Almost no experiments have classical Kirkwood-Dirac representations

Technical Summary: "Almost no experiments have classical Kirkwood-Dirac representations"

Problem Statement
A central challenge in quantum information is delineating the boundary between classical and quantum phenomena. In the quasiprobability framework, a state is deemed "classical" if it admits a representation as a positive probability distribution. While the Wigner function has been pivotal in this regard, the Kirkwood-Dirac (KD) distribution has emerged as a versatile tool for discrete-variable quantum information, offering applications in metrology, weak values, thermodynamics, and contextuality.

The KD distribution $Q(\rho)$ is defined relative to two arbitrary bases, $\{|a_i\rangle\}$ and $\{|b_j\rangle\}$, associated with observables $A$ and $B$. A state $\rho$ is KD-positive (classical) if all entries of $Q(\rho)$ are real and non-negative. A critical open problem is the full characterization of the set of KD-positive states, denoted $E_{KD+}$. Unlike the Wigner function, where the set of positive states is known to be limited to Gaussian states (for pure states), the structure of $E_{KD+}$ for KD distributions is complex. It is not necessarily a polytope, and its extremal points may include mixed states that are not convex combinations of pure KD-positive states (so-called "exotic states"). This complexity makes it difficult to identify nonclassicality or simulate quantum circuits efficiently using KD distributions.

Methodology
The authors employ a probabilistic geometric approach to characterize the sets of KD-positive states, KD-real observables, and KD-positivity-preserving unitaries.

1. Random Basis Selection: Instead of analyzing specific, potentially pathological cases, the authors consider two sets of basis projectors, $A = \{|a_i\rangle\langle a_i|\}$ and $B = \{|b_j\rangle\langle b_j|\}$, chosen randomly according to the Haar measure on the unitary group $U(d)$.
2. Linear Algebraic Formulation: The problem is mapped to the study of the transition matrix $U$ (where $U_{ij} = \langle a_i | b_j \rangle$). The authors define a linear map $C_U$ acting on the space of real matrices. They demonstrate that the space of KD-real observables, $V_{KDr}$, is isomorphic to the kernel of this map, $\text{Ker}(C_U)$.
3. Dimensional Analysis: The authors establish that the convex hull of the basis projectors, $\text{conv}(A \cup B)$, is always contained within $E_{KD+}$. The core of the proof involves showing that for almost all unitary matrices $U$ (specifically those with no zero entries), the dimension of the space of KD-real observables is minimal ($2d-1$). This implies that no "extra" KD-positive states or observables exist beyond the convex hull and real span of the basis projectors.
4. Measure Theory: By constructing a specific unitary matrix $U_\star$ for which the kernel dimension is minimal and utilizing the fact that the determinant of the relevant linear map is a non-zero polynomial in the matrix elements of $U$, the authors invoke Proposition 3 (a result regarding the vanishing Haar measure of zero sets of real analytic functions) to prove that this minimal structure holds with probability 1.

Key Contributions and Results
The paper presents Theorem 1, which asserts that for almost all choices of two $d$-dimensional bases (with $d > 2$), the sets of classical objects in the KD framework are minimal and explicitly characterized:

• KD-Positive States ($E_{KD+}$): The set of KD-positive states is exactly the convex hull of the two bases:
  $$E_{KD+} = \text{conv}(A \cup B)$$
  This set is a polytope of dimension $2(d-1)$ with exactly $2d$ known vertices (the projectors of the two bases). There are no "exotic" mixed extremal states for almost all KD distributions.
• KD-Real Observables ($V_{KDr}$): An observable is KD-real if and only if it lies in the real vector space spanned by the two bases:
  $$V_{KDr} = \text{span}_{\mathbb{R}}(A \cup B)$$
• KD-Positive Measurements ($M_{KD+}$): Any measurement element with a positive KD representation must be a linear combination of the basis projectors with non-negative coefficients.
• KD-Positivity-Preserving Unitaries: The only unitary transformations that map KD-positive states to KD-positive states are trivial global phases:
  $$T = e^{i\alpha}I, \quad \alpha \in \mathbb{R}$$

Significance and Implications
The authors argue that these results fundamentally simplify the understanding of classicality in the KD framework for "almost all" experimental setups:

1. Robustness of Nonclassicality: Since the set of classical states is a simple polytope with known facets, constructing witnesses for nonclassicality becomes straightforward. One can easily lower-bound the distance of a given state from the classical set using the known geometry of $E_{KD+}$.
2. Contextuality: The results clarify the conditions for generalized contextuality. Since the set of input states yielding classical (noncontextual) behavior is minimal and well-defined for almost all observables, the detection of contextuality is robust against the specific choice of bases.
3. Absence of Exotic States: The paper demonstrates that "exotic" KD-positive states (mixed states that cannot be decomposed into KD-positive pure states) are rare phenomena, existing only in tailored, non-generic KD distributions (e.g., specific spin-1 or multi-qubit systems).
4. Classical Simulation: The results imply severe limitations on the efficiency of classical simulations of quantum circuits using KD distributions. Since the only KD-positive circuits involve initialization in $\text{conv}(A \cup B)$ followed immediately by measurement in $M_{KD+}$, almost all non-trivial quantum circuits will exhibit KD non-positivity, rendering them difficult to simulate classically via Monte Carlo sampling methods that rely on positive quasiprobabilities.

In conclusion, the paper establishes that the complexity often associated with KD distributions is the exception rather than the rule. For the vast majority of experimental configurations, the classical-quantum boundary is sharp, simple, and fully characterized by the convex geometry of the measurement bases.