Temporal Quasiprobability Distributions

• Temporal quasiprobability distributions (TQDs) are quasidistributions that encode multi-time quantum correlations with potentially negative or complex values, reflecting noncommuting observables.
• They employ frameworks like Kirkwood–Dirac, Margenau–Hill, and Keldysh formalisms to model sequential events and quantify work statistics in coherent quantum processes.
• TQDs offer operational insights for temporal state tomography and weak measurement schemes, distinguishing classical from quantum temporal correlations.

Searching arXiv for recent and foundational papers on temporal quasiprobability distributions and closely related KD/MH temporal formalisms. Temporal quasiprobability distributions (TQDs) are quasidistributions that aim to play the role of joint probability distributions for outcomes recorded at different times for noncommuting observables. Unlike classical joint probabilities, they can be negative or complex, are linear in the quantum state, have marginals that reproduce the correct single-time probabilities, and encode both the dynamical disturbance due to earlier measurements and the incompatibility between observables and initial state (Bizzarri et al., 2024). Across current formulations, TQDs appear as quasiprobabilities over sequences of events built from ordered operator products, as Kirkwood–Dirac (KD) and Margenau–Hill (MH) temporal distributions, as Keldysh quasiprobability distributions (KQPDs), and as quasiprobability distributions of work associated with two-time energy changes in coherent quantum thermodynamics (Francica, 2023). Recent work further treats informationally complete TQDs as complete descriptions of multi-time quantum processes, thereby linking TQDs directly to temporal state tomography and to broader temporal-state formalisms (Jia, 4 May 2026, Jia et al., 8 Jan 2026).

1. Definition and formal scope

Temporal quasiprobability distributions are defined for quantities associated with more than one time. In the most general event-based formulation, an event is represented not necessarily by a projector but by a general effect $E$, and for two events $E,F$ one defines a real-valued function $v(E,F)$ satisfying reality, normalization against the identity, and linearity in each argument; under sequential continuity this yields

$$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$

For three or more events the same axioms no longer fix a unique joint quasiprobability; the result depends on the grouping or temporal parenthesization of events, so different ordered products such as $EFG$, $FEG$, and $EGF$ can correspond to distinct temporal quasiprobabilities (Francica, 2023).

A complementary general framework is the Keldysh quasiprobability distribution. For observables $\hat A_l$ probed at times $\tau_l$, its characteristic function is written on the Keldysh contour as

$$\Lambda(\boldsymbol{\lambda}) = \mathrm{Tr}\left\{ \hat{\mathcal{T}}_K \exp\left[ -\frac{i}{2}\int_K d\tau\,\sum_l f_l(\tau)\lambda_l\hat A_l(\tau) \right] \hat\rho_0 \right\},$$

and the TQD is the multidimensional Fourier transform of $E,F$0. This construction is measurement-independent, naturally accommodates multi-time problems, and yields moments with a specific operator ordering fixed by the contour structure (Hofer, 2017).

In recent multi-time process formulations, a TQD is a function

$E,F$1

whose marginals reproduce the single-time phase-space representations and which satisfies temporal Kolmogorov consistency. In this sense, TQDs generalize classical consistent families of joint distributions to quasiprobabilistic, multi-time quantum settings (Jia et al., 8 Jan 2026). This suggests that “temporal quasiprobability distribution” is best understood as a structural notion rather than a single unique formula.

2. Principal constructions: KD, MH, and Keldysh forms

A central class of TQDs is obtained by extending the Kirkwood–Dirac quasiprobability to time-ordered processes. For a two-time process $E,F$2, the right two-time KD quasiprobability is

$E,F$3

while the left version is

$E,F$4

These are related by complex conjugation, and their real part defines the temporal Margenau–Hill quasiprobability (Jia et al., 8 Jan 2026).

The same work defines left, right, and doubled temporal KD quasiprobabilities for arbitrary multi-time quantum processes. The doubled construction introduces projectors on both bra and ket sides, and left/right KD distributions arise as marginals of the doubled KD distribution. Their real parts define doubled temporal MH quasiprobabilities. These constructions satisfy temporal Kolmogorov consistency and recover the correct single-time measurement statistics as marginals (Jia et al., 8 Jan 2026).

In the two-time weak sequential setting, the Margenau–Hill quasiprobability for observables $E,F$5 at $E,F$6 and $E,F$7 at $E,F$8 is

$E,F$9

It is the real part of the Kirkwood–Dirac quasiprobability, is linear in $v(E,F)$0, normalized, reproduces $v(E,F)$1 and $v(E,F)$2 as marginals, and can become negative, signalling nonclassicality (Bizzarri et al., 2024).

The KQPD provides a broader temporal language. For $v(E,F)$3 instantaneous probes, its characteristic function can be written as

$v(E,F)$4

and the corresponding KQPD is its Fourier transform. This is a temporal quasiprobability over the values of observables at multiple times, with negativity arising from interference between different forward and backward histories on the Keldysh contour (Hofer, 2017).

3. Temporal quasiprobabilities of work

A major application of TQDs is the statistics of quantum work in the presence of initial coherence in the energy basis. For a thermally isolated system driven by a time-dependent Hamiltonian

$v(E,F)$5

with unitary

$v(E,F)$6

the average work is identified with the average energy change

$v(E,F)$7

For coherent initial states, the two-projective-measurement scheme no longer gives the correct average work without disturbing coherence, so a class of quasiprobability distributions of work is introduced: $v(E,F)$8 This family obeys $v(E,F)$9, gives the correct average work for any $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$0, has a $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$1-independent second moment, and reduces to the standard TPM probability distribution for incoherent initial states (Francica, 2021).

The temporal character is explicit: $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$2 encodes correlations between energy at time $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$3 and energy at time $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$4 through the initial eigenvalues $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$5, final eigenvalues $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$6, and the unitary $$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$7. The associated characteristic function is

$$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$8

which makes clear that work TQDs are built from two-time quantum correlation functions (Francica, 2021).

A related KD formulation defines the quasiprobability of work as

$$v(E,F)=\operatorname{Re}\operatorname{Tr}\{EF\rho\}.$$9

with

$EFG$0

so that the characteristic function becomes

$EFG$1

This construction was experimentally reconstructed in an electron-nuclear spin system associated with a nitrogen-vacancy center in diamond, where the full complex KD distribution of work was accessed interferometrically (Hernández-Gómez et al., 2024).

For many-body dynamics, the quantum Ising model exhibits a sharp distinction between global and local quenches. For a global quench there is a symmetric non-contextual representation with a Gaussian probability distribution of work, whereas for a local quench a negative fourth moment of the work can occur, signalling contextuality (Francica et al., 2023). This suggests that the degree to which a work TQD retains irreducibly quasiprobabilistic structure depends strongly on whether the drive is extensive or localized.

4. Measurement schemes and experimental access

TQDs are operationally accessible through several distinct schemes. In weak sequential measurements, a first projective measurement is replaced by a weak POVM induced through coupling the system to an ancilla pointer via a modular unitary,

$EFG$2

with Kraus operators

$EFG$3

and POVM elements

$EFG$4

The resulting two-time joint outcome statistics are

$EFG$5

which directly links weak measurement data to the Margenau–Hill temporal quasiprobability (Bizzarri et al., 2024).

That scheme was realized experimentally with photonic qubits encoded in polarization, where the first measurement was a weak POVM and the second a projective measurement. The experiment reconstructed weak commensurate and weak Margenau–Hill quasiprobabilities and showed how negativity persists only in an intermediate window of measurement strength (Bizzarri et al., 2024).

A second route is ancilla interferometry. For work quasiprobabilities, a detector can be coupled at the beginning and end of the protocol through

$EFG$6

so that appropriate off-diagonal elements of the detector’s final density matrix encode $EFG$7 (Francica, 2021). Closely related interferometric schemes reconstruct the KD quasiprobability distribution of work from the characteristic function in solid-state spin platforms (Hernández-Gómez et al., 2024).

More generally, temporal KD characteristic functions for arbitrary multi-time processes can be measured interferometrically by embedding the system dynamics into controlled evolutions $EFG$8 and $EFG$9 on a control qubit. Measuring Pauli $FEG$0 and $FEG$1 on the control yields the real and imaginary parts of the characteristic function, from which the temporal KD distribution follows by inverse Fourier transform (Jia et al., 8 Jan 2026).

The most general recent operational statement is that any TQD can be obtained via classical post-processing of measurement outcomes generated by a fixed set of quantum instruments. In this “quantum snapshotting” approach, one performs a fixed time-ordered sequence of instruments, estimates the trajectory distribution $FEG$2, reconstructs the TQD through a linear map

$FEG$3

and then reconstructs the temporal state via a dual-frame expansion (Jia, 4 May 2026).

5. Negativity, contextuality, and temporal nonclassicality

Negativity is the most widely used indicator of TQD nonclassicality. In work quasiprobabilities $FEG$4, negative values arise from off-diagonal elements of $FEG$5 in the initial energy basis together with nontrivial mixing of energy eigenstates under the dynamics. In operator form, the contributions are bounded, with negative terms possible down to $FEG$6, and the resulting negativity is interpreted as temporal nonclassicality analogous to spatial nonclassicality in Wigner functions (Francica, 2021).

A closely related issue is generalized contextuality. For work, a no-go theorem states that one cannot at the same time have a genuine nonnegative probability distribution of work, linearity in the initial state, reduction to TPM for incoherent states, and average work equal to the average energy change. Allowing quasiprobabilities circumvents this by permitting negativity while retaining the other properties (Francica, 2021). In this sense, work TQDs are not optional reformulations but the objects required when initial energy coherence is present.

In weak-value scenarios, the conditional KQPD has support only on the eigenvalue spectrum of the measured observable, so an anomalous weak value lying outside that spectrum requires that the underlying KQPD be negative (Hofer, 2017). In Leggett–Garg settings, if the three-time KQPD were everywhere nonnegative, it would provide a macrorealist, non-invasive hidden-variable model and the Leggett–Garg inequality could not be violated; hence LGI violation requires KQPD negativity (Hofer, 2017).

Recent many-body work sharpened this dynamical viewpoint through the first-time negativity (FTN) of the Margenau–Hill quasiprobability. For sequential local measurements in an interacting Ising chain, the negativity measure

$FEG$7

and the onset time

$FEG$8

track when temporal quasiprobabilities first become genuinely nonclassical. FTN discriminates clearly between interaction-dominated and field-dominated regimes, is systematically reshaped by temperature, and reveals a characteristic spatio-temporal structure when measurements are performed on different sites (Shukla et al., 1 Jan 2026). This suggests that negativity onset is itself a dynamical observable of many-body temporal coherence and operator spreading.

6. Time reversal, temporal states, and tomography

Time reversal in TQDs is subtle. A formal time reversal obtained by transforming states and events gives a symmetry

$FEG$9

provided quasiprobabilities are invariant under the corresponding conjugation and work is odd under time reversal (Francica, 2023). However, an operational time reversal defined by reversing a concrete detector-based measurement protocol can lead to a different backward quasiprobability $EGF$0. These two notions coincide only in special cases such as $EGF$1, whereas for generic $EGF$2 the formal and operational reversals differ (Francica, 2023). This corrects a common misconception that time reversal of a TQD is uniquely fixed by reversing the sign of the temporal variable.

The most developed recent unification treats informationally complete TQDs as complete descriptions of multi-time quantum processes. For a Markovian process

$EGF$3

one introduces a temporal state $EGF$4 on

$EGF$5

with temporal marginals given by partial traces. Informationally complete temporal KD distributions then satisfy a generalized temporal Born rule

$EGF$6

and the temporal state can be reconstructed by a dual-frame expansion

$EGF$7

(Jia, 4 May 2026).

This viewpoint turns TQDs into the basic data structure for temporal state tomography (TST). In that framework, one first estimates a trajectory distribution using a fixed instrument basis, reconstructs the TQD by linear post-processing, and then reconstructs $EGF$8. The sample complexity obeys

$EGF$9

and, when the instrument decomposition is informationally complete with $\hat A_l$0,

$\hat A_l$1

(Jia, 4 May 2026). Temporal KD and MH quasiprobabilities thereby become not merely interpretive tools but operationally complete representations of multi-time dynamics.

7. Conceptual implications and open directions

Several broad implications follow from the current TQD literature. First, TQDs are not restricted to one physical domain. They unify two-time measurement statistics, weak values, fluctuating work, Leggett–Garg-type temporal correlations, and multi-time process representations within a common quasiprobabilistic language (Hofer, 2017). Second, different TQD constructions are not interchangeable. KD, MH, KQPD, commensurate quasiprobabilities, and work-specific $\hat A_l$2-families share a common logic but differ in ordering, operational reconstruction, and in which classical properties are preserved (Bizzarri et al., 2024, Francica, 2021).

Third, current results indicate that nonclassicality in time is not exhausted by single-time quasiprobability negativity. A system may have a classical single-time $\hat A_l$3 while still exhibiting nonclassical two-time behavior in $\hat A_l$4, and conversely a nonclassical single-time $\hat A_l$5 need not imply nonclassical temporal correlations (Alexanian, 2016). This suggests that temporal quasiprobabilities capture a distinct layer of quantum structure.

Fourth, recent unification results imply that many temporal-state formalisms can be viewed as operator representations derived from temporal KD or MH correlators through Bloch tomography. In particular, the doubled KD state coincides with the spatiotemporal doubled density operator, and the two-time MH temporal state coincides with the pseudo-density operator (Jia et al., 8 Jan 2026). A plausible implication is that future debates about “states over time” may increasingly be phrased in terms of which TQD and which operator ordering are operationally natural, rather than which formalism is uniquely correct.

The main limitations remain explicit. Several works focus on closed dynamics, two-time statistics, or finite-dimensional settings (Francica, 2021, Bizzarri et al., 2024). For more than two events, quasiprobabilities are not uniquely fixed by general axioms and depend on temporal grouping (Francica, 2023). Extensions to open quantum systems, full temporal process tensors, continuous-variable settings, and indefinite causal order are identified as open directions (Jia et al., 8 Jan 2026, Jia, 4 May 2026). Even so, the present literature already establishes TQDs as a technically precise and experimentally accessible framework for representing temporal quantum correlations beyond classical stochastic descriptions.