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arXiv:2511.04028

Anomalous heat flow and quantum Otto cycle with indefinite causal order

Qing-Feng Xue, Qi Zhang, Xu-Cai Zhuang, Yun-Jie Xia, Enrico Russo, Giulio Chiribella, Rosario Lo Franco, Zhong-Xiao Man

quant-phSubmitted Nov 6, 2025• Announced Nov 7, 2025• 0 citations• Revised Nov 7, 2025 (v2)
v1 · Nov 6, 2025v2 · Nov 7, 2025
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Abstract

The principle that heat spontaneously flows from higher temperature to lower temperature is a cornerstone of classical thermodynamics, often assumed to be independent of the sequence of interactions. While this holds true for macroscopic systems at equilibrium, here we show that, when the order of interactions between two identical thermalization channels is indefinite, an anomalous heat flow emerges, whereby heat can sometime flow from a colder entity to a hotter one. Taking advantage of this anomalous heat flow, we design a quantum Otto cycle with indefinite causal order, which not only achieves refrigeration but also generates work. The anomalous heat flow and the quantum Otto cycle are experimentally simulated in a photonic quantum setup, which provides a proof-of-principle demonstration of the theory.

Comments
6 + 7 pages, 6 + 6 figures

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The principle that heat spontaneously flows from higher temperature to lower temperature is a cornerstone of classical thermodynamics, often assumed to be independent of the sequence of interactions. While this holds true for macroscopic systems at equilibrium, here we show that, when the order of interactions between two identical thermalization channels is indefinite, an anomalous heat flow emerges, whereby heat can sometime flow from a colder entity to a hotter one. Taking advantage of this anomalous heat flow, we design a quantum Otto cycle with indefinite causal order, which not only achieves refrigeration but also generates work. The anomalous heat flow and the quantum Otto cycle are experimentally simulated in a photonic quantum setup, which provides a proof-of-principle demonstration of the theory.
In classical thermodynamics, heat must spontaneously flow from hot to cold systems. In quantum thermodynamics, the same law applies when considering multipartite product thermal states evolving unitarily. If initial correlations are present, anomalous heat flow can happen, temporarily making cold thermal states colder and hot thermal states hotter. Such an effect can happen due to entanglement but also because of classical randomness, hence lacking a direct connection with nonclassicality. In this work, we introduce scenarios in which anomalous heat flow have a direct link to nonclassicality, defined as the failure of noncontextual models to explain experimental data. We start by extending known noncontextuality inequalities to a setup in which sequential transformations are considered. We then show a class of quantum prepare-transform-measure protocols, characterized by a time interval ( 0 , τ c ) for a given critical time τ c , where anomalous heat flow happens only if a noncontextuality inequality is violated. We also analyze a recent experiment from and find the critical time τ c based on the authors’ experimental parameters. We conclude by investigating heat flow in the evolution of two-qudit systems, showing that our findings are not an artifact of using two-qubit systems.
In the classical world, physical events always happen in a fixed causal order.However, it was recently revealed that quantum mechanics allows events to occur with indefinite causal order (ICO).In this study, we use an optical quantum switch to experimentally investigate the application of ICO in thermodynamic tasks.Specifically, we simulate the working system interacting with two identical thermal reservoirs in an ICO, observing the quantum heat extraction even though they are in thermal equilibrium where heat extraction is inaccessible by traditional thermal contact.Using such a process, we simulate an ICO refrigeration cycle and investigate its properties.We also show that by passing through the ICO channel multiple times, one can extract more heat per cycle and thus obtain a higher refrigeration performance.Our results suggest that the causal nonseparability can be a powerful resource for quantum thermodynamic tasks.
We propose a thermodynamic refrigeration cycle which uses indefinite causal orders to achieve nonclassical cooling. The cycle cools a cold reservoir while consuming purity in a control qubit. We first show that the application to an input state of two identical thermalizing channels of temperature T in an indefinite causal order can result in an output state with a temperature not equal to T. We investigate the properties of the refrigeration cycle and show that thermodynamically, the result is compatible with unitary quantum mechanics in the circuit model but could not be achieved classically. We believe that this cycle could be implemented experimentally using tabletop photonics. Our result suggests the development of a new class of thermodynamic resource theories in which operations are allowed to be performed in an indefinite causal order.
Abstract The quantum effect plays an important role in quantum thermodynamics, and recently the application of an indefinite causal order to quantum thermodynamics has attracted much attention. Based on two trapped ions, we propose a scheme to add an indefinite causal order to the isochoric cooling stroke of an Otto engine through reservoir engineering. Then, we observe that the quasi-static efficiency of this heat engine is far beyond the efficiency of a normal Otto heat engine and may reach one. When the power is its maximum, the efficiency is also much higher than that of a normal Otto heat engine. This enhancement may originate from the non-equilibrium of the reservoir and the measurement on the control qubit.
The quantum switch has seen multiple applications in quantum information and thermodynamical tasks. As it is constructed by placing two quantum channels in a coherent superposition of alternating causal orders, a composition known as indefinite causal order, these enhancements are often attributed to this indefinite causality and coherent superposition. However, as recent works have shown that the quantum switch also features non-Markovian effects that can contribute to the enhancement of communication capacities and work extraction, we attempt to show in this work that these non-Markovian effects can also enhance heat extraction tasks. In particular, we compare the quantum switch to the superposition of independent channels where two quantum channels are placed in a superposition, which have no non-Markovian effects, and show that the quantum switch can only outperform the superposition of independent channels in the prethermalization regimes, which also depends on the presence and amount of non-Markovianity. Our work reveals that positive heat extraction is still possible even when the working body is at a higher temperature than the interacting baths, allowing us to construct a refrigeration cycle utilizing this feature.
The quantum Otto cycle serves as a bridge between the macroscopic world of heat engines and the quantum regime of thermal devices composed from a single element. We compile recent studies of the quantum Otto cycle with a harmonic oscillator as a working medium. This model has the advantage that it is analytically trackable. In addition, an experimental realization has been achieved, employing a single ion in a harmonic trap. The review is embedded in the field of quantum thermodynamics and quantum open systems. The basic principles of the theory are explained by a specific example illuminating the basic definitions of work and heat. The relation between quantum observables and the state of the system is emphasized. The dynamical description of the cycle is based on a completely positive map formulated as a propagator for each stroke of the engine. Explicit solutions for these propagators are described on a vector space of quantum thermodynamical observables. These solutions which employ different assumptions and techniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables the study of finite time cycles limiting time on the adiabatic and the thermalization times. Explicit finite time solutions are found which are frictionless (meaning that no coherence is generated), and are also known as shortcuts to adiabaticity.The transition from frictionless to sudden adiabats is characterized by a non-hermitian degeneracy in the propagator. In addition, the influence of noise on the control is illustrated. These results are used to close the cycles either as engines or as refrigerators. The properties of the limit cycle are described. Methods to optimize the power by controlling the thermalization time are also introduced. At high temperatures, the Novikov–Curzon–Ahlborn efficiency at maximum power is obtained. The sudden limit of the engine which allows finite power at zero cycle time is shown. The refrigerator cycle is described within the frictionless limit, with emphasis on the cooling rate when the cold bath temperature approaches zero.