Dissipated Work in Quantum Systems

• Dissipated work in quantum systems is defined as the irrecoverable energy loss from nonadiabatic processes, measurement backaction, and coherence-induced effects.
• Fluctuation theorems like the Jarzynski equality and Crooks relation mathematically connect dissipated work with entropy production and free-energy differences.
• Understanding dissipated work is crucial for optimizing quantum thermal machines, where minimizing quantum friction and non-Markovian losses enhances performance.

Dissipated Work in Quantum Systems

Dissipated work in quantum systems refers to the portion of the total energetic input that does not translate into useful work but is instead irreversibly lost—typically as heat—due to quantum and statistical fluctuations, nonadiabatic effects, quantum friction, system-bath correlations, and measurement backaction. Unlike in classical macroscopic systems, where fluctuations and irreversibility are dominated by thermal effects, dissipated work at the quantum scale arises from a richer interplay of coherence, measurement disturbance, and the structure of quantum open-system dynamics. Understanding and quantifying dissipated work is fundamental for designing and evaluating quantum engines, refrigerators, and information-processing devices, as well as for exploring the foundations of quantum statistical mechanics (Millen et al., 2015).

1. Quantum Work Definitions and Fluctuation Theorems

Quantum work is not a Hermitian observable; it is operationally defined via statistical protocols, most notably the two-point projective measurement (TPM) scheme. For a closed quantum system initialized in a Gibbs state $\rho_0 = e^{-\beta H(0)}/Z_0$ and subjected to unitary evolution under a time-dependent Hamiltonian %%%%1%%%%, work is assigned by measuring $H(0)$ at $t=0$, evolving unitarily, and then measuring $H(\tau)$ at $t=\tau$. The work value for a single trajectory is $W = E_m^\tau - E_n^0$, with joint probability $P(n,m)$ determined by the initial thermal population and the quantum transition amplitude.

Quantum fluctuation theorems constrain the distribution of dissipated work. The quantum Jarzynski equality,

$$\langle e^{-\beta W} \rangle = e^{-\beta \Delta F},$$

where $\Delta F$ is the free-energy difference, is valid under the TPM scheme both in Hermitian and certain non-Hermitian systems (the latter when the spectrum is real) (Gardas et al., 2015). The Crooks relation,

$\frac{P_F(W)}{P_R(-W)} = e^{\beta (W - \Delta F)},$

links the forward and reverse work distributions and provides a direct connection between the probability of realizing negative (dissipative) work and the free-energy change. These relations tie average dissipated work to entropy production:

$$\langle W_\text{diss} \rangle = \langle W \rangle - \Delta F \geq 0,$$

with equality only for quasistatic (reversible) protocols (Millen et al., 2015).

2. Dissipation Mechanisms: Quantum Origins

Quantum systems possess irreducible sources of fluctuations and dissipation distinct from classical stochasticity. Chief among these are:

• Quantum Measurement Backaction: Any projective energy measurement disturbs the state, introducing additional fluctuations in work and heat statistics due to non-commutativity and wavefunction collapse.
• Coherence-Induced Dissipation: The presence of off-diagonal elements in the energy eigenbasis leads to so-called “quantum friction.” Nonadiabatic driving of coherent superpositions generates excess work required to return the state to equilibrium, with the difference being dissipated (Millen et al., 2015). Off-diagonal coherences can contribute to the entropy production and thus to dissipated work, especially during fast driving or nonideal cycles.
• Entanglement and System-Bath Correlations: Strong coupling to a bath produces entanglement between system and environment, which can enhance or suppress dissipation mechanisms. In open systems, the energy change not due to unitary evolution (so not stored as work) is identified as heat (Esposito et al., 2014). The resulting dissipated work under irreversible transformations appears in the energy balance through the heat term.
• Nonadiabatic and Non-Markovian Effects: Fast driving, lack of separation between system and bath timescales, and strong memory effects result in irreversibility and extra dissipation, causing deviations from ideal work extraction. For instance, quantum friction arises whenever the system is driven faster than its internal relaxation timescale, and the resulting excitations are ultimately dissipated as heat (Millen et al., 2015, Esposito et al., 2014).

3. Operational Decomposition and Resource-Theoretic Perspectives

The energetic balance of quantum processes is captured by a first-law-like relation, where various contributions can be precisely separated:

$\Delta U = W + Q,$

with $W$ the work done (typically associated with changes in $H(t)$), and $Q$ the heat exchanged, identified with system-bath energy changes. In a more refined operational approach, the change in internal energy can be split into ergotropy (extractable work by unitary transformations), adiabatic work, and heat as associated with population changes of the so-called passive state (Binder et al., 2014): | Term | Operational Meaning | Transformation | |-----------------|----------------------------------------|------------------------| | Ergotropy | Cyclic unitary work (coherent/noneq.) | $\rho \mapsto \rho'$ | | Adiabatic Work | Level change w/o population reshuffle | $H \to H'$ (adiabatic) | | Heat (Q) | Irreversible energy exchange (pop.) | Pop. mixing via CPTP |

Dissipated work is directly associated with the nonzero heat and decrease in ergotropy during nonadiabatic and CPTP (possibly non-unital) maps. Resource-theoretic treatments further identify dissipated work as the part not extractable as deterministic or single-shot work, and link it to entropy production via the monotonicity of the free energy under thermal operations (Binder et al., 2014, Vinjanampathy et al., 2015).

4. Quantum Thermal Machines and Dissipation

Quantum heat engines and refrigerators operate via cycles in which quantum working media exchange energy with hot/cold baths, subject to explicit driving protocols. Dissipated work limits engine efficiency and sets the irreversible entropy production per cycle. In the quantum Otto cycle, for example, nonadiabatic transitions (quantum friction) during expansion/compression strokes generate dissipated work, lowering efficiency below the ideal value $\eta = 1 - \omega_1/\omega_2$ (Millen et al., 2015). Shortcut-to-adiabaticity schemes and reservoir engineering can minimize but not eliminate dissipated work.

For open-system engines described by Lindblad master equations or nonequilibrium Green's function methods, entropy production is quantified by the difference between the actual heat flow and the change in system entropy, and is strictly nonnegative (Esposito et al., 2014, Millen et al., 2015). The degree of dissipated work can serve as a figure of merit for thermodynamic performance and the approach to reversibility.

5. Information, Coherence, and Dissipation

In quantum information thermodynamics, dissipated work is closely connected to the cost of logical operations, erasure, and feedback protocols. Quantum versions of Landauer's principle show that erasure of a qubit generates a minimum of $k_B T \ln 2$ of dissipated heat, a lower bound that is achieved only for reversible operations (Millen et al., 2015). Measurement-induced dissipation—through projective operations and quantum feedback—contributes to the overall irreversibility and is now actively investigated as a source of thermodynamic cost at the quantum level.

Moreover, in protocols that exploit quantum correlations (e.g., in Maxwell demon scenarios), information gain and feedback reduce average dissipated work, bounded by generalized second laws incorporating information-theoretic terms (e.g., mutual information), but never violating the total nonnegativity of entropy production (Millen et al., 2015).

6. Outlook and Open Problems

Despite significant progress in quantifying and mitigating dissipated work in quantum systems, several open questions persist:

• Precise definitions and operational detection of dissipated work in strongly coupled or non-Markovian regimes remain challenging.
• The interplay of entanglement, coherence, and quantum correlations with dissipation is only partially characterized.
• The extension of fluctuation theorems and work statistics to nonprojective measurements and arbitrary quantum channels is ongoing.

Investigating dissipated work remains central to advancing quantum thermodynamics, both for deepening fundamental understanding and for optimizing quantum technologies approaching the ultimate limits of control and energy efficiency (Millen et al., 2015).