Frozen and Growing Quantum Work under Noise: Coherence and Correlations as Key Resources

Abstract

We investigate the decomposition of ergotropy into incoherent and coherent contributions for quantum systems subject to typical Markovian noise channels. The incoherent part originates from population inversion in the energy eigenbasis after dephasing, while the coherent part captures the role of quantum coherence in work extraction. For single-qubit systems, we derive explicit conditions for freezing and enhancement of coherent ergotropy and obtain an analytical upper bound, showing that it cannot exceed one half of the state's quantum coherence. We then study two classes of separable two-qubit states under local noise. For Bell-diagonal states, which are locally completely passive and possess no local coherence, we prove that the total extractable work equals the average of geometric quantum and classical correlations. In this case, coherent ergotropy cannot be enhanced, although freezing occurs under specific noise conditions. By contrast, for separable states with local coherence, coherent ergotropy can increase under all considered noise channels, including phase-flip and depolarizing noise. Extending the analysis to multipartite systems, we show that both the magnitude and range of noise-induced enhancement grow with the number of qubits, indicating collective reinforcement. Finally, we demonstrate through an explicit example that entanglement does not prevent this enhancement: coherent ergotropy may increase under noise even for entangled states. Our results reveal that noise can assist energy storage, challenging the conventional view of noise as purely detrimental and suggesting compatibility between noise-assisted enhancement and fast entanglement-based charging mechanisms in quantum batteries.

Figures (12)

• Figure 1: Three-dimensional plot of the noise strength bound resulting in coherent ergotropy enhancement for (a) bit flip Eq. \ref{['q_frozen_growing_bit_flip_single']} and (b) bit-phase flip Eq. \ref{['q_frozen_growing_bit_phase_flip_single']}.
• Figure 2: Coherent ergotropy of single qubit system under (a) bit flip Eq. \ref{['coerg_bit_flip_single']} and (b) bit-phase flip. The vertical dashed lines pinpoint the noise strength for which the enhancement of coherent ergotropy with respect to its noiseless value occurs, i.e. $\mathcal{W}^{\rm C}(q) \geq \mathcal{W}^{\rm C}(0)$. The Bloch vector parameters ($n_{1}$, $n_{2}$, $n_{3}$), respectively for salmon, green and blue curves are ($0.6$, $0.5$, $0.4$), ($0.4$, $0.3$, $0.6$), and ($0.1$, $0.5$, $0.2$).
• Figure 3: (a) Coherent ergotropy (solid) and quantum coherence (dashed) of single qubit system under amplitude damping channel, given by Eq. \ref{['coerg_amplitude_damping_single']}. The shaded areas show the regions for which the coherent ergotropy grows beyond its noise-free value. The Bloch vector parameters ($n_{1}$, $n_{2}$, $n_{3}$) are (0.2, 0.3, -0.9), (0.7, 0.5, -0.4), (0.1, 0.3, -0.4), and (0.6, 0.7, -0.2) for green, purple, salmon and blue curves, respectively. (b) The difference in coherent ergotropy, with and without amplitude damping. The areas for which $\Delta \mathcal{W}^{\rm C}>0$ show the enhancement of coherent ergotropy of single qubit system subjected to amplitude damping channel. For this panel, we fix $n_{1}=0.5$ and $n_{2}=0.3$.
• Figure 4: Total extractable work $\mathcal{W}$ (solid) and the average of geometrical quantum and classical correlations $\mathcal{I}$ (dashed) of a two-qubit Bell-diagonal state under (a) bit flip, (b) bit-phase flip, (c) phase flip, and (d) amplitude damping. For the amplitude damping case also the average of correlations are computed using Eqs. \ref{['GQC2']} and \ref{['GCC2']}, even though we emphasize that the formulas are not able to properly evaluate the correlation content under amplitude damping, due to non-unital nature of this channel.
• Figure 5: Coherent contribution of extractable energy under amplitude damping.