Quasiprobability Work Distribution

• Quasiprobability distributions of work are defined as quantum-statistical frameworks that incorporate initial coherence and allow negative or complex values absent in classical probability.
• They preserve linearity in the initial state and reproduce the correct average energy change while reducing to the TPM scheme for incoherent states.
• Operational approaches using characteristic and Wigner functions enable measurement of these distributions, linking their negativity to contextuality and quantum advantage.

The quasiprobability distribution of work is a quantum-statistical description of work fluctuations for driven processes in which the initial state may possess coherence in the energy basis. In contrast with the two-projective-measurement (TPM) scheme, which yields a genuine probability distribution but destroys such coherence at the initial measurement, a work quasiprobability is designed to remain linear in the initial state, reproduce the average energy change of the unmeasured process, and reduce to TPM for incoherent initial states, at the cost of allowing negative or, in some formulations, complex values (Francica, 2021, Francica, 2022).

1. Conceptual origin

In classical thermodynamics, work fluctuations are defined from trajectories and admit an ordinary probability distribution. In quantum mechanics, by contrast, work is not associated with a universally accepted Hermitian operator, while the natural identification of average work with the average energy change,

$$\langle w\rangle=\operatorname{Tr}\!\big[(H^{(H)}(\tau)-H(0))\rho_0\big],$$

does not by itself fix a fluctuation theory when the initial state has coherence in the eigenbasis of $H(0)$ (Francica, 2021).

The standard TPM construction measures $H(0)$, evolves unitarily, and measures $H(\tau)$. For initial states diagonal in the initial energy basis, TPM is non-invasive and its average coincides with the physical energy change. For coherent initial states, however, the first projective measurement replaces $\rho_0$ by its dephased state $\Delta(\rho_0)$, so TPM describes work for the dephased preparation rather than for the actual coherent state (Francica, 2021).

This obstruction is formalized by a no-go theorem associated with Perarnau-Llobet and collaborators: there is no scheme that yields a genuine probability distribution of work which is linear in $\rho_0$, gives the correct average work, and reduces to TPM for incoherent states, unless one allows quasiprobabilities (Francica, 2021). A quasiprobability distribution of work is therefore not an optional reformulation but the standard response to the incompatibility between linearity, TPM consistency, and coherence-sensitive energetics.

2. General families and axiomatic characterizations

A broad class of work quasiprobabilities is obtained for a driven Hamiltonian $$H(t)=\sum_k \epsilon_k(t)\lvert \epsilon_k(t)\rangle\langle \epsilon_k(t)\rvert$$ and unitary evolution $U_{\tau,0}$ by

$$p_q(w)=\sum_{k,j,i}\operatorname{Re}\!\Big\{\langle \epsilon_i\vert\rho_0\vert\epsilon_j\rangle \langle \epsilon_j\vert U_{\tau,0}^\dagger\vert\epsilon_k'\rangle \langle \epsilon_k'\vert U_{\tau,0}\vert\epsilon_i\rangle\Big\} \delta\!\big(w-\epsilon_k'+q\epsilon_i+(1-q)\epsilon_j\big),$$

with real parameter $H(0)$0 and symmetry $H(0)$1 (Francica, 2021). Every member of this family is normalized, yields the correct average work, and has a $H(0)$2-independent second moment,

$H(0)$3

while higher moments depend on $H(0)$4 (Francica, 2021).

For incoherent initial states, only diagonal matrix elements survive and $H(0)$5 collapses exactly to TPM, independently of $H(0)$6 (Francica, 2021). In this sense, the family interpolates continuously among several earlier definitions. The choices $H(0)$7 and $H(0)$8 recover the Allahverdyan construction, while $H(0)$9 reproduces the Solinas–Gasparinetti definition (Francica, 2021).

A complementary result shows that, under a quasi-Gleason framework together with three work conditions—TPM reduction for incoherent states, correct first moment, and correct second moment—the most general real quasiprobability of work is precisely the class $H(0)$0, together with affine combinations of such distributions (Francica, 2022). This establishes $H(0)$1 not merely as an example but as a characterization theorem for real, linear, TPM-compatible work quasiprobabilities.

A different selection principle singles out one particular member. When one adds the requirements of time-reversal symmetry, positivity of the second-order moment, and a support condition restricting work values to energy differences $H(0)$2, the only admissible definition is the Margenau–Hill quasiprobability of work (Pei et al., 2023).

Family	Defining feature	Status
$H(0)$3	One-parameter real quasiprobability class	Correct first and second moments; TPM for incoherent states (Francica, 2021)
Margenau–Hill	Symmetric ordering, effectively $H(0)$4 under extra support constraints	Unique under added requirements (Pei et al., 2023)
Kirkwood–Dirac	Ordered product, generally complex	Related but not real in general (Gherardini et al., 2024)

The time-reversal problem sharpens this distinction. For the $H(0)$5 family, a formal symmetry $H(0)$6 can be defined, but compatibility between formal and operational time reversal while retaining TPM reduction survives only for $H(0)$7 and $H(0)$8 (Francica, 2023). This reinforces the special role of the Margenau–Hill-type endpoints within the broader class.

3. Initial coherence, fluctuation relations, and generalized thermodynamics

Initial coherence enters $H(0)$9 exclusively through off-diagonal matrix elements $H(\tau)$0 in the initial energy basis. These produce interference terms weighted at shifted work values

$H(\tau)$1

and are the origin of quasiprobabilistic negativity (Francica, 2021).

The coherence content can be quantified by the relative entropy of coherence,

$H(\tau)$2

and promoted to a stochastic variable through a coherence distribution $H(\tau)$3 whose average equals $H(\tau)$4 and satisfies $H(\tau)$5 (Francica, 2021). The same work then introduces a joint quasidistribution $H(\tau)$6 whose marginals reproduce the work quasiprobability and the coherence distribution.

This joint object yields fluctuation relations that explicitly incorporate coherence. For work alone,

$H(\tau)$7

and for the joint work–coherence quasidistribution,

$H(\tau)$8

When the initial populations are thermal, this becomes a Jarzynski-type equality,

$H(\tau)$9

from which follows the generalized second law

$\rho_0$0

(Francica, 2021).

Related formulations preserve this thermodynamic role of coherence while changing the operational starting point. The operational quasiprobability (OQ) reproduces the Jarzynski equality and yields average work consistent with the classical definition; for coherent initial states it modifies the Jarzynski relation by an explicit coherence-dependent factor and splits the work difference relative to TPM into a free-energy contribution plus a term proportional to the relative entropy of coherence (Jae et al., 5 Oct 2025). This suggests that coherence is not merely a correction to work statistics but part of the irreversible bookkeeping of quantum thermodynamic transformations.

4. Characteristic functions, measurement schemes, and alternative representations

For the class $\rho_0$1, the characteristic function

$\rho_0$2

admits an operator expression involving symmetrized insertions of $\rho_0$3 and $\rho_0$4, and its moments are obtained by differentiation at $\rho_0$5 (Francica, 2021). A detector-based scheme measures $\rho_0$6 without direct projective energy measurements by coupling the system to an ancilla observable $\rho_0$7 through impulsive interactions at the beginning and end of the protocol. The detector coherences then encode the traces entering $\rho_0$8, so the initial coherence of the system is preserved rather than erased (Francica, 2021).

A different operational construction uses the Wigner function of a continuous-variable apparatus. In the single-measurement setting, the final Wigner function $\rho_0$9 of the apparatus defines a quasiprobability distribution of work in phase space. Its $\Delta(\rho_0)$0-marginal reproduces a Gaussian-convoluted TPM distribution and approaches TPM in the sharp-pointer limit, while fixed-$\Delta(\rho_0)$1 slices encode the full mean energy difference for coherent initial states. Negativity and interference fringes in $\Delta(\rho_0)$2 are directly tied to initial energy-basis coherence (Cerisola et al., 2023).

The Kirkwood–Dirac approach instead defines a generally complex quasiprobability

$\Delta(\rho_0)$3

with work distribution $\Delta(\rho_0)$4 and characteristic function

$\Delta(\rho_0)$5

An interferometric protocol aided by an auxiliary system reconstructs this characteristic function and, by Fourier transformation, the work quasiprobability. This scheme was experimentally demonstrated in an electron–nuclear spin system associated with a nitrogen-vacancy center in diamond (Hernández-Gómez et al., 2024).

The tutorial literature presents these schemes—weak two-point measurements, Ramsey-like interferometry, and detector-assisted methods—as a common operational layer for multitime quasiprobabilities in thermodynamics, including work and heat (Gherardini et al., 2024). Across these representations, the central invariant is that the quasiprobability captures the unperturbed statistics of incompatible observables, rather than the disturbed statistics of a sequential projective protocol.

5. Negativity, contextuality, and work extraction

Negativity is the defining nonclassical feature of a work quasiprobability. For the family $\Delta(\rho_0)$6, individual operator contributions $\Delta(\rho_0)$7 satisfy the bound

$\Delta(\rho_0)$8

so each elementary contribution can be as low as $\Delta(\rho_0)$9 (Francica, 2021). Negative values require both off-diagonal components of $\rho_0$0 and dynamics that mix energy levels, and they preclude an interpretation in terms of a classical distribution over trajectories.

This nonclassicality is closely linked to contextuality. In the general framework of work quasiprobabilities, negativity of $\rho_0$1 indicates that the protocol cannot be represented by a universally noncontextual hidden-variable model of the relevant operational statistics (Francica, 2022). The broader quasiprobability literature treats KD and MH distributions as direct carriers of anomalous weak values and contextuality in multitime measurements (Gherardini et al., 2024).

Recent operational results sharpen this link. For binary unbiased qubit measurements, the real part of the Kirkwood–Dirac quasiprobability and the operational quasiprobability are equivalent, and they are nonnegative if and only if the measurements are jointly measurable (Jae et al., 5 Oct 2025). In that setting, non-joint measurability can increase extractable work beyond the classical bound imposed by jointly measurable measurements, so negativity becomes a witness of a specifically measurement-incompatibility-enabled thermodynamic advantage (Jae et al., 5 Oct 2025).

Negativity, however, is not a faithful scalar resource measure in every setting. In a three-level nitrogen-vacancy-center system, the operational quasiprobability and the Kirkwood–Dirac quasiprobability can display different amounts of negativity while enabling the same work extraction (Jae et al., 5 Oct 2025). A plausible implication is that negativity certifies nonclassicality but does not by itself order thermodynamic usefulness across inequivalent quasiprobability representations.

The same theme appears in quantum batteries. If the symmetric quasiprobability $\rho_0$2 becomes negative over some time interval during the charging process, then quantum advantage in charging is guaranteed (Francica, 23 Mar 2025). In autonomous work-extraction models with an explicit weight, coherent ergotropy extraction can even reduce work fluctuations for appropriate non-classical states of the work reservoir, but total unlocking of coherent energy leads to diverging fluctuations, in sharp contrast with incoherent ergotropy extraction, which can be completed with finite work fluctuations (Łobejko, 2021).

6. Representative models and extensions

Concrete models show that work quasiprobabilities are not restricted to abstract measurement theory. In a driven qubit with Hamiltonian

$\rho_0$3

the coherence contribution to $\rho_0$4 is proportional to $\rho_0$5, where $\rho_0$6 is the initial coherence and $\rho_0$7 encodes the relevant Heisenberg-picture rotation. In the sudden-quench and adiabatic limits, $\rho_0$8, so the quasiprobability collapses to TPM; coherence affects work statistics only at intermediate driving speeds (Francica, 2021).

In the transverse-field Ising model, the thermodynamic limit separates two regimes. For a global quench there exists a symmetric non-contextual representation with a Gaussian probability distribution of work, whereas for a local quench the fourth moment of work can become negative, signaling contextuality (Francica et al., 2023). The same work emphasizes critical behavior near the quantum phase transition and interprets initial coherence as a useful resource (Francica et al., 2023).

Thermal machines offer another arena. In a measurement-based quantum Otto engine with an anisotropic two-spin working substance, a global non-selective measurement in the Bell basis creates coherence in the energy eigenbasis before a unitary work stroke. The full-counting-statistics work distribution then becomes negative for some work values; in a standard Otto engine between two heat baths, where the corresponding states are energy-diagonal, such negativity does not occur (Purkait et al., 2024).

Relativistic quantum thermodynamics generalizes the concept further. A covariant formulation treats work as a four-vector $\rho_0$9 and defines a Wigner–Margenau–Hill quasiprobability whose characteristic function symmetrizes initial and final four-momentum operators while keeping the components at each hypersurface inside a single exponential. This real, not necessarily positive distribution yields the correct average work four-vector, satisfies a covariant Jarzynski equality, and shows that a full Crooks theorem for the joint work four-vector fails in general, while the scalar rest-frame energy component still obeys a Crooks-type relation (Pei et al., 12 Jun 2026).

Taken together, these developments present the quasiprobability distribution of work as a unifying framework for coherence-sensitive work statistics. Its central role is to preserve linearity, correct energy moments, and operational meaning in regimes where ordinary probabilities are excluded by noncommutativity. The resulting negativity is not an anomaly to be removed; it is the signature that the work process genuinely belongs to quantum thermodynamics rather than to a classical stochastic surrogate.