Quantum Anomalous Heat Flow

• Quantum anomalous heat flow is defined as the reversal or redirection of traditional hot-to-cold energy transfer, driven by quantum coherence, correlations, and additional conserved charges.
• It demonstrates that local thermal imbalances can be overridden in closed and mesoscopic systems through engineered quantum interactions, leading to phenomena like energy reversal and anomalous thermal Hall response.
• Experimental and theoretical models, including two-qubit and three-qubit setups, show that manipulating initial quantum correlations and coherence produces measurable deviations from classical heat transport laws.

Quantum anomalous heat flow denotes a family of quantum-thermodynamic transport phenomena in which the naive hot-to-cold direction of heat transfer, or the naive local universality of heat transport, fails once quantum resources or nonclassical transport channels are included. In finite closed systems, the term usually refers to correlation- or coherence-driven reversal of energy exchange between subsystems that are locally thermal, so that a colder subsystem can lose energy and a hotter subsystem can gain it under energy-conserving dynamics. In mesoscopic and topological settings, it also covers anomalous suppression, enhancement, or transverse redirection of heat currents, including nonquantized local edge heat flow and Berry-curvature-driven thermal Hall response. In all of these usages, the effect is not treated as a violation of thermodynamics; rather, the thermodynamic accounting is refined by mutual information, coherence, additional conserved charges, or geometric response functions (Naghdi et al., 2022, Guan et al., 6 Jun 2025, Sukhorukov et al., 25 Oct 2025, Sengupta et al., 2018).

1. Thermodynamic meaning and information-theoretic structure

The standard information-theoretic formulation begins with two subsystems $A$ and $B$ in local Gibbs states, evolving unitarily under an interaction that performs no work. In that setting, uncorrelated initial states satisfy the ordinary Clausius direction,

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$

so if $A$ is colder than $B$, then $A$ absorbs heat. With initial correlations, the inequality is modified to

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$

and a sufficiently negative $\Delta \mathcal{I}(A:B)$ permits $\langle \mathcal{Q}_A\rangle<0$, meaning the colder subsystem becomes colder and the hotter subsystem becomes hotter. A work-inclusive version,

$$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$

makes explicit that correlation consumption can substitute for work as the resource enabling the reversal (Comar et al., 2024, Lipka-Bartosik et al., 2023).

A broader classification for initial states in local equilibrium identifies three resources that can generate anomalous heat transfer: $B$0 Here $B$1 captures correlation consumption, $B$2 captures intrasystem temperature inhomogeneity, and $B$3 captures intrasystem interaction energy. For qubit systems with only two-body intersystem interactions, initial quantum coherence is necessary for anomalous heat transfer in the instantaneous regime, so classical correlation alone is insufficient in that setting (Ma et al., 2022).

A later global-local thermodynamic formulation places this within a charge-symmetric framework. For pure energy conservation and initially uncorrelated product Gibbs states, the exact local expression

$B$4

implies a no-go theorem for anomalous energy flow, because both terms on the right-hand side are nonnegative. In systems with multiple conserved charges, however, the corresponding balance law contains additional drag terms from non-energy charges, and these can reverse the sign of the energy flow without any initial correlations (Guan et al., 6 Jun 2025).

The concept also interfaces with a more radical proposal about the definition of quantum heat itself. A 2025 proposal argues that the conventional quantum-transport identification

$B$5

misses a free-energy-flow term $B$6, and that the corrected formula

$B$7

is needed to avoid a low-temperature entropy-flow divergence. This proposal treats the anomaly as a failure of the conventional definition of heat in the quantum limit rather than as a reversal of the hot-to-cold direction (Jimenez-Valencia et al., 31 Aug 2025).

2. Correlation-driven reversal in minimal many-body systems

The cleanest microscopic realizations use locally thermal finite-dimensional systems with energy-conserving exchange dynamics. In a two-qubit formulation, the initial state is written as

$B$8

with $B$9 and $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$0 thermal and $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$1 chosen so that the marginals remain thermal. The short-time criterion for anomalous exchange is $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$2, so the effect requires a nontrivial interplay between initial correlations and the interaction Hamiltonian. In the explicit two-qubit construction, the relevant coherence lies in the $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$3 sector, and the phase mismatch between the interaction and the correlation matrix controls whether the initial energy difference grows or decreases. This provides the basic mechanism behind a correlation-fueled heat pump: mutual information and entanglement are consumed to drive energy from the colder qubit toward the hotter one (Holdsworth et al., 2022).

A minimal many-body extension is the three-qubit spin chain $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$4-$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$5-$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$6, where each qubit has local Hamiltonian

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$7

each reduced state is Gibbsian, and the interaction is a nearest-neighbor Dzyaloshinskii–Moriya coupling

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$8

The dynamics satisfy

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge 0,$$9

so local energy changes are identified as heat. The initial correlations are inserted through nearest-neighbor blocks

$A$0

with $A$1 off-diagonal in the $A$2 sector and $A$3, ensuring that local marginals remain Gibbsian (Naghdi et al., 2022).

This three-site model exhibits several distinct anomalous regimes. In the uncorrelated reference case with $A$4, heat flows classically as $A$5. With suitable negative nearest-neighbor correlations, the entire chain reverses and the observed direction becomes $A$6. Because the system is closed, the reversal is transient and oscillatory rather than relaxational, and the hottest and coldest qubits reach their extrema out of phase. The middle qubit does not merely relay heat; it acts as conduit and temporary storage, which makes the three-body dynamics qualitatively richer than the two-qubit case (Naghdi et al., 2022).

The same geometry also supports local-only reversal and directional pumping. If only one adjacent pair is appreciably correlated, only that local exchange reverses while the other side remains classically directed, albeit modified by the altered state of the middle qubit. If $A$7 but the correlations are unequal, left-right symmetry is broken even though the end temperatures are equal, and heat is preferentially routed through the more strongly correlated side. This is the sense in which unequal initial quantum correlations act as a directional heat pump or routing mechanism (Naghdi et al., 2022).

3. Coherence, contextuality, and operational diagnostics

An important development is the isolation of coherence as a resource distinct from pre-existing system-environment correlations. In a multipartite cascade collision model, a thermal ancilla $A$8 sequentially interacts with system spins $A$9 through

$B$0

with the bath particle refreshed after each collision. The system and bath begin in a product state, but the system carries internal coherence between degenerate levels,

$B$1

Only zero-mode coherence between degenerate sectors affects heat flow. The subsystem heat current takes the form

$B$2

with the one-way coherence contribution

$B$3

and the associated apparent temperature

$B$4

In this framework, heat flows from higher apparent temperature to lower apparent temperature, and the phase dependence enters through $B$5: $B$6 enhances ordinary flow, $B$7 reverses it, and later spins in the cascade show the strongest effect (Huang et al., 1 Jan 2026).

A complementary foundational result links short-time anomalous heat flow to generalized contextuality. For resonant two-qubit interactions with thermal marginals, the average heat contains a thermal population term of order $B$8 and a coherence-sensitive term of order $B$9. The noncontextuality inequalities derived for sequential prepare-transform-measure scenarios constrain any noncontextual model by a bound scaling as $A$0, so whenever anomalous heat flow appears in the interval $A$1, the data must violate a noncontextuality inequality. In the Micadei et al. parameter regime, the estimated critical time is

$A$2

and analogous conclusions extend to a two-qutrit partial-SWAP model, showing that the effect is not a peculiarity of two-qubit kinematics (Comar et al., 2024).

Operationally, these ideas have been probed on gate-based quantum hardware using mid-circuit measurements. In a two-qubit IBM implementation, the quantum heat $A$3 is compared with a two-point-measurement semiclassical reference $A$4, and the witness

$A$5

detects violation of the semiclassical bound. A positive $A$6 witnesses negativity in the real part of the underlying Kirkwood–Dirac quasiprobability distribution. Mid-circuit measurements are essential because they implement the two-point-measurement protocol, but they also introduce disturbance and energy leakage, which the paper models explicitly as a measurement-induced population bias (Mallik et al., 2024).

4. Fundamental limits, catalysts, and causal-structure extensions

The correlation-driven picture does not imply that every initial correlation is operationally available. For closed, reversible, energy-preserving dynamics on a bipartite system $A$7, the exact optimal anomalous energy transfer is obtained by minimizing the final local energy over unitaries satisfying $A$8. The resulting theorem shows that only the block-diagonal components of the state inside degenerate total-energy subspaces can affect local energy transfer; coherences between nondegenerate total-energy sectors are inert under strictly energy-preserving dynamics. The optimal protocol is a blockwise passive rearrangement that places larger eigenvalues on basis states with lower local energy in the target subsystem. This gives a tight fundamental bound $A$9 for noncatalytic anomalous energy flow (Lipka-Bartosik et al., 2023).

The same work shows that catalysts can exceed this noncatalytic bound. An ancillary system $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$0 is introduced with exact return condition

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$1

and the catalytic process can unlock correlations that were inaccessible in the bare $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$2 dynamics, including correlations associated with nondegenerate sectors. In a Tavis–Cummings implementation with two atoms and a cavity, the catalytic protocol yields $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$3, and in regimes where $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$4, catalysis can activate anomalous energy flow altogether (Lipka-Bartosik et al., 2023).

A different result, however, proves that catalysis does not rescue anomalous energy flow when the initial state is an uncorrelated product Gibbs state and only energy is conserved. In that case, the exact local identity

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$5

already forbids AEF, and the corresponding catalytic analysis shows that neither strict nor correlating catalysts can make the final quantity $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$6 negative. The same paper then shows how this no-go result is circumvented in systems with multiple conserved charges: normal transport of a non-energy charge can drag the energy flow against the temperature gradient, yielding AEF without any initial correlations. The generalized criterion for anomalous charge flow of $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$7 is

$$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$8

and the mechanism is again a competition between information production and cross-charge transport (Guan et al., 6 Jun 2025).

A further extension replaces correlation or charge drag by indefinite causal order. In a quantum switch of two identical thermalizing channels $$(\beta_A-\beta_B)\,\langle \mathcal{Q}_A\rangle \ge \Delta \mathcal{I}(A:B),$$9 and $\Delta \mathcal{I}(A:B)$0, the control qubit is prepared in a superposition of orders and then measured in the $\Delta \mathcal{I}(A:B)$1 basis. The conditioned system states

$\Delta \mathcal{I}(A:B)$2

can exhibit branch-conditioned anomalous heat flow even though the channels are both at temperature $\Delta \mathcal{I}(A:B)$3. For balanced control, the $\Delta \mathcal{I}(A:B)$4 branch can heat a system using colder channels when $\Delta \mathcal{I}(A:B)$5, while the $\Delta \mathcal{I}(A:B)$6 branch can cool a system using hotter channels in a complementary window. Averaged over control outcomes, ordinary thermodynamics is recovered. The same branch-conditioned effect is then used to construct a quantum Otto cycle that simultaneously refrigerates and outputs work, with proof-of-principle photonic simulation (Xue et al., 6 Nov 2025).

5. Chiral, topological, and locally nonuniversal heat transport

A second major usage of the term concerns heat carried by ballistic or chiral quantum channels. The baseline is the universal quantum limit for a single perfectly transmitting channel,

$\Delta \mathcal{I}(A:B)$7

measured in integer quantum Hall edge channels using noise thermometry. This establishes the single-channel benchmark against which anomalous chiral heat transport is interpreted (Jezouin et al., 2015).

Once realistic edge environments are included, however, local heat flow need not match the naive quantized value. In equilibrium drift-diffusion models of integer quantum Hall edges coupled nonlocally to reservoirs or chirality-breaking diffusive channels, a local probe can detect a heat flux larger than the universal value

$\Delta \mathcal{I}(A:B)$8

even though the full cross section still carries only the equilibrium amount allowed by thermodynamics. The anomalous enhancement is traced to interaction-induced correlations and backaction, and the effect survives when dissipative reservoirs are replaced by energy-conserving mesoscopic capacitors (Stäbler et al., 2022).

The converse anomaly is the “missing heat flux” problem at quantum Hall edges. For a chiral bosonic edge mode,

$\Delta \mathcal{I}(A:B)$9

but coupling to a disordered compressible strip generates a correction

$\langle \mathcal{Q}_A\rangle<0$0

In a gapped local dielectric regime this yields a universal negative $\langle \mathcal{Q}_A\rangle<0$1 correction,

$\langle \mathcal{Q}_A\rangle<0$2

while a hydrodynamic strip produces a crossover to $\langle \mathcal{Q}_A\rangle<0$3 and $\langle \mathcal{Q}_A\rangle<0$4 laws with a sign change. The crucial point is that the plasmon-carried heat current can be suppressed while the total heat current of the edge-plus-strip system remains conserved; the apparent deficit is interpreted as induced energy flow in the strip, i.e. heat drag between chiral and nonchiral modes (Sukhorukov et al., 25 Oct 2025).

A topological-transverse version appears in Floquet-engineered $\langle \mathcal{Q}_A\rangle<0$5-$\langle \mathcal{Q}_A\rangle<0$6 borophene. Off-resonant circularly polarized light generates a mass term

$\langle \mathcal{Q}_A\rangle<0$7

which produces Berry curvature

$\langle \mathcal{Q}_A\rangle<0$8

In a longitudinal temperature gradient, this yields an anomalous transverse thermal Hall conductivity

$\langle \mathcal{Q}_A\rangle<0$9

whose sign is reversed by flipping the optical helicity. Here “anomalous heat flow” means Berry-curvature-driven transverse heat transport in zero magnetic field rather than reversal of hot-to-cold exchange (Sengupta et al., 2018).

6. Nonequilibrium devices, heat pumping, and mediated transport

A third usage concerns driven mesoscopic devices in which heat currents reverse relative to naive thermal intuition under nonequilibrium biasing. In a single-level quantum dot with

$$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$0

and aligned driving $$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$1, $$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$2, reverse heat transport is identified by

$$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$3

In the relevant regime, particles still move left to right, $$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$4, but because the level lies below the chemical potentials, the right reservoir loses heat. The reversible energy level

$$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$5

separates absolute negative mobility from reverse heat transport, and the entropy production

$$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$6

remains nonnegative, so the second law is respected (Zhang et al., 2024).

The correlation-driven two-qubit reversal mechanism can also be converted into an explicit thermal machine. In the isolated stage, entangled locally thermal qubits exhibit anomalous energy exchange through the coherent $$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$7 block. When each qubit is then attached to its own bath, the hotter qubit becomes overexcited relative to its bath and the colder qubit underexcited, so the device pumps heat from the cold bath to the hot bath. The performance is assessed using an efficiency notion based on mutual information, and the pump is interpreted as being driven entirely by quantum correlation as fuel (Holdsworth et al., 2022).

Related thermocoherent transport appears in repeated-interaction models. A target qubit bombarded by two-qubit projectiles with heat-exchange coherence $$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$8 obeys a heat current

$$\Delta E_A(\beta_A-\beta_B)\ge \Delta \mathcal{I}-\beta_B W,$$9

At equal initial system and bath temperatures, the ordinary thermal term vanishes and the remaining coherence term drives a genuine heat current. In the high-temperature regime this leads to quantum Onsager relations and coherent analogs of Peltier and Seebeck effects (Pusuluk et al., 2020).

Finally, anomalously large longitudinal heat flow can also arise when heat transfer is mediated by a quantum degree of freedom. In a generalized quantum Langevin treatment of two reservoirs coupled through a harmonic mediator, the exact steady-state current depends nontrivially on the coupling strength and bath cutoff, and the model was used to explain an STM tip–substrate heat current roughly ten orders of magnitude larger than blackbody radiation through a CO-mediated channel. Here the anomaly is not directional reversal but unexpectedly large quantum-mediated heat transport (Panasyuk et al., 2012).

Taken together, these results establish that quantum anomalous heat flow is not a single mechanism but a structured class of phenomena. Depending on context, it may be driven by correlation depletion, degenerate coherence, internal interaction energy, temperature inhomogeneity, additional conserved charges, indefinite causal order, nonlocal edge backaction, Berry curvature, or externally biased energy filtering. The unifying statement is that microscopic heat transport is not determined by local temperatures alone. Global state structure, conservation laws, and the operational definition of the transported quantity can all be thermodynamically decisive.