Kirkwood-Dirac Quasiprobability Distributions

• Kirkwood-Dirac quasiprobability distributions are quantum state representations defined via two eigenbases, normalized to recover Born-rule marginals yet permitting negative or complex values.
• They enable state and operator reconstruction and quantify nonclassicality through measures like negativity and coherence, with direct applications in metrology and tomography.
• The geometric analysis of KD-positive states delineates classical from quantum regimes, highlighting its role in understanding measurement disturbance and resource-theoretic advantages.

Searching arXiv for recent and foundational papers on Kirkwood-Dirac quasiprobability distributions. Searching kirkwood-dirac quasiprobability distributions geometry positivity coherence metrology The Kirkwood–Dirac (KD) quasiprobability distribution is a representation of quantum states relative to the eigenbases of two observables, typically denoted $A$ and $B$, that assigns to each pair of basis labels a complex number $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$. It is normalized, reproduces the Born-rule marginals for both observables, and is informationally complete whenever all overlaps $\langle a_i|b_j\rangle$ are nonzero. Unlike a classical joint probability distribution, however, the KD distribution can assume negative or nonreal values. These departures from positivity have become central to contemporary analyses of nonclassicality, quantum advantage, weak values, measurement disturbance, metrology, thermodynamics, scrambling, and contextuality (Arvidsson-Shukur et al., 2024).

1. Definition, normalization, and reconstruction

Let $A=\sum_i a_i|a_i\rangle\langle a_i|$ and $B=\sum_j b_j|b_j\rangle\langle b_j|$ be two nondegenerate observables on a $d$-dimensional Hilbert space, with orthonormal eigenbases $\{|a_i\rangle\}$ and $\{|b_j\rangle\}$. For a density operator $\rho$, the KD quasiprobability matrix is defined by

$B$0

with $B$1 and $B$2 (Langrenez et al., 2023).

This array satisfies three probability-like identities: $B$3 Thus the total weight is normalized and the marginals reproduce the outcome distributions of $B$4 and $B$5 (Langrenez et al., 2024).

When $B$6, the KD representation is informationally complete. One reconstruction formula is

$B$7

and more generally the KD symbol of an operator $B$8 is

$B$9

This makes the KD formalism not only a state representation but also a symbol calculus for operators (Langrenez et al., 2023).

A common misconception is that KD quasiprobabilities are merely classical joint distributions written in unusual notation. They are not: the defining ordered product of projectors is generally non-Hermitian, and this is precisely why the entries can be negative or complex.

2. Classicality, nonclassicality, and the meaning of KD positivity

A state is called KD-positive, or KD-classical, with respect to the chosen bases if all entries are real and nonnegative: $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$0 In that case the KD distribution is an honest joint probability distribution on the product label set (Langrenez et al., 2024).

In general, $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$1 may be negative or nonreal. Several papers in the recent literature connect these features to nonclassicality. Negative or nonreal KD entries have been linked to quantum advantage in weak measurements and contextuality, enhanced sensitivity in quantum metrology, signatures of quantum chaos, scrambling, and thermodynamic irreversibility (Langrenez et al., 2023). The real part can be negative even when one passes to the Margenau–Hill distribution, while the imaginary part is associated with measurement disturbance and thermodynamic nonclassicality (Arvidsson-Shukur et al., 2020).

For pure states, one exact criterion for KD classicality is phase factorization. Writing

$$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$2

a pure state $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$3 is KD classical if and only if there exist phases $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$4 and $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$5 such that

$$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$6

for every relevant pair $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$7 on the supports of the amplitudes (Xu, 2022). This criterion formalizes the statement that positivity requires global phase compatibility across all nonzero KD entries.

The literature also clarifies that noncommutation alone does not suffice for KD nonclassicality. Pairwise noncommutation of $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$8, $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\langle a_i|\rho|b_j\rangle$$9, and $\langle a_i|b_j\rangle$0 can occur while the KD distribution remains classical, and tighter counting conditions are required to force negativity or nonreality (Arvidsson-Shukur et al., 2020). This corrects an older intuition that any incompatibility of observables automatically implies nonpositive KD values.

3. Geometry of KD-positive states

The set of KD-positive states,

$\langle a_i|b_j\rangle$1

is a convex subset of the density-operator space. Its geometry has become a central theme because it marks the boundary between states admitting a classical-probability interpretation in a chosen KD representation and those exhibiting KD nonpositivity (Langrenez et al., 2023).

A major result is that in several regimes this set is exactly the convex hull of the eigenprojectors of the two reference bases: $\langle a_i|b_j\rangle$2 where $\langle a_i|b_j\rangle$3 and $\langle a_i|b_j\rangle$4. This equality holds under the weak-incompatibility condition

$\langle a_i|b_j\rangle$5

in the following cases: $\langle a_i|b_j\rangle$6; an open and dense set of bases in $\langle a_i|b_j\rangle$7; discrete-Fourier-transform bases in prime dimension; and any pair of bases sufficiently near one of these cases (Langrenez et al., 2023).

For random pairs of orthonormal bases chosen independently according to Haar measure, the situation is even sharper: with probability one, the KD-positive set is the convex hull of the $\langle a_i|b_j\rangle$8 basis projectors. In that case the free-state set is a $\langle a_i|b_j\rangle$9-dimensional simple polytope with exactly $A=\sum_i a_i|a_i\rangle\langle a_i|$0 vertices (Langrenez et al., 2024). This shows that, generically, KD classicality is “minimal” in the sense that no additional extreme states appear beyond the obvious basis eigenstates.

The following summary captures the principal geometric regimes described in the literature.

Regime	Characterization of $A=\sum_i a_i|a_i\rangle\langle a_i|$1	Source
$A=\sum_i a_i|a_i\rangle\langle a_i|$2 with $A=\sum_i a_i|a_i\rangle\langle a_i|$3	$A=\sum_i a_i|a_i\rangle\langle a_i|$4	(Langrenez et al., 2023)
$A=\sum_i a_i|a_i\rangle\langle a_i|$5, open dense set of bases	$A=\sum_i a_i|a_i\rangle\langle a_i|$6	(Langrenez et al., 2023)
Prime $A=\sum_i a_i|a_i\rangle\langle a_i|$7, DFT overlap matrix	$A=\sum_i a_i|a_i\rangle\langle a_i|$8	(Langrenez et al., 2023)
Haar-random bases	With probability one, convex hull of the $A=\sum_i a_i|a_i\rangle\langle a_i|$9 eigenprojectors	(Langrenez et al., 2024)

This geometry has direct resource-theoretic significance. If the KD-positive set is just a small polytope, then almost every state lies outside it; the corresponding KD negativity is then generic rather than exceptional (Langrenez et al., 2024).

4. Pure-state structure, Fourier pairs, and mixed extremal states

The pure-state sector has a particularly rich classification. For KD-classical pure states, support sizes in the two bases play a decisive role. If

$B=\sum_j b_j|b_j\rangle\langle b_j|$0

then a general block-structure theorem constrains the allowed supports and recovers the support-uncertainty bound $B=\sum_j b_j|b_j\rangle\langle b_j|$1, with $B=\sum_j b_j|b_j\rangle\langle b_j|$2, after a suitable decomposition of the relevant submatrix of the transition matrix $B=\sum_j b_j|b_j\rangle\langle b_j|$3 into nonoverlapping nonnegative blocks (Xu, 2022).

For mutually unbiased bases, especially discrete-Fourier-transform pairs,

$B=\sum_j b_j|b_j\rangle\langle b_j|$4

the classification simplifies dramatically. In that setting a pure state is KD classical if and only if

$B=\sum_j b_j|b_j\rangle\langle b_j|$5

This resolves the conjecture of De Bièvre et al. for DFT bases in arbitrary dimension (Xu, 2022).

A related later result studies DFT-based KD classicality through a directed-graph construction. In dimension $B=\sum_j b_j|b_j\rangle\langle b_j|$6, the full set of KD-classical states equals the convex hull of KD-classical pure states. More generally, for arbitrary $B=\sum_j b_j|b_j\rangle\langle b_j|$7, along any path in the directed graph associated with a factorization structure of $B=\sum_j b_j|b_j\rangle\langle b_j|$8, the intersection of the KD-classical state set with the real span of the path-associated KD-classical pure states is exactly the convex hull of those pure states (Cai et al., 14 Mar 2026). This suggests a refined combinatorial organization of KD-classical sectors in composite dimensions.

Not all dimensions and basis pairs are so rigid. When the overlap matrix $B=\sum_j b_j|b_j\rangle\langle b_j|$9 is real orthogonal in dimension $d$0, the real-linear space of operators with real KD symbol has dimension $d$1, which is strictly larger than $d$2. In this regime one can have

$d$3

so mixed extreme KD-positive states must exist (Langrenez et al., 2023).

The explicit spin-1 example is particularly instructive. Taking $d$4 as the $d$5 eigenbasis and $d$6 as the eigenbasis of a rotated spin component $d$7 about $d$8 by angle $d$9, the real orthogonal overlap matrix is

$\{|a_i\rangle\}$0

In this case there exist KD-positive mixed states of the form

$\{|a_i\rangle\}$1

that remain KD-positive for an open interval of $\{|a_i\rangle\}$2 yet lie outside $\{|a_i\rangle\}$3. Hence they are mixed extreme points that are not convex combinations of pure KD-positive states (Langrenez et al., 2023).

This phenomenon matters conceptually because it distinguishes “KD-positive” from “mixtures of pure KD-positive states.” The two notions coincide generically, but not universally.

5. Quantifying nonpositivity, coherence, and related witnesses

A widely used scalar quantifier is the total KD nonpositivity

$\{|a_i\rangle\}$4

or equivalently $\{|a_i\rangle\}$5 depending on convention. It equals its classical minimum exactly when the KD distribution is real and nonnegative (Burkat et al., 17 Feb 2025, Arvidsson-Shukur et al., 2020). For $\{|a_i\rangle\}$6 projective measurements, the maximum total nonclassicality is

$\{|a_i\rangle\}$7

achieved if and only if each pair of successive bases is mutually unbiased and the state is pure with equal overlap $\{|a_i\rangle\}$8 with every vector in the first and last bases (Arvidsson-Shukur et al., 2020).

Recent work has developed convex-roof constructions to extend pure-state witnesses to mixed states. The convex roof of support uncertainty,

$\{|a_i\rangle\}$9

provides a nonfaithful witness for the convex hull of pure KD-positive states, while the convex roof of the total KD nonpositivity,

$\{|b_j\rangle\}$0

is faithful for that convex hull: $\{|b_j\rangle\}$1 if and only if $\{|b_j\rangle\}$2 lies in the convex hull of pure KD-positive states (Langrenez et al., 2024).

Moment-based detection criteria provide another route. Defining raw moments

$\{|b_j\rangle\}$3

classical positivity implies

$\{|b_j\rangle\}$4

More generally, the Hankel matrices $\{|b_j\rangle\}$5 built from the moments must be positive semidefinite if the KD table is pointwise nonnegative, so $\{|b_j\rangle\}$6 certifies KD nonpositivity (Chakrabarty et al., 9 Jun 2025). Explicit examples show that higher-order determinants can detect nonpositivity even when the second-versus-third-order test fails.

KD nonclassicality has also been turned into a coherence monotone. For a fixed incoherent basis $\{|b_j\rangle\}$7, the KD-nonclassicality coherence is defined by

$\{|b_j\rangle\}$8

It obeys

$\{|b_j\rangle\}$9

where $\rho$0, and for pure states the upper bound is saturated: $\rho$1 The same framework yields uncertainty-based lower bounds via the Maassen–Uffink relation and trade-off relations between KD-nonclassicality coherences in different bases (Budiyono et al., 2023).

6. Operational roles in measurement, tomography, metrology, thermodynamics, and scrambling

The KD distribution has become a unifying operational language across several quantum-information subfields. In measurement theory, the imaginary part of the KD distribution governs disturbance. Under the unitary $\rho$2, the rate of change of a later measurement probability is

$\rho$3

so nonreal conditional KD values are directly tied to measurement back-action (Arvidsson-Shukur et al., 2024).

In weak measurement theory, the weak value of an observable is a conditional KD average. Anomalous weak values require KD nonpositivity, and “strange” weak values imply contextuality in hidden-variable descriptions (Arvidsson-Shukur et al., 2024, Budiyono et al., 2023). Direct experimental access to KD values can be obtained without post-selection using ancilla-assisted cycle-test circuits that estimate the relevant Bargmann invariants. In these schemes, the real and imaginary parts of $\rho$4 are obtained from ancilla expectation values after a controlled cyclic SWAP, and the same toolkit extends to OTOCs and post-selected Fisher information (Wagner et al., 2023).

In finite-dimensional tomography, KD enjoys several properties emphasized as comparative advantages over many other quasiprobabilities. It vanishes off the grid of actual eigenvalue pairs, obeys correct marginals, satisfies a Cauchy–Schwarz bound

$\rho$5

and in spin-$\rho$6 and spin-1 systems two noncommuting observables suffice to determine the state completely from the KD distribution (Umekawa et al., 2023).

In quantum metrology, KD negativity and imaginarity have been linked to enhanced performance. Standard Fisher-information expressions can be written in terms of conditional KD distributions, and post-selected Fisher information can be expressed as a quasiprobability covariance of an extended KD distribution; KD negativity enables unbounded compressive concentration of Fisher information into selected trials (Arvidsson-Shukur et al., 2024). The 2020 bounds on achievable KD nonclassicality were explicitly motivated by metrological advantage, including the observation that only the negative-real part can underlie nonclassical Fisher-information boosts, whereas the imaginary part encodes disturbance (Arvidsson-Shukur et al., 2020).

In quantum thermodynamics, KD-based work quasiprobabilities replace the two-point-measurement protocol when initial coherence makes ordinary work statistics inadequate. For a driven Hamiltonian,

$\rho$7

This quasiprobability reproduces the unperturbed average work and its higher moments while allowing complex values that quantify noncommutativity (Hernández-Gómez et al., 2024). Interferometric experiments in an NV-center electron-nuclear spin platform reconstructed the characteristic function and extracted the quasiprobability distribution of work, including its first and second moments (Hernández-Gómez et al., 2024). Related KDQ constructions have also been developed for coherent collision models, where negativities and imaginary parts diagnose quantum traits in energy exchange and work statistics (Pezzutto et al., 10 Mar 2025).

In scrambling and chaos, the out-of-time-ordered correlator can be expressed as a moment of an extended KD distribution. The resulting quasiprobability exhibits negative and nonreal behavior, distinguishes chaotic from integrable dynamics, and develops symmetry-breaking “pitchfork” bifurcations at the onset of scrambling (Halpern et al., 2017). This operational role is now one of the standard applications summarized in the broader review literature (Arvidsson-Shukur et al., 2024).

7. Relations to other quasiprobabilities and broader conceptual developments

The KD distribution is frequently contrasted with the Wigner function. Unlike the Wigner formalism, which is tied to specific phase-space structures, KD can be defined for arbitrary observables or bases (Arvidsson-Shukur et al., 2024). For finite-state systems, one argument for its special status is that it stays supported only on the actual spectral grid and can encode the full density operator from as few as two noncommuting observables, with the imaginary part essential for full distinguishability in qubits (Umekawa et al., 2023).

There are also direct structural links between KD and discrete Wigner theory. For KD distributions defined in qudit Fourier bases, the discrete Fourier transform of the KD table yields expectation values of Weyl–Heisenberg operators, and a further symplectic Fourier transform gives the odd-dimensional discrete Wigner distribution. This provides a KD-to-Wigner mapping without reconstructing the density matrix (Burkat et al., 17 Feb 2025).

Resource-theoretic and simulation-theoretic questions have sharpened these comparisons. In KD representations based on mutually unbiased bases, pure KD-positive states always have uniform distributions on their support of size $\rho$8, “in full analogy to the stabilizer states” (Burkat et al., 17 Feb 2025). At the same time, positivity preservation under unitary evolution need not coincide with stochastic quasiprobability evolution: there exist unitaries that preserve KD positivity of every state while inducing superoperators with negative entries, and classical sampling algorithms can become exponentially inefficient even when the state remains KD positive throughout the evolution (Burkat et al., 17 Feb 2025). This suggests that KD positivity alone does not determine classical simulability.

More abstractly, KD has been situated between classical and “postquantum” quasiprobabilities. Under mild assumptions one obtains a strict hierarchy

$\rho$9

together with bounds such as

$B$00

and extensions of the $B$01-norm bound to arbitrarily many measurements (Liu et al., 12 Apr 2025). These results formalize the statement that KD enlarges classical probability theory, but not arbitrarily.

A further characterization singles out KD among all Born-compatible quasiprobability representations on the joint spectrum of two observables. For such representations, each observable $B$02 has an associated conditional expectation given $B$03; among them, only the left-KD representation has the property that this conditional expectation coincides with the best predictor of $B$04 by a function of $B$05 for all $B$06 (Spriet et al., 3 Nov 2025). This identifies a precise statistical-optimality property that is unique to KD.

Taken together, these developments present the KD quasiprobability distribution as a flexible but highly structured framework. Its central mathematical tension is that it retains normalization, correct marginals, and, in many settings, informational completeness, while relaxing positivity and reality. Its central physical tension is that those relaxed features are precisely what permit it to encode disturbance, incompatibility, coherence, contextuality, and operational quantum advantage.