We characterize the impact that the application of two consecutive quantum channels or their quantum superposition (thus, without a definite causal order) has on ergotropy, i.e. the maximum work that can be extracted from a system through a cyclic unitary transformation. First of all, we show that commutative channels always lead to a non-negative gain; non-commutative channels, on the other hand, can entail both an increase and a decrease in ergotropy. We then perform a thorough analysis for qubit channels and provide general conditions for achieving a positive gain on the incoherent part of ergotropy. Finally, we extend our results to d-dimensional quantum systems undergoing a pair of completely depolarizing channels.
We characterize the impact that the application of two consecutive quantum channels or their quantum superposition (thus, without a definite causal order) has on ergotropy, i.e. the maximum work that can be extracted from a system through a cyclic unitary transformation. First of all, we show that commutative channels always lead to a non-negative gain; non-commutative channels, on the other hand, can entail both an increase and a decrease in ergotropy. We then perform a thorough analysis for qubit channels and provide general conditions for achieving a positive gain on the incoherent part of ergotropy. Finally, we extend our results to d-dimensional quantum systems undergoing a pair of completely depolarizing channels.
We characterize the impact that the application of two maps in a quantum-controlled order has on the process of work extraction via unitary cycles and its optimization. The control is based on the quantum switch model that applies maps in an order not necessarily compatible with the underlying causal structure and, in principle, can be implemented experimentally. First, we show that the activation of quantum maps through the quantum switch model always entails a non-negative gain in ergotropy compared to their consecutive application. We also establish a condition that the maps should fulfill in order to achieve a nonzero ergotropic gain. We then perform a thorough analysis of maps applied to a two-level system and provide general conditions for achieving a positive gain on the incoherent part of ergotropy. Our results are illustrated with several examples and applied to qubit and d-dimensional quantum systems. In particular, we demonstrate that a nonzero work can be extracted from a system thermalized by two coherently controlled reservoirs.
The quantum switch implements indefinite causal order via a coherent control of the orderings of multiple quantum operations, leading to various advantages and applications, one of which being in work extraction where enhancements to daemonic ergotropy is possible. Motivated by recent developments in the connections between non-Markovianity and the quantum switch, we construct a non-Markovian process that reduces to the two-party quantum switch in the fully non-Markovian limit. By controlling the amount of non-Markovianity in the process, we identified two operational regimes with differing behaviors. One has its daemonic ergotropy dependent on the presence and amount of non-Markovianity, achieving the maximum in the quantum switch case of full non-Markovianity. The other regime, however, has no advantages from non-Markovianity. We compare this non-Markovian process with the case of a superposition of independent channels, where two channels are placed in a coherent superposition without indefinite causal order, uncovering the advantages of non-Markovianity. Finally, the conditions required for the production of positive daemonic ergotropy are also derived for the case of fully non-Markovian and fully Markovian limits, where we compare against the conditions required for the superposition of independent channels.
Quantum compositions of channels can grant advantages over classical compositions in quantum information processing or thermodynamical tasks. One of such quantum composition is one with indefinite causal order, where multiple quantum channels are placed in a superposition of different causal or operation orders. The advantages of indefinite causal order were demonstrated in an experimentally realizable setup known as the quantum switch and were often attributed to this new quantum resource of indefinite causality, but some doubts and disagreements still remains as compositions without indefinite causal order can achieve the same advantages. In this work, we proposed that the quantum switch's uniqueness lies in its ability to exhibit coherent non-Markovianity, where memory effects acts across different paths of superposition. We showed that the quantum switch violates completely positive or CP-divisibility and thus has intrinsic non-Markovianity in its evolution, which allows for non-Markovian backflow of information from the environment. This backflow of information can then play a role in the many advantages seen in the quantum switch, and by extending the quantum switch operation such that the non-Markovianity is controllable, we showed that these advantages depend on the presence and amount of non-Markovianity, achieving the quantum switch case when the system is fully non-Markovian. Particularly, we demonstrated this for the quantum switch's advantages in communication capacities, work extraction via daemonic ergotropy, and refrigeration via heat extraction. Other quantum compositions that are often compared to indefinite causal order, such as the superposition of independent channels and superposition of trajectories, are also captured by this extended construction of the quantum switch, and we showed that their difference with the quantum switch lies in the presence or absence of non-Markovianity. Our work suggests that non-Markovianity is an important recipe for the quantum switch or indefinite causal order, and is a nonnegligible factor for its advantages.
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.
If two identical copies of a completely depolarizing channel are put into a superposition of their possible causal orders, they can transmit nonzero classical information. Here we study how well we can transmit classical information with N depolarizing channels put in superposition of M causal orders via a quantum switch. We calculate the Holevo quantity if the superposition uses only cyclic permutations of channels and find that it increases with M and it is independent of N. For a qubit it never reaches 1 if we are increasing M. On the other hand, the classical capacity decreases with the dimension d of the message system. Further, for N=3 and 4 we studied the superposition of all causal orders and uniformly superposed causal orders belonging to different cosets created by a cyclic permutation subgroup.
In recent years, quantum battery (QB) models have emerged as crucial platforms for exploring the connections between quantum effects and energy within the framework of quantum thermodynamics. Inspired by the physical implementation of quantum switch (QS), we, in this work, present a simple QB model in which a two-level system as battery, under a series of coherent chargers, performs the indefinite causal order (ICO) charging by introducing a control qubit as QS. Within the energy-preserving collision interaction framework, we analyze the charging ICO and conventional definite causal order (DCO) dynamics, respectively, and discover the underlying connection between the final charged states of battery in ICO and DCO charging modes. That is, the ICO charged state can be understood as a split of the DCO charged state, accompanied by an offset caused by the interference component characterizing the ICO effect. That provides direct theoretical evidence that the ICO effect positively influences the ergotropy (i.e., the maximum extractable work) of QB only when quantum coherence is present in the charging process, and it is independent of the average stored energy of the QB. By combining numerical simulations, we examine that quantum coherence in our model is a strong quantum resource contributing to the ergotropy of QB, while the ICO effect plays a catalyst-like role, assisting quantum coherence in further enhancing the ergotropy of the battery. In addition, we also demonstrate that increasing the number of chargers also benefits the optimal ergotropy of QB. Finally, we reveal that, when the coherent chargers are subject to the dephasing channel, the ICO charging mode is more robust against dephasing noise than the DCO charging mode. This investigation deepens our understanding of quantum coherence and the ICO effect in the context of quantum thermodynamics.