^B). The parameter (\theta) can be tuned to maximize the extracted energy. The energy output is defined as (E_{\text{out}}=E_A-E_B), where (E_A) and (E_B) are the system energies after Alice’s measurement and after Bob’s correction, respectively. For each eigenstate the output is given analytically (Eq. 11). The authors show that a single (\theta) cannot simultaneously maximize the output for all eigenstates; consequently, for mixed initial states the output is dominated by the eigenstate with the largest population.

To treat mixed states the paper introduces an “X‑state” density matrix with real parameters (a,b,c,d,\chi,\delta) and phases. The output reduces to a simple form (E_{\text{out}}=D\sin2\theta - F(1-\cos2\theta)) where (D) and (F) are linear combinations of the X‑state parameters and the system Hamiltonian coefficients. The optimal (\theta) satisfies (\tan2\theta = D/F) and the maximal output is (\sqrt{D^2+F^2}-F). This result highlights that both population imbalance and coherence contribute to the extractable energy.

The core of the work is the inclusion of environmental effects via the Bloch‑Redfield master equation. Each qubit couples to its own reservoir, which may be bosonic or fermionic, characterized by temperature (T_{A,B}) and chemical potential (\mu_{A,B}). Unlike the Lindblad approach, the Redfield equation retains non‑secular terms, allowing accurate description of nonequilibrium steady states (NESS). The authors systematically vary three key parameters: (1) detuning (\Delta), (2) temperature difference (\Delta T = T_A - T_B), and (3) chemical‑potential difference (\Delta\mu = \mu_A - \mu_B).

Key findings are:

• For bosonic baths, any temperature gradient suppresses QET output, because thermal noise degrades coherence and skews transition rates unfavorably.
• For fermionic baths, a moderate temperature difference can enhance output. The asymmetry of the Fermi‑Dirac distribution redistributes occupation probabilities, increasing the weight of higher‑energy eigenstates that contribute positively to (E_{\text{out}}).
• Chemical‑potential bias has a strong, non‑monotonic effect. When the average chemical potential is far above or below the qubit energy scales, the system is driven into regimes with suppressed transitions, reducing output. When the bias brings the reservoirs’ electrochemical potential close to the qubit energies, transition rates are optimized and QET output can be significantly amplified; the effect is most pronounced when (|\Delta\mu|) is comparable to (\varepsilon_{A,B}).
• Larger detuning (|\Delta|) magnifies the influence of both temperature and chemical‑potential differences, because the energy asymmetry makes the system more sensitive to the directionality of bath‑induced transitions.

Overall, the analysis shows that for mixed initial states the QET output is largely governed by the most populated eigenstate, but nonequilibrium reservoirs can be engineered to shift the population distribution and transition dynamics such that the total output exceeds what would be obtained in equilibrium. This demonstrates that environmental engineering—adjusting temperature gradients, chemical‑potential biases, and qubit detuning—can turn the surrounding baths from a detrimental source of decoherence into a resource that boosts carrier‑free energy transfer.

The paper concludes that incorporating realistic environmental interactions via the Redfield formalism provides a quantitative framework for optimizing QET in practical quantum devices, such as quantum processors, quantum heat engines, or quantum networks, where carrier‑free energy transport could be exploited for low‑loss power distribution or for novel thermodynamic cycles.