Kirkwood–Dirac Quasiprobabilities

• Kirkwood–Dirac quasiprobability distributions are complex frameworks representing quantum states and measurement statistics, generalizing classical joint probabilities.
• They diagnose quantum nonclassicality through negativity and nonreal values, providing a quantifiable resource for computational and metrological advantages.
• Their structure, based on operator products and support uncertainties, informs resource theories and experimental techniques across finite and infinite-dimensional quantum systems.

The Kirkwood–Dirac (KD) quasiprobability distribution is a complex-valued mathematical framework used to represent quantum states and quantum measurement statistics, generalizing classical joint probability distributions to quantum theory. Unlike classical joint distributions, KD distributions may be negative or nonreal and are crucial for operationally diagnosing and quantifying nonclassicality, resourcefulness, and contextuality in quantum information processing, metrology, and the foundations of quantum mechanics. Formally, given a quantum state and two (typically noncommuting) observables with respective orthonormal bases, the KD distribution encodes both amplitude and phase relationships, reproduces the correct quantum marginals, and exposes the full informational content of the state—even in the presence of coherence and incompatible measurements.

1. Foundational Definition and Mathematical Structure

For a quantum state $|\psi\rangle$ in $d$ dimensions and two orthonormal bases $A = \{|a_j\rangle\}$ and $B = \{|b_k\rangle\}$, the standard KD distribution is

$$Q_{j k}(|\psi\rangle) = \langle a_j|\psi\rangle \langle\psi|b_k\rangle \langle b_k|a_j\rangle.$$

For a density matrix $\rho$,

$$Q_{j k}(\rho) = \langle a_j|\rho|b_k\rangle \langle b_k|a_j\rangle.$$

More general operator-valued constructions use products of projectors, e.g., $\mathrm{Tr}[\pi_{p} \pi_{x} \rho]$ for phase-space variables $x, p$ (Lundeen et al., 2013). The KD distribution is “quasi” because entries can be negative or complex, even though its marginals $\sum_j Q_{j k}$ and $\sum_k Q_{j k}$ return the standard quantum probabilities in either basis.

In continuous-variable systems, the KD distribution is defined in terms of the Fourier transform of ordered products of unitaries, and for finite-state systems (e.g., qubits or qutrits) it is supported only on the physically realizable combinations of eigenvalues of the measured observables (Umekawa et al., 2023).

2. Positivity, Nonclassicality, and the Geometry of KD States

A state is “KD-positive” if all entries of its KD distribution are real and nonnegative, allowing interpretation as a classical probability distribution. Otherwise, negativity or nonreality constitutes “KD nonclassicality.” Quantitatively, the total nonpositivity is given by

$\mathcal{N}(Q(\rho)) = \sum_{i,j} |Q_{i j}(\rho)|,$

with $\mathcal{N} = 1$ if and only if the distribution is classical (Langrenez et al., 5 Jul 2024).

The geometry of KD-positive states is generically minimal in Hilbert space: for two randomly chosen observables (i.e., almost all choices), the set of all KD-positive states $\mathcal{C}_{\mathrm{KD}}$ is the polytope

$$\mathcal{C}_{\mathrm{KD}} = \mathrm{conv}\left(\{ |a_n\rangle\langle a_n| \} \cup \{ |b_m\rangle\langle b_m| \} \right),$$

the convex hull of the $2d$ basis projectors, of dimension $2d-1$ (Langrenez et al., 27 May 2024). Thus, most quantum states lie outside this polytope and thus display KD negativity; only the basis states (and their convex combinations) are KD-positive for generic bases. Extensions to three-state (qutrit) systems or to special cases like discrete Fourier transform (DFT) bases in prime dimension confirm this minimal polytope as the entire KD-positive set (Langrenez et al., 2023).

For pure states, fine-grained classification is possible. Given support sizes $n_A$ (in basis $A$) and $n_B$ (in basis $B$),

• In the DFT (mutually unbiased) case, KD classicality is equivalent to $n_A \cdot n_B = d$ (Xu, 2022).
• For completely incompatible (COINC) bases, a state is KD-positive only if the support uncertainty $n_A + n_B = d+1$ (Bievre, 2021).

In general, for arbitrary observables, the set of KD-positive states can possess mixed-state extremal points that are not convex combinations of pure KD-positive states, especially in specific structured low-dimensional cases (Langrenez et al., 2023).

3. Noncommutativity, Measurement Incompatibility, and Bounds

KD nonclassicality requires, but is not implied by, operator noncommutation. Rigorous criteria refine this relation:

• Noncommutation is necessary but not sufficient; sufficient “incompatibility” is required, quantified in sharp inequalities involving the state’s structure and overlap with the measurement bases (Arvidsson-Shukur et al., 2020).
• For pure states with nondegenerate observables, KD nonclassicality arises if $$2N_A + 2N_B > 3d + n_\parallel - 3\bar{n}_\parallel$$, where $N_A$ ($N_B$) counts the nonzero overlaps, and $n_\parallel, \bar{n}_\parallel$ encode alignment properties.
• Noncommuting projectors in the ordered product (e.g., $\pi_p \pi_x$) are essential for KD negativity: compatible (commuting) measurements can never produce negativity (Tan et al., 7 Jan 2024).

Support uncertainty—encoding the “spread” across both bases—serves as both an operational and a geometric witness, and convex-roof constructions of support uncertainty or total nonpositivity generalize KD nonclassicality detection to mixed states (Langrenez et al., 5 Jul 2024). For example, the convex roof of total KD nonpositivity

$$N(\rho) = \inf_{\{ \lambda_i, |\psi_i\rangle \}} \sum_i \lambda_i N_t(|\psi_i\rangle),$$

equals 1 only for convex combinations of pure KD-positive states, providing a faithful mixed-state nonclassicality witness.

4. Operational Meaning: Resource Theories, Quantum Advantage, and Experimental Probes

KD nonclassicality is both an identifier and a resource for quantum advantage in computation, metrology, and thermodynamics. In rebit-based quantum computation, KD nonpositivity (i.e., negativity or complex values) is necessary for universal (quantum-advantageous) computation; all circuits that remain in the KD-positive regime are efficiently classically simulable (Thio et al., 9 Jun 2025).

The KD “mana,” defined as $$\mathrm{Mana}(\rho) = \log \sum_{g, \chi} |Q_{g, \chi}(\rho)|$$, is monotonic under free operations and vanishes on KD-positive states, serving as a quantitative resource monotone (Thio et al., 9 Jun 2025).

Any quantum resource (e.g., coherence, entanglement, contextuality) can be “revealed” by appropriate KD-type distributions, with KD negativity certifying “resourcefulness”—there always exists an incompatible measurement set separating classical from resourceful states (Tan et al., 7 Jan 2024). Negativity in the distribution corresponds to geometric separation from the classical set and is also detected via anomalous (non-spectral) weak values in experiment.

In quantum thermodynamics, the KD distribution of work or energy fluctuations provides unperturbed means and variances, retains full information in presence of quantum coherence, and its imaginary part quantifies noncommutativity between initial and final Hamiltonians (Hernández-Gómez et al., 31 May 2024, Pezzutto et al., 10 Mar 2025).

KD distributions can be reconstructed in experiment through interferometric schemes (such as Ramsey protocols with ancillas), weak joint measurements (e.g., measuring position weakly and momentum strongly in photons), or using direct quantum circuits measuring Bargmann invariants (Lundeen et al., 2013, Wagner et al., 2023).

5. Applications: Metrology, Chaos, Scrambling, and Coherence

KD distributions appear widely:

• In quantum metrology, KD negativity can enable Fisher information enhancements and precision beating classical limits; anomalous weak values (implied by negative KD elements) permit amplified signal responses in parameter estimation protocols (Arvidsson-Shukur et al., 2020, Wagner et al., 2023).
• In quantum chaos and scrambling, extended KD distributions underlie the out-of-time-ordered correlator (OTOC), with symmetry breaking and bifurcations in the extended KD structure diagnosing many-body chaos and scrambling dynamics (Halpern et al., 2017).
• In quantum thermodynamics, KD-based work distributions capture coherent and noncommutative energy fluctuations; their real and imaginary parts provide access to nonclassical features with direct impact on energy and entropy trade-offs (Hernández-Gómez et al., 31 May 2024, Pezzutto et al., 10 Mar 2025).
• In quantum coherence theory, KD-based nonclassicality is a faithful coherence quantifier, upper-bounded by state purity and connected to entropy-based uncertainty relations (Budiyono et al., 2023).

KD negativity and nonreality are also essential for contextuality: nonclassical KD values are necessary for quantum scenarios not simulable by classical hidden-variable models (Budiyono et al., 2023, Arvidsson-Shukur et al., 27 Mar 2024).

6. Bounds, Hierarchies, and Simulability

KD quasiprobabilities relax classical probability axioms, yielding a strict hierarchy: classical probabilities $\subset$ KD quasiprobabilities $\subset$ postquantum quasiprobabilities, with the outer boundary corresponding to normalization and modulus constraints unconstrained by quantum mechanics (Liu et al., 12 Apr 2025). Pointwise,

$$|p_{x y}| \leq \min(p_x, p_y), \quad |q_{x y}| \leq \max(p_x, p_y), \quad |l_{x y}| \leq 1,$$

with global sums of absolute values bounded as $\sum_{x, y} |q_{x y}| \leq \sqrt{N}$ for an $N$-dimensional system.

In simulation, KD positivity does not ensure efficient classical simulability unless the dynamical superoperators are stochastic in the KD representation. KD-positive circuits can be exponentially hard to sample-classically under non-stochastic gates, in stark contrast to discrete Wigner theory where Clifford evolutions guarantee stochasticity and efficient simulation (Burkat et al., 17 Feb 2025).

7. Extensions and Group-Theoretic Generalizations

The KD representation is not limited to finite-dimensional Hilbert spaces. It can be defined over second countable locally compact abelian groups $G$, where for an operator $A$ with kernel $k_A(g, g')$,

$$KD_A(g, \chi) = \overline{\chi(g)} \int k_A(g, g') \chi(g') d\mu_G(g')$$

connects KD distributions to the Kohn–Nirenberg quantization (Spriet, 31 Jul 2025). In this framework, pure states with positive KD distributions correspond to Haar measures on closed subgroups, up to Weyl–Heisenberg symmetry. The classical “fragment” (the set of KD-positive states and KD-real observables) is nontrivial if and only if the group has a compact connected component.

This group-theoretic perspective ties the KD distribution to the Wigner–Weyl symmetric quantization and underpins phase-space–like representations relevant for quantum information, signal processing, and beyond.

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In summary, Kirkwood–Dirac quasiprobability distributions provide an informationally complete, operationally meaningful, and mathematically precise generalization of classical joint probabilities to quantum theory, directly exposing the signatures, witnesses, and resources of quantum nonclassicality. Their structure, geometry, and negativity are central to contemporary quantum information protocols, foundational diagnostics, resource theories, and experimental probes spanning finite- and infinite-dimensional quantum systems.