Kirkwood-Dirac Representations in Quantum Systems

• Kirkwood-Dirac representations are informationally complete quasiprobability distributions that generalize classical joint probabilities by encoding quantum observables—including noncommuting pairs—with potential negative or complex values.
• They enable optimal prediction and conditional expectation in quantum measurement theory, serving as operational witnesses to nonclassicality, contextuality, and computational quantum advantage.
• Their convex geometric structure, characterized by KD positivity criteria, provides practical bounds for quantum state tomography and resource quantification in diverse measurement settings.

The Kirkwood-Dirac (KD) representation is a family of informationally complete quasiprobability distributions that encode joint statistics of quantum observables, including noncommuting pairs, with deep implications for quantum measurement theory, resource quantification, and quantum information processing. KD representations generalize classical joint probabilities to the quantum domain, capturing negative and even complex entries that serve as operational witnesses of quantum nonclassicality, contextuality, and computational advantage.

1. Foundational Definition and Formalism

The core structure of the KD representation is as follows. Given a finite-dimensional Hilbert space $\mathcal{H}$, fix two orthonormal bases $\mathcal{A} = \{\ket{a_i}\}$ and $\mathcal{B} = \{\ket{b_j}\}$—often chosen as eigenbases of observables $A$ and $B$—with unitary overlap matrix $U_{ij} = \langle a_i | b_j \rangle$. For any density operator $\rho$, the Kirkwood-Dirac distribution is defined by

$$Q_{ij}(\rho) = \langle a_i | \rho | b_j \rangle \langle b_j | a_i \rangle.$$

This array, which may have negative or nonreal values, satisfies normalization and correctly reproduces Born-rule marginals: $$\sum_j Q_{ij}(\rho) = \langle a_i | \rho | a_i \rangle, \quad \sum_i Q_{ij}(\rho) = \langle b_j | \rho | b_j \rangle, \quad \sum_{ij} Q_{ij}(\rho) = 1.$$ The KD distribution generalizes to arbitrary quantum measurements (including POVMs and multi-step projective scenarios) as

$$Q_{\rho}(a_1, \ldots, a_n) = \mathrm{Tr}[\,\Pi^{(n)}_{a_n} \cdots \Pi^{(1)}_{a_1} \rho\,],$$

where $\{\Pi^{(\ell)}_{a_\ell}\}$ are rank-1 projections (Lostaglio et al., 2022, Wagner et al., 2023). This structure enables a unified approach to quantum measurement statistics, correlators, and conditional expectations (Spriet et al., 3 Nov 2025).

Unlike classical joint probabilities, which must be nonnegative, some $Q_{ij}(\rho)$ will in general be negative or complex—features directly linked to measurement incompatibility and noncommutativity (Bievre, 2022, Liu et al., 18 Nov 2024).

2. KD Classicality, Support-Uncertainty, and Nonclassicality

A state $\rho$ is called KD-positive (or KD-classical) with respect to $(\mathcal{A}, \mathcal{B})$ if $Q_{ij}(\rho) \geq 0$ (and real) for all $i, j$; otherwise it is KD-nonclassical. The set of all such states is convex: $$\mathrm{KDC}(\mathcal{A},\mathcal{B}) = \{\rho: Q_{ij}(\rho) \geq 0 \; \forall\,i,j\}.$$ For pure states $\ket{\psi}$, a key witness of nonclassicality is the support-uncertainty: $$n_{\mathcal{A}}(\psi) = |\{ i: \langle a_i | \psi \rangle \neq 0 \}|, \quad n_{\mathcal{B}}(\psi) = |\{ j: \langle b_j | \psi \rangle \neq 0 \}|,$$ with the Donoho-Stark bound: $$n_{\mathcal{A}}(\psi) n_{\mathcal{B}}(\psi) \geq \left(\max_{i,j} |U_{ij}| \right)^{-2}.$$ When $\mathcal{A}$ and $\mathcal{B}$ are mutually unbiased bases (MUBs), $$|U_{ij}| = 1/\sqrt{d} \Rightarrow n_{\mathcal{A}}(\psi) n_{\mathcal{B}}(\psi) \geq d$$ (Yang et al., 2023).

For the discrete Fourier transform (DFT) matrix, the exact KD classicality criterion is $n_{\mathcal{A}}(\psi) n_{\mathcal{B}}(\psi) = d$; states with $n_{\mathcal{A}} n_{\mathcal{B}} > d$ are strictly KD-nonclassical (Yang et al., 2023). The uncertainty diagram catalogs which $(n_{\mathcal{A}}, n_{\mathcal{B}})$ pairs are realizable; for the DFT, there are no "holes" above $n_{\mathcal{A}} + n_{\mathcal{B}} = d+1$, so all such support pairs occur (completing the classification and settling a conjecture of [Phys. Rev. Lett. 127, 190404 (2021)]).

3. Geometry of KD-Positive States and Convex Structure

The set of KD-positive states is, in general, a convex polytope. Its extreme points are pure KD-positive states, which are characterized for various choices of bases:

• For qubits ($d=2$), generic qutrits ($d=3$), and prime-dimensional DFT matrices, the only pure KD-positive states are the basis states—i.e., projectors onto $|a_i\rangle$ or $|b_j\rangle$ (Langrenez et al., 2023).
• In these cases, $$\mathrm{KDC} = \mathrm{conv}\big(\{|a_i\rangle\langle a_i|, |b_j\rangle\langle b_j|\}\big)$$.
• For real orthogonal transformations or certain composite $d$, the polytope strictly contains this hull; explicit mixed-state KD-positive points exist that cannot be written as convex combinations of pure KD-positive states (Langrenez et al., 2023, Langrenez et al., 5 Jul 2024).

For finite abelian group settings, as discussed in the context of DFTs, every pure KD-positive state is a translate or modulate of a computational basis state ("stabilizer-type"), and in prime-power order all KD-positive states are convex combinations thereof (Bièvre et al., 21 Jan 2025). For non-prime-power groups, KD-positive mixed states can exist outside the convex hull of pure ones.

The convex roof of the total nonpositivity $\sum_{ij}|Q_{ij}(\rho)|$ serves as a faithful witness: $N_{\mathrm{cr}}(\rho) = 1$ if and only if $\rho$ is a convex combination of pure KD-positive states (Langrenez et al., 5 Jul 2024). The support-uncertainty roof is a less faithful, but operationally valuable, criterion.

4. KD Representations, Conditional Expectation, and Optimal Prediction

Among all "Born-compatible" quasiprobability representations—those reproducing the correct marginals for observables $A$ and $B$—the KD representation is uniquely distinguished by its conditional expectation property: for any observable $X$ and state $\rho$, the "conditional expectation of $X$ given $B=b$" built from KD coincides with the best mean-square predictor of $X$ among all functions of $B$: $$f^*(B) = \mathbb{E}_{\mathrm{KD},\rho}[X|B] = \sum_b \frac{\langle b | X \rho | b \rangle}{\langle b | \rho | b \rangle} |b\rangle\langle b|.$$ This variational optimality property singles out the KD representation as the quantum generalization of classical conditional expectation (Spriet et al., 3 Nov 2025). Other quasiprobabilities (e.g., Wigner, symmetrically ordered, Margenau–Hill) do not share this optimal prediction property.

Furthermore, the dual frame operators in the KD representation encode weak values: for any POVM element $E$, the KD dual $$\xi(E|i,i')=\langle a'_{i'}|E|a_i\rangle/\langle a'_{i'}|a_i\rangle$$ is the weak value of $E$ with pre-selection $|a_i\rangle$ and post-selection $|a'_{i'}\rangle$ (Schmid et al., 7 May 2024).

5. Operational Significance, Nonclassicality, and Quantum Resources

KD negativity or non-reality is tightly linked to the manifestation of quantum resources:

• KD nonpositivity signals contextuality, coherence, and computational quantum advantage (Thio et al., 9 Jun 2025, Tan et al., 7 Jan 2024, Liu et al., 18 Nov 2024).
• In resource theories, every resourceful (nonfree) state can be identified by the existence of a KD representation corresponding to some pair of incompatible measurements, in which the distribution is negative at at least one outcome (Tan et al., 7 Jan 2024). Total negativity is proportional to the Frobenius norm distance to the closest free state.
• Measurement incompatibility is a necessary requirement for KD negativity: if all measurements commute, the KD is automatically positive (Tan et al., 7 Jan 2024).

KD-nonclassicality is a resource monotone for quantum computation: classical simulation is efficient for circuits remaining within the KD-positive subtheory, and KD negativity is necessary to achieve computational quantum advantage (Thio et al., 9 Jun 2025). In the rebit model, the convex hull of CSS stabilizer states characterizes the set of KD-positive states, strictly smaller than the qubit stabilizer polytope.

For metrology and weak measurement, KD nonclassicality underlies the possibility of anomalous weak values and trade-offs in quantum Fisher information (Lostaglio et al., 2022, Liu et al., 18 Nov 2024). Coherence monotones can be defined in terms of the sum of the imaginary parts of KD entries, maximized over mutually unbiased bases (Liu et al., 18 Nov 2024).

6. Generalizations, Dualities, and Informativeness

KD representations are informationally complete: knowledge of $Q_{ij}(\rho)$ for all $i, j$ determines $\rho$ (Arvidsson-Shukur et al., 27 Mar 2024). In particular, in finite-state systems, the KD table for two noncommuting observables suffices for full quantum state or process tomography (Umekawa et al., 2023, Schmid et al., 7 May 2024).

KD distributions can be generalized to arbitrary POVMs or to higher-order correlators; they underlie quantum features such as scrambling (out-of-time-order correlators), thermodynamic fluctuation relations, and contextually robust witnesses of nonclassicality (Lostaglio et al., 2022, Arvidsson-Shukur et al., 27 Mar 2024).

KD representations admit an extension beyond quantum states to channels and instruments. The correspondence is strictly functorial: compositions of physical processes commute with the corresponding KD superoperators, and the KD representation of a POVM is precisely a table of weak values relative to the chosen bases (Schmid et al., 7 May 2024).

7. Limits, Boundaries, and Comparative Structure

The KD quasiprobability forms an intermediate region between classical and postquantum quasiprobabilities:

• Classical joint probabilities: $p_{xy} \in [0,1]$, sum to 1, and $|p_{xy}| \leq \min(p_x,p_y)$.
• KD distributions: $q_{xy}$ may be negative or complex, always satisfy $\sum_{x,y} q_{xy} = 1$, and $|q_{xy}|^2 \leq p_x p_y$ (Liu et al., 12 Apr 2025).
• Postquantum bounds: arrays $l_{xy}$ with $|l_{xy}| \leq 1$, sum normalization.

Universal $\ell^2$ and $\ell^1$-norm bounds, as well as support-uncertainty inequalities, provide rigorous constraints delimiting the KD region (Liu et al., 12 Apr 2025). For instance, $\sum_{x,y}|q_{xy}|^2 \leq 1$, with similar bounds for higher moments and chain products of measurements.

8. Research Frontiers and Classification for Abstract Settings

For second-countable locally compact abelian (LCA) groups, the KD representation generalizes via the Fourier transform, enabling a phase-space analysis for arbitrary $G \times \widehat{G}$ (Spriet, 31 Jul 2025). The set of KD-positive states is determined by Haar measures supported on closed subgroups or cosets; the classical fragment is nontrivial if and only if $G$ has a compact connected identity component.

In the DFT setting, the structure of KD-real and KD-positive operators is resolved: any KD-positive state with respect to the DFT of any finite dimension can be expressed as a real linear combination of pure KD-positive projectors (Xu, 22 Dec 2024). For prime-dimensional DFTs, only the computational basis states are KD-positive (Xu, 22 Dec 2024, Bièvre et al., 21 Jan 2025).

Table: Key Properties of the KD Representation

Property	Characterization	Reference
Normalization	$\sum_{ij} Q_{ij}(\rho) = 1$	(Lostaglio et al., 2022)
Marginals	Born rule: $\sum_j Q_{ij} = \langle a_i|\rho|a_i\rangle$, $\sum_i Q_{ij} = \langle b_j|\rho|b_j\rangle$	(Yang et al., 2023)
KD positivity	$Q_{ij}(\rho) \geq 0$ for all $(i,j)$	(Langrenez et al., 2023)
Conditional expectation	Optimal predictor property (unique to KD)	(Spriet et al., 3 Nov 2025)
Informational completeness	KD fully determines $\rho$ if all $Q_{ij}$ known	(Arvidsson-Shukur et al., 27 Mar 2024)
Support-uncertainty	KD classical iff $n_\mathcal{A} n_\mathcal{B} = d$ (DFT case)	(Yang et al., 2023)
Geometry	Simplex/polytope: basis projectors or more general depending on $(\mathcal{A},\mathcal{B})$	(Langrenez et al., 2023)
Nonclassicality witness	KD negativity or nonreal values	(Tan et al., 7 Jan 2024, Bievre, 2022)

Concluding Synthesis

Kirkwood-Dirac representations unify the probabilistic and quantum-theoretic descriptions of measurement outcomes, operationalize optimal prediction and conditional inference in the quantum setting, and provide an analytically tractable and experimentally accessible means of diagnosing and quantifying quantum resources—contextuality, coherence, magic, and metrological advantage. Their geometry, boundaries, and criteria of positivity and nonclassicality have been classified for a wide array of bases, groups, and measurement scenarios. The formalism extends to quantum channels and processes, and ongoing work explores their structure in infinite-dimensional, group-theoretic, and postquantum generalizations. The KD framework is foundational to understanding classical simulability, contextuality, resource theory, and the operational core of quantum advantages in information processing.