Kirkwood–Dirac Quasiprobability Representation

• The KD quasiprobability representation is a quantum framework that maps state statistics to pairs of observables, capturing noncommutativity and contextuality through potential negative or complex values.
• It relies on normalization, marginal recovery, and an inversion formula via dual operator bases, enabling complete state reconstruction and offering insights into measurement disturbances and weak values.
• Generalizations of the KD framework extend to multi-measurement scenarios, discrete Fourier transforms, and resource theories, highlighting its versatility in quantum metrology, thermodynamics, and foundational studies.

The Kirkwood–Dirac (KD) quasiprobability representation is a quasiprobability description of a quantum state relative to two observables, introduced by Kirkwood in 1933 and Dirac in 1945 and now used across quantum metrology, chaos, thermodynamics, weak-value theory, and foundations (Arvidsson-Shukur et al., 2020). For nondegenerate observables with eigenbases $\{|a_i\rangle\}_{i=1}^d$ and $\{|b_j\rangle\}_{j=1}^d$, the standard KD array of a state $\rho$ is

$$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$

It is normalized, reproduces the Born marginals of both observables, and is informationally complete when $\langle a_i|b_j\rangle\neq 0$ for all $i,j$, but its entries can be negative or nonreal. Those departures from classical probability are precisely what make the representation useful: they encode noncommutativity, disturbance, anomalous weak values, and several operational quantum advantages (Arvidsson-Shukur et al., 2024).

1. Definition, reconstruction, and generalizations

The KD representation is indexed by outcome pairs of two measurements. Its basic structural properties are the same across the recent literature: normalization,

$\sum_{i,j}Q_{ij}(\rho)=1,$

marginal recovery,

$$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$

and, when $\langle b_j|a_i\rangle\neq 0\ \forall i,j$, the inversion formula

$$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$

Equivalently, $\{|b_j\rangle\}_{j=1}^d$0 forms a dual operator basis to $\{|b_j\rangle\}_{j=1}^d$1 (Zhang et al., 25 Jan 2026).

This dual-basis viewpoint places KD within the general theory of quasiprobability representations. For any operator $\{|b_j\rangle\}_{j=1}^d$2, one may introduce the dual symbol

$\{|b_j\rangle\}_{j=1}^d$3

so that

$\{|b_j\rangle\}_{j=1}^d$4

The formal role is therefore analogous to a classical expectation value, except that the “distribution” $\{|b_j\rangle\}_{j=1}^d$5 is generally complex (Arvidsson-Shukur et al., 2024).

The standard two-basis KD distribution admits several extensions. One is the ordered $\{|b_j\rangle\}_{j=1}^d$6-measurement or extended KD array,

$\{|b_j\rangle\}_{j=1}^d$7

which is central in scrambling and multi-time problems (Arvidsson-Shukur et al., 2024). Another is the POVM form

$\{|b_j\rangle\}_{j=1}^d$8

which preserves normalization and marginals while allowing nonprojective measurements (Liu et al., 12 Apr 2025). A further generalization replaces finite-dimensional basis pairs by Fourier-dual structures on second countable locally compact abelian groups, where

$\{|b_j\rangle\}_{j=1}^d$9

with $\rho$0 the Weyl operator; in that setting the KD transform is a unitary map $\rho$1 (Spriet, 31 Jul 2025).

A recurrent theme in the literature is that, unlike Wigner-type constructions, KD is arbitrary-basis rather than canonically tied to position–momentum structure. This suggests why it has become useful in discrete systems, generalized measurements, and basis-dependent resource theories (Arvidsson-Shukur et al., 2024).

2. Classicality, nonclassicality, and what KD values mean

A state is called KD-positive, or KD-classical, when all entries satisfy $\rho$2. In that case the KD table is a bona fide joint probability distribution over the two outcome sets. If some entry is negative or nonreal, the state is KD-nonclassical (Langrenez et al., 2023).

The recent literature distinguishes two kinds of nonclassicality. Negative KD elements signify nonclassicality of the “Metrological” type: they bound how weak-value amplification can exceed classical metrology limits, and they signal operator-scrambling in out-of-time-ordered correlators. Nonreal elements encode a different kind of nonclassicality: they quantify measurement back-action and “thermodynamic” contextuality, for example in fluctuation-theorem settings involving successive noncommuting energy measurements (Arvidsson-Shukur et al., 2020).

A common misconception is that noncommutation alone guarantees KD nonclassicality. That claim is false. Pairwise noncommutation of $\rho$3, $\rho$4, and $\rho$5 does not suffice: explicit examples exist in which $\rho$6, $\rho$7, and $\rho$8 all pairwise fail to commute while the KD array remains real and nonnegative (Arvidsson-Shukur et al., 2020). The sufficient-nonclassicality theorem for pure states therefore uses finer support data. If

$\rho$9

where $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$0 and $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$1 count support sizes in the two bases and $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$2 record aligned and orthogonal overlaps, then at least one KD entry is negative or nonreal (Arvidsson-Shukur et al., 2020). In particular, if no KD entry vanishes, the KD array is automatically nonclassical (Arvidsson-Shukur et al., 2020).

Several quantitative measures separate real negativity from nonreality. The paper “Conditions tighter than noncommutation needed for nonclassicality” defines

$$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$3

and proves that, over all pure states and sequences of $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$4 bases in dimension $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$5,

$$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$6

This maximum is achieved iff each consecutive pair of bases is mutually unbiased and the initial and final bases have uniform overlap with the state (Arvidsson-Shukur et al., 2020).

KD also fits into a broader hierarchy of admissible quasiprobabilities. Classical joint distributions form a strict subset of KD quasiprobabilities, and KD quasiprobabilities form a strict subset of a larger postquantum set defined only by normalization and entrywise modulus bounds (Liu et al., 12 Apr 2025). For KD tables $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$7, one has $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$8, $$Q_{ij}(\rho)=\langle b_j|a_i\rangle\,\langle a_i|\rho|b_j\rangle =\mathrm{Tr}\!\bigl(\Pi_{b_j}\Pi_{a_i}\rho\bigr).$$9, and, more strongly,

$\langle a_i|b_j\rangle\neq 0$0

with the same $\langle a_i|b_j\rangle\neq 0$1 bound extending to arbitrary-length KD chains (Liu et al., 12 Apr 2025).

3. Geometry of KD-positive states

The set of KD-positive states,

$\langle a_i|b_j\rangle\neq 0$2

is a convex subset of the state space. By Krein–Milman,

$\langle a_i|b_j\rangle\neq 0$3

Two natural subsets of pure states always lie inside $\langle a_i|b_j\rangle\neq 0$4: the projectors onto the $\langle a_i|b_j\rangle\neq 0$5-eigenbasis and onto the $\langle a_i|b_j\rangle\neq 0$6-eigenbasis (Langrenez et al., 2023).

A central geometric question is whether these basis projectors already generate the entire KD-positive set. Langrenez, Arvidsson-Shukur, and De Bièvre show that this occurs exactly when

$\langle a_i|b_j\rangle\neq 0$7

They identify three regimes where only convex combinations of the $\langle a_i|b_j\rangle\neq 0$8- and $\langle a_i|b_j\rangle\neq 0$9-eigenprojectors are KD-positive: $i,j$0 with $i,j$1; an open dense subset of $i,j$2, equivalently full Haar measure, in dimension $i,j$3; and discrete-Fourier-transform bases in prime dimension. They also prove a stability statement: if $i,j$4 is close to such a $i,j$5, then the same convex-hull description persists (Langrenez et al., 2023).

In these minimal regimes, $i,j$6 is the $i,j$7-dimensional convex polytope with exactly $i,j$8 vertices, namely the $i,j$9- and $\sum_{i,j}Q_{ij}(\rho)=1,$0-projectors, and its facets are labeled by leaving out one $\sum_{i,j}Q_{ij}(\rho)=1,$1-projector and one $\sum_{i,j}Q_{ij}(\rho)=1,$2-projector (Langrenez et al., 2023). Outside these regimes the geometry is richer. For a specific spin-$\sum_{i,j}Q_{ij}(\rho)=1,$3 example with

$\sum_{i,j}Q_{ij}(\rho)=1,$4

the KD-positive set contains mixed extreme points that are not convex combinations of pure KD-positive states. Writing

$\sum_{i,j}Q_{ij}(\rho)=1,$5

one finds $\sum_{i,j}Q_{ij}(\rho)=1,$6 for $\sum_{i,j}Q_{ij}(\rho)=1,$7, while $\sum_{i,j}Q_{ij}(\rho)=1,$8 for $\sum_{i,j}Q_{ij}(\rho)=1,$9 (Langrenez et al., 2023).

For pure states, the DFT case is especially rigid. In the characterization of KD-classical pure states, the DFT relation implies that the only KD-classical pure states satisfy

$$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$0

where $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$1 and $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$2 are the support sizes of $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$3 in the two bases. This resolves the conjecture of De Bièvre cited there (Xu, 2022). A later DFT study extends this to mixed states in prime-power dimensions, proving that the KD-classical set is the convex hull of KD-classical pure states and organizing the general $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$4-dimensional structure by a directed graph on coprime factorizations of $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$5 (Cai et al., 14 Mar 2026).

4. Conditional expectation, weak values, and measurement dynamics

The KD representation is tightly connected to conditional averages and weak values. In the standard two-basis setting, a weak value of $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$6 under postselection on $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$7 can be written as a KD-conditional average,

$$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$8

with $$\sum_j Q_{ij}(\rho)=\langle a_i|\rho|a_i\rangle,\qquad \sum_i Q_{ij}(\rho)=\langle b_j|\rho|b_j\rangle,$$9 the conditionalized KD weights (Arvidsson-Shukur et al., 2024). This is one reason negative or nonreal KD values are directly associated with anomalous weak values.

Recent work sharpens that connection by characterizing KD among all Born-compatible quasiprobability representations on the joint spectrum of $\langle b_j|a_i\rangle\neq 0\ \forall i,j$0 and $\langle b_j|a_i\rangle\neq 0\ \forall i,j$1. For any such representation, one can define a conditional expectation of $\langle b_j|a_i\rangle\neq 0\ \forall i,j$2 given $\langle b_j|a_i\rangle\neq 0\ \forall i,j$3. What is special about KD is uniqueness: only the KD representation has the property that its associated conditional expectation of $\langle b_j|a_i\rangle\neq 0\ \forall i,j$4 given $\langle b_j|a_i\rangle\neq 0\ \forall i,j$5 coincides with the best predictor of $\langle b_j|a_i\rangle\neq 0\ \forall i,j$6 by a function of $\langle b_j|a_i\rangle\neq 0\ \forall i,j$7, for all $\langle b_j|a_i\rangle\neq 0\ \forall i,j$8 (Spriet et al., 3 Nov 2025). In the left-KD case,

$\langle b_j|a_i\rangle\neq 0\ \forall i,j$9

This least-squares characterization distinguishes KD from other Born-compatible frames (Spriet et al., 3 Nov 2025).

The role of KD in measurement theory has also been extended beyond the weak-measurement limit. A von Neumann pointer model with coupling

$$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$0

and Gaussian pointer width $$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$1 yields the decoherence factor

$$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$2

In the resulting reduced dynamics, the time-dependent KD array obeys the linear interpolation

$$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$3

where

$$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$4

Thus $$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$5 gives the full complex KD distribution in the weak regime, whereas $$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$6 yields the real nonnegative Margenau–Hill or “classical Wigner” formula in the strong regime, while informational completeness is maintained throughout the transition (Zhang et al., 25 Jan 2026).

This measurement-strength interpolation suggests a unifying reading of KD: it is not merely a weak-measurement artifact, but a representation whose deformation under pointer-induced decoherence continuously connects weak values, ABL-type conditional averages, and projective statistics (Zhang et al., 25 Jan 2026).

5. Operational roles in quantum information, thermodynamics, and foundations

The KD representation has become operationally significant because its negative and nonreal values track concrete tasks and phenomena. In measurement-disturbance theory, the imaginary part of KD controls the change of probabilities under unitary generation: $$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$7 The imaginary sector therefore quantifies disturbance in a basis-resolved way (Arvidsson-Shukur et al., 2024).

In direct state measurement, a weak measurement of position $$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$8 followed by a strong measurement of momentum $$\rho=\sum_{i,j}Q_{ij}(\rho)\,\frac{|a_i\rangle\langle b_j|}{\langle b_j|a_i\rangle}.$$9 yields

$\{|b_j\rangle\}_{j=1}^d$00

so scanning $\{|b_j\rangle\}_{j=1}^d$01 directly reconstructs a wavefunction or density matrix in KD form (Arvidsson-Shukur et al., 2024).

In thermodynamics, KD provides a quasiprobability of work that preserves coherence discarded by two-point-measurement protocols. For instantaneous energy measurements,

$\{|b_j\rangle\}_{j=1}^d$02

Interferometric reconstruction with an ancilla accesses the characteristic function $\{|b_j\rangle\}_{j=1}^d$03, whose real and imaginary parts are read out through $\{|b_j\rangle\}_{j=1}^d$04 and $\{|b_j\rangle\}_{j=1}^d$05, and Fourier transformation yields the KD work distribution (Hernández-Gómez et al., 2024). Its first moment equals the standard average work,

$\{|b_j\rangle\}_{j=1}^d$06

while the imaginary part of the second moment is

$\{|b_j\rangle\}_{j=1}^d$07

which is also the commutator term entering the Robertson–Schrödinger uncertainty relation (Hernández-Gómez et al., 2024).

In chaos and scrambling, extended KD quasiprobabilities underlie out-of-time-ordered correlators, with negativity and nonreality distinguishing scrambling from decoherence (Arvidsson-Shukur et al., 2024). In foundations, KD negativity appears in Leggett–Garg inequality violations, consistent-histories interference terms, and contextuality. One recent coherence paper states that whenever any weak value $\{|b_j\rangle\}_{j=1}^d$08 is “strange,” meaning it has a negative real part or nonzero imaginary part, it is known to witness quantum contextuality; since $\{|b_j\rangle\}_{j=1}^d$09 iff at least one such strange weak value appears, the corresponding KD-based coherence measure is directly tied to contextuality (Budiyono et al., 2023).

A broader resource-theoretic claim is also available: for every quantum resource defined by a closed convex set of nonresourceful states, there exists a KD-type quasiprobability distribution whose negativity reveals that resourceful state, and one can choose the construction so that the negative value is proportional to the Frobenius distance to the closest nonresourceful state (Tan et al., 2024). This suggests that KD-type negativity is not confined to one operational niche.

6. Resource-theoretic, temporal, and abstract extensions

Several lines of work turn KD nonclassicality into standalone resource measures. One defines basis-relative KD-nonclassicality coherence by

$\{|b_j\rangle\}_{j=1}^d$10

This quantity vanishes iff $\{|b_j\rangle\}_{j=1}^d$11 is diagonal in the reference basis, is upper bounded by $\{|b_j\rangle\}_{j=1}^d$12, and for pure states satisfies equality,

$\{|b_j\rangle\}_{j=1}^d$13

(Budiyono et al., 2023). A complementary construction based only on the imaginary part defines

$\{|b_j\rangle\}_{j=1}^d$14

proves convexity and several monotonicity properties, shows

$\{|b_j\rangle\}_{j=1}^d$15

and establishes equality with $\{|b_j\rangle\}_{j=1}^d$16 for a single qubit (Budiyono et al., 2023).

For bipartite systems, KD nonreality over orthonormal product bases yields measures $\{|b_j\rangle\}_{j=1}^d$17 and $\{|b_j\rangle\}_{j=1}^d$18 of general quantum correlations. These vanish exactly on classical–quantum or classical–classical states, are convex and locally unitary invariant, and on pure states provide faithful witnesses of entanglement and measurement-induced nonlocality (Budiyono et al., 2022).

Recent work has also produced experimentally economical witnesses of KD nonpositivity using moments. Defining

$\{|b_j\rangle\}_{j=1}^d$19

one has, for any KD-positive table,

$\{|b_j\rangle\}_{j=1}^d$20

More generally, the Hankel matrix $\{|b_j\rangle\}_{j=1}^d$21 built from $\{|b_j\rangle\}_{j=1}^d$22 must satisfy $\{|b_j\rangle\}_{j=1}^d$23, so any violation of $\{|b_j\rangle\}_{j=1}^d$24 detects KD nonpositivity (Chakrabarty et al., 9 Jun 2025).

The temporal extension is structurally significant. For a multi-time process, one defines right, left, and doubled temporal KD quasiprobabilities, with $\{|b_j\rangle\}_{j=1}^d$25 and real parts equal to temporal Margenau–Hill quasiprobabilities. These objects are experimentally accessible through interferometric schemes and feed directly into temporal Bloch tomography. The associated reconstructed “temporal states” recover, as special cases, consistent histories, pseudo-density operators, process tensors, quantum combs, superdensity operators, and doubled-density-operator formalisms (Jia et al., 8 Jan 2026).

Finally, the abstract harmonic-analysis formulation on second countable locally compact abelian groups shows that KD is the natural standard-ordering, or Kohn–Nirenberg, quantization on $\{|b_j\rangle\}_{j=1}^d$26. In that setting, generalized pure KD-positive states are, up to Weyl–Heisenberg action, Haar measures on closed subgroups; the classical fragment is nontrivial iff the group has a compact connected component; and for connected compact abelian groups, KD-positive states are exactly classical mixtures of Fourier-basis projectors (Spriet, 31 Jul 2025). This places the finite-dimensional geometry of KD positivity within a broader phase-space and representation-theoretic framework.

The resulting picture is coherent across these developments. KD behaves like a joint probability distribution to the extent allowed by Born marginals and normalization, but its negative and nonreal sectors encode the obstructions to a classical joint model. Those obstructions are not incidental defects of the formalism; they are the features through which KD connects measurement disturbance, weak values, contextuality, coherence, work statistics, temporal correlations, and the geometry of the quantum state space.