Operational Quasiprobability: Measurement Framework

• Operational quasiprobability is defined from experimentally specified, sequential or selective measurement procedures that allow direct comparison with classical probability models.
• It reveals nonclassical features such as negativity and violation of classical assumptions (e.g., NSIT and joint measurability) in qudit and continuous-variable systems.
• OQ underpins applications in entanglement witnessing, quantum thermodynamics, and quantum scrambling by linking measured statistics to explicit classical benchmarks.

Operational quasiprobability (OQ) denotes a class of quasidistributions defined from operationally specified measurement procedures rather than from a state representation alone. In the canonical formulations for qudits and continuous-variable systems, OQ is constructed from selective or sequential measurements and is designed to be commensurate with a classical probability description, so that negativity has a direct operational meaning: it signals incompatibility with a hidden-variable or macrorealist account satisfying assumptions such as locality, noninvasive measurability, no-signaling-in-time (NSIT), or joint measurability (Ryu et al., 2012, Jae et al., 2017). Subsequent work extended this logic to conjunction measurements of incompatible POVMs, weak-sequential protocols, entanglement witnesses, metrological models, quantum thermodynamic work distributions, and network nonclassicality (Jae et al., 2023, Bizzarri et al., 2024, Ryu et al., 2022, Jae et al., 5 Oct 2025, Onggadinata et al., 2023). In a broader literature, several phase-space quasiprobabilities also acquire operational meaning through POVMs, weak values, Gaussian channels, or resource theory, but those constructions are generally adjacent to OQ rather than identical with it (Harada, 2011, Linowski et al., 2023, Tan et al., 2019).

1. Conceptual basis and commensurability

A defining motivation for OQ is the criticism that standard quasiprobabilities such as the Wigner, Husimi, and Glauber–Sudarshan distributions are often incommensurate with classical probabilities: the same formal integral need not refer to the same physical observation in the quantum and classical descriptions. The qudit formulation therefore introduces OQ as a quasiprobability built from the statistics of selective, consecutive measurements, so that quantum and hidden-variable models can be compared on the same operational footing (Ryu et al., 2012). In this framework, the quasiprobability is not a representation of the state alone; it depends on the state, the chosen observables, and the measurement order.

For discrete systems, the classical benchmark is a hidden-variable model with locality and noninvasive measurability. The paper on qudits states that the quasiprobability function becomes positive semidefinite if consecutive measurement results are described by such a model, and becomes negative otherwise (Ryu et al., 2012). For continuous variables, the analogous benchmark is expressed through NSIT together with the arrow-of-time (AoT) condition. In that case the OQ reduces to the sequential joint probability itself and is nonnegative; negative OQ is therefore a sufficient condition for violating NSIT, though not a necessary one (Jae et al., 2017).

This commensurability requirement distinguishes OQ from state-based phase-space representations. The latter may still be highly useful, but their negativity is not automatically interpretable as failure of a classical model for the same experimentally specified statistics. OQ was introduced precisely to make that comparison operationally explicit (Ryu et al., 2012, Jae et al., 2017).

2. Canonical constructions for qudits and continuous variables

For qudits, the basic construction starts from $K$ possible observables $A_k$, each with $D$ outcomes $a_k\in\{0,1,\dots,D-1\}$, and from a selection string $\mathbf n=(n_1,\dots,n_K)$ indicating which measurements are performed. With $\omega=e^{2\pi i/D}$, the characteristic function is defined by

$\chi(\mathbf{n}) = \mbox{Tr} \left[\mathcal{ T}\prod_{k=1}^K \left(\delta_{n_k,0} \mathcal{ I} + (1-\delta_{n_k,0})\sum_{a_k=0}^{D-1} \omega^{n_k a_k} \mathcal{ A}_k(a_k) \right)\hat{\rho}\right],$

where $$\mathcal A_k(a_k)(\hat\rho)=\hat A_k(a_k)\hat\rho\hat A_k^\dagger(a_k)$$ for general POVM/Kraus measurements, and $\mathcal T$ denotes chronological ordering. The operational quasiprobability is the discrete Fourier transform

$$\mathcal{W}(\mathbf{a}) \equiv\frac{1}{D^K} \sum_{\mathbf{n}=0}^{D-1} \omega^{- \mathbf{a} \cdot \mathbf{n}} \chi(\mathbf{n}),$$

with inverse

$A_k$0

The one-variable marginals reproduce the actual single-measurement probabilities, and the full object is real-valued and normalized (Ryu et al., 2012).

For complementary qudit observables defined by mutually unbiased bases, the construction simplifies to a generalized Bloch form,

$A_k$1

where $A_k$2 is the generalized Bloch vector of the state and $A_k$3 encodes the selected basis elements (Ryu et al., 2012). This formula makes explicit that negativity depends jointly on the state and on the measured observables.

For continuous variables, the construction is formulated for two sequential measurements $A_k$4 at time $A_k$5 and $A_k$6 at time $A_k$7, with four experimental configurations: no measurement, only $A_k$8, only $A_k$9, and the sequential measurement $D$0 then $D$1. Because the outcomes are unbounded, the probability densities are expanded in Hermite polynomials, and the resulting OQ for continuous variables is

$D$2

Its marginals are the single-measurement distributions,

$D$3

and its negative volume is quantified by

$D$4

In the optical realization of the paper, the measurements are coherent-state POVMs, so the relevant examples are quadrature variables of light (Jae et al., 2017).

3. Sequential, weak, and incompatibility-based variants

A major later strand of OQ generalizes the original sequential-measurement logic to incompatible POVMs. For two $D$5-outcome measurements $D$6 and $D$7, together with a conjunction measurement $D$8, the operational quasiprobability is defined directly from measured probabilities by

$D$9

Equivalently, in operator form,

$a_k\in\{0,1,\dots,D-1\}$0

with

$a_k\in\{0,1,\dots,D-1\}$1

The associated HOVM satisfies the marginality relations

$a_k\in\{0,1,\dots,D-1\}$2

so OQ functions as a joint-like model for incompatible measurements (Jae et al., 2023).

A weak-sequential variant appears in the two-time qudit protocol in which the first measurement is a finite-strength POVM and the second remains projective. There the experimentally accessible weak sequential statistics satisfy

$a_k\in\{0,1,\dots,D-1\}$3

linking the observed distribution to the strong TPM distribution $a_k\in\{0,1,\dots,D-1\}$4, the final marginal $a_k\in\{0,1,\dots,D-1\}$5, and the Margenau–Hill quasiprobability $a_k\in\{0,1,\dots,D-1\}$6. The corresponding weak commensurate quasiprobability is

$a_k\in\{0,1,\dots,D-1\}$7

showing explicitly that weak-sequential OQ interpolates between disturbance-based and incompatibility-based descriptions (Bizzarri et al., 2024).

A different weak-value-based line constructs quasiprobabilities from Aharonov weak values of projection operators,

$a_k\in\{0,1,\dots,D-1\}$8

and from the more general $a_k\in\{0,1,\dots,D-1\}$9-parameterized family

$\mathbf n=(n_1,\dots,n_K)$0

This framework is strongly adjacent to OQ because it is operationally tied to weak measurement and postselection, although it is not the canonical commensurate OQ construction of the qudit and continuous-variable papers (Fukuda et al., 2016).

4. Negativity, positivity, and classical benchmarks

In OQ, negativity is not merely a representational feature; it is a statement about the impossibility of a specified classical account. For qudits, positivity follows if the sequential statistics admit a hidden-variable model with locality and noninvasive measurability, while negativity rules that out (Ryu et al., 2012). For continuous variables, NSIT and AoT imply

$\mathbf n=(n_1,\dots,n_K)$1

so $\mathbf n=(n_1,\dots,n_K)$2 is a sufficient witness of NSIT violation (Jae et al., 2017).

In the incompatibility-based metrological construction, positivity is state-dependent: even when the HOVM is not a POVM, there may be states for which all scalar OQ values are nonnegative. In that regime the paper defines the Fisher information of OQ,

$\mathbf n=(n_1,\dots,n_K)$3

and proves that if there exists a state for which $\mathbf n=(n_1,\dots,n_K)$4, then incompatibility of the underlying measurements is necessary (Jae et al., 2023). In other words, incompatibility is the resource, and OQ is the statistical model that makes that resource visible.

A closely related criterion appears in the thermodynamic OQ of work for two-level systems. There the paper proves that the real part of the Kirkwood–Dirac quasiprobability and the OQ are equivalent in two dimensions, and that for binary unbiased measurements they are nonnegative if and only if the measurements are jointly measurable (Jae et al., 5 Oct 2025). This identifies OQ negativity with non-joint measurability in that qubit setting.

These results show that OQ negativity can witness several distinct failures of classicality: noninvasive hidden-variable models, NSIT, joint measurability, or positive joint models for incompatible measurements. The precise benchmark depends on the operational construction.

5. Applications

One application is entanglement witnessing. For two qudits, the marginal quasiprobability

$\mathbf n=(n_1,\dots,n_K)$5

is nonnegative for all separable states, so its negativity is a sufficient entanglement witness (Ryu et al., 2012). A later silicon double quantum dot study used a marginal OQ as an entanglement indicator for two spin qubits under charge noise, calling

$\mathbf n=(n_1,\dots,n_K)$6

the entanglement strength. In that work, the OQ approach required 4 measurement configurations, versus 16 for the negativity/tomographic route, and the pattern in the OQ-driven entanglement strength remained comparatively invariant under exchange-fluctuation noise $\mathbf n=(n_1,\dots,n_K)$7 (Ryu et al., 2022).

A second application is quantum scrambling. The quasiprobability behind an out-of-time-ordered correlator can be reconstructed operationally from ancilla-assisted circuits with three and four sequential measurements. For qubit observables $\mathbf n=(n_1,\dots,n_K)$8 and $\mathbf n=(n_1,\dots,n_K)$9, the extended Kirkwood–Dirac quasiprobability $\omega=e^{2\pi i/D}$0 underlies the OTOC decomposition, and its total nonclassicality

$\omega=e^{2\pi i/D}$1

serves as a scrambling witness (Mohseninia et al., 2019). This is not labeled OQ in that paper, but it is structurally an operational quasiprobability reconstructed from sequential measurement statistics.

A third application is quantum thermodynamics. In a recent thermodynamic formulation, OQ is used directly as a work distribution. For initial and final energy measurements $\omega=e^{2\pi i/D}$2 and $\omega=e^{2\pi i/D}$3, reconstructed from end-point measurement and TPM data, the OQ is

$\omega=e^{2\pi i/D}$4

This work distribution preserves both marginals, reproduces TPM for incoherent initial states, reproduces Jarzynski equality for Gibbs initial states, and yields the average work consistent with the classical difference of average energies. In the same framework, non-joint measurability can increase the amount of extractable work beyond the classical bound imposed by jointly measurable measurements (Jae et al., 5 Oct 2025).

Finally, the OQ logic has been extended to network nonclassicality. In an operational treatment of bilocality, one replaces hidden-variable models by joint probability distributions over all outcomes, and, when positivity fails, by joint quasi-probability distributions and quasi-stochastic processes. Within that framework, violations of bilocality require negativity somewhere in the operational decomposition (Onggadinata et al., 2023).

6. Relation to phase-space quasiprobabilities and adjacent operational formalisms

The term “operational quasiprobability” is sometimes used more broadly than the strict sequential-measurement sense. A coherent-state-based operational meaning of the Husimi distribution arises when a generalized coherent-state family defines a POVM

$\omega=e^{2\pi i/D}$5

so that $\omega=e^{2\pi i/D}$6 is the probability distribution of an approximately simultaneous measurement. This is highly relevant background for OQ, but it is not the same as the commensurate sequential OQ of the qudit and continuous-variable frameworks (Harada, 2011).

A related phase-space line shows that the Cahill–Glauber chain $\omega=e^{2\pi i/D}$7 can be realized operationally by a concrete Gaussian channel

$\omega=e^{2\pi i/D}$8

for which

$\omega=e^{2\pi i/D}$9

This gives an operational bridge among standard quasiprobability representations, but again it does not define OQ from sequential measurement statistics (Linowski et al., 2023).

The resource-theoretic paper on phase-space negativity is also adjacent rather than identical to OQ. There the negativity of the Glauber–Sudarshan $\chi(\mathbf{n}) = \mbox{Tr} \left[\mathcal{ T}\prod_{k=1}^K \left(\delta_{n_k,0} \mathcal{ I} + (1-\delta_{n_k,0})\sum_{a_k=0}^{D-1} \omega^{n_k a_k} \mathcal{ A}_k(a_k) \right)\hat{\rho}\right],$0-function is defined consistently even for singular distributions and shown to equal the robustness of nonclassicality,

$\chi(\mathbf{n}) = \mbox{Tr} \left[\mathcal{ T}\prod_{k=1}^K \left(\delta_{n_k,0} \mathcal{ I} + (1-\delta_{n_k,0})\sum_{a_k=0}^{D-1} \omega^{n_k a_k} \mathcal{ A}_k(a_k) \right)\hat{\rho}\right],$1

so negativity acquires an operational meaning through noise robustness rather than through an OQ construction (Tan et al., 2019). Likewise, noise-adaptive Bell-type tests based on Cahill–Glauber quasiprobabilities remain phase-space ordering frameworks, even though they are tied to explicit measurements and to a noise-dependent order parameter (Lee et al., 2020).

This broader landscape suggests two distinct but overlapping uses of “operational.” In the strict OQ literature, quasiprobability is reconstructed from experimentally specified measurement contexts—typically sequential or conjunction measurements—and negativity is interpreted against explicit classical constraints such as noninvasiveness or joint measurability (Ryu et al., 2012, Jae et al., 2017, Jae et al., 2023). In adjacent phase-space and weak-value literatures, “operational” more often means that a quasidistribution is linked to a POVM, a conditioned measurement protocol, a Gaussian channel, or a resource-theoretic task (Harada, 2011, Linowski et al., 2023, Tan et al., 2019). Both senses are important, but they address different questions.

Operational quasiprobability therefore occupies a specific place within the larger theory of quasidistributions. It is the branch in which quasiprobability is defined from measurement procedures themselves, with marginals tied to observed statistics and with negativity interpreted as failure of a designated classical operational model. That feature makes OQ especially useful wherever incompatible measurements, sequential disturbance, or nonclassical conditional statistics are themselves the central objects of study.