Born-Compatible Quasiprobability Representations

• Born-compatible quasiprobability representations are mathematical structures that assign quasiprobabilities to quantum states and effects while exactly reproducing Born rule outcome probabilities.
• They employ constructions such as SIC-POVMs and dual operator frames to bridge classical probability with nonclassical features like negativity and complex-valued entries.
• Their operational significance extends to quantum simulation, resource theories, and foundational interpretations by highlighting conditions for classical simulability and quantum speedup.

Born-compatible quasiprobability representations are mathematical structures for expressing quantum states and measurement probabilities such that all Born rule predictions are preserved at the level of marginals. These representations provide unified frameworks that interpolate between classical probability theory and quantum theory, with an explicit mapping from density operators to (typically nonclassical, possibly negative or complex) “quasiprobabilities” over some structured space. The concept of Born compatibility—reproducing all outcome probabilities as per the Born rule—places sharp restrictions on admissible representations, and underpins multiple foundational, operational, and resource-theoretical developments across quantum information theory, contextuality analysis, simulation, and the interpretation of probability itself.

1. Definitions and Foundational Motivation

A quasiprobability representation (QPR) of quantum theory is a pair of affine, linear maps from density operators and measurement effects to (real or complex) functions over a finite index set or continuous space $\Lambda$, often subject to additional frame-theoretic constraints. The central requirement of Born compatibility is that the representation must reproduce the outcome probabilities predicted by the Born rule. Explicitly, for a Hilbert space $\mathcal{H}$, density operator $\rho$, and effect (measurement operator) $E$: $\Tr[\rho E] = \int_\Lambda d\lambda \;\mu_\rho(\lambda)\;\xi_{E}(\lambda),$ where $\mu_\rho$ and $\xi_E$ are the representations of the state and effect, potentially allowing $\mu_\rho(\lambda), \xi_E(\lambda)$ outside the interval $[0,1]$.

Born compatibility is equivalently characterized by the marginalization property: any marginalization over some subset of the indices (e.g., measurement outcomes or ontic variables) yields the correct Born probabilities for relevant observables. Importantly, the representation may be non-unique and may yield negative or complex “quasiprobabilities,” reflecting quantum interference and nonclassicality (DeBrota et al., 2020, Wallman et al., 2012, Wagner et al., 13 Sep 2025).

2. Construction: SIC-based and Frame-based Approaches

A particularly significant construction of a Born-compatible QPR is based on symmetric informationally complete positive operator-valued measures (SIC-POVMs). For a Hilbert space of dimension $d$, assume existence of a SIC $\{\Pi_i\}_{i=1}^{d^2}$ such that

$\Pi_i = \frac{1}{d}|\psi_i\rangle\langle\psi_i|, \qquad \Tr(\Pi_i\Pi_j) = \frac{d\delta_{ij} + 1}{d^2(d+1)}.$

Any subjective probability assignment $p(H_i)$ to the SIC outcomes determines a unique density operator via the affine reconstruction formula: $$\rho = \sum_{i=1}^{d^2} \left[ (d+1)p(H_i) - \frac{1}{d} \right]\Pi_i.$$ Given another POVM $\{D_j\}$, the Born rule in this representation is: $\Pr(D_j) = (d+1)\sum_{i=1}^{d^2} p(H_i)\left(\frac{1}{d}\Tr[\Pi_i D_j]\right) - 1,$ where $\frac{1}{d}\Tr[\Pi_i D_j]$ is interpreted as a conditional probability $p(D_j|H_i)$ for post-SIC collapse (DeBrota et al., 2020).

More generally, QPRs can be described via dual operator frames $\{F_i\}, \{D_i\}$ such that: $\mu_i(\rho) = \Tr[F_i \rho], \qquad \mu_i(E) = \Tr[E D_i],$ and the biorthonormality condition $\Tr[F_i D_j] = \delta_{ij}$ ensures Born compatibility: $\Tr[E \rho] = \sum_{i}\mu_i(\rho)\mu_i(E).$ All Born-compatible QPRs correspond to such frame-dual pairs, and the structure theorem (Wagner et al., 13 Sep 2025) classifies all such representations in finite-dimensional and tomographically-local settings.

3. Characterization Theorems and Uniqueness Properties

Born compatibility, while restrictive, admits multiple distinct QPRs—including real-valued (e.g., Wigner) and complex-valued (e.g., Kirkwood–Dirac) distributions. For a pair of nondegenerate observables $\hat A$ and $\hat B$, any QPR on their joint spectrum that reproduces the Born marginals is Born-compatible if

$$\sum_{b} Q_{a,b}(\rho) = \langle a|\rho|a\rangle, \qquad \sum_{a} Q_{a,b}(\rho) = \langle b|\rho|b\rangle.$$

The general construction involves operator frames on $\mathcal{B}(\mathcal{H})$, with the Born compatibility conditions enforcing that the frame sums marginalize appropriately to the projectors onto the spectra of $\hat A$ and $\hat B$ (Spriet et al., 3 Nov 2025).

Among all such representations, the Kirkwood–Dirac (KD) distribution is uniquely distinguished by its conditional expectation property: only the KD QPR,

$$Q_{\rm KD}(a,b) = \langle b|a\rangle \langle a|\rho|b\rangle,$$

ensures the marginal-conditional expectation of any observable given $\hat B$ coincides with the best (orthogonal) predictor onto the algebra generated by $\hat B$. No other Born-compatible QPR enjoys this pullthrough or optimal predictor property (Spriet et al., 3 Nov 2025).

4. Negativity, Geometry, and Classical Limits

A central feature of Born-compatible QPRs is the appearance of negativity or complex-valued entries in the representation of at least some quantum states or measurement effects. Notably, in the SIC-based representation, the quasi-probability weights $(d+1)p(H_i)-1/d$ may become negative for $p(H_i) < 1/[d(d+1)]$. Basis changes between different SICs need not be stochastic, and the allowed region of probability vectors forms a convex "qplex"—a restricted convex set within the full $d^2$-simplex with inner product constraints corresponding to quantum compatibility (DeBrota et al., 2020).

For qubit systems, no Born-compatible QPR can be nonnegative for more than four orthonormal bases; these must correspond, up to rotation, to the eight corners of a right cuboid on the Bloch sphere, permuted by the Pauli group. In higher dimensions, at most $2^{d^2}$ pure states can be nonnegative in any Born-compatible QPR, and they must satisfy strong geometric and combinatorial constraints (Wallman et al., 2012).

The advent of negativity marks the boundary between classical, noncontextual ontological subtheories (where simulation by classical probability theory is possible) and the full nonclassical quantum structure, and underpins "magic" resource theories and quantum speedup.

5. Operational and Foundational Significance

Born-compatible QPRs enable a spectrum of operational and interpretational advances:

• Normative Justification: Within QBist/Bayesian frameworks, the Born rule emerges as an extension of Dutch-book coherence, elevated from a mere empirical law to a normative rule enforcing consistency among an agent's probabilistic beliefs under quantum constraints (DeBrota et al., 2020).
• Conditional Inference: The KD representation uniquely supports conditional expectation formulas that are internally consistent with Hilbert–Schmidt projections; thus, only KD-based inferences reproduce quantum Bayesian updating without ambiguity (Spriet et al., 3 Nov 2025).
• Simulability and Resource Theory: Negative-value subtheories are necessary for universal quantum computation and for protocols that transcend classical simulability. Subtheories admitting nonnegative Born-compatible QPRs are classically tractable (Wallman et al., 2012).
• Foundations and Extensions: The frame-theoretic and structure-theorem approaches characterize all Born-compatible QPRs for finite-dimensional quantum theory and generalize to arbitrary tomographically-local GPTs (Wagner et al., 13 Sep 2025).

6. Applications, Extensions, and Limitations

• Quantum Measurement and Temporal Correlations: For sequential measurements, standard Born rule assignments may fail to yield consistent joint probabilities. Correction terms lead to a quasiprobability that can exhibit negativity, and unifies spatiotemporal quantum predictions via a single operator $\varrho_{AB}$ (Fullwood et al., 22 Jul 2025).
• Complex-valued QPRs and Generalized Theories: The structure theorem for complex-valued, Born-compatible QPRs shows all such representations decompose via canonical functorial operations between states/effects and the complexification of the theory (Wagner et al., 13 Sep 2025).
• Consistency and Classical Limits: In classical regimes or specially restricted subtheories, Born-compatible QPRs may be everywhere nonnegative, but such theories are inevitably highly sparse and structurally constrained. Nonclassical features are quantified by the necessity (and sometimes the structure) of negativity.

7. Summary Table: Main Variants and Their Properties

Construction/Representation	Key Features	Uniqueness/Constraint
SIC-based affine form	Informational completeness, minimal twist of total probability, explicit negativity	Exists if SICs available in dimension; supports affine extension of Dutch-book argument
Kirkwood–Dirac (KD)	Complex-valued, captures best predictor conditional expectations, supports weak-value theory	Unique among Born-compatible QPRs for optimal conditionality property
Wigner/discrete Wigner	Real-valued, phase space-based, admits negativity	Born-compatible; not uniquely determined by conditional expectations
Frame–dual frame (general theorem)	All Born-compatible QPRs, accommodates complex cases	All relation boils down to biorthonormality via frames

Born-compatible quasiprobability representations thus serve as a precise unifying language for quantifying, modeling, and interpreting the boundary between classical and quantum probabilistic structure, the status of the Born rule, and the operational architectures of quantum processing and inference. The constraints imposed by Born compatibility sharply delineate the scope of classically simulable quantum phenomena, the geometry of quantum state spaces, and the nature of negative and complex probability in quantum theory.