Complex-Valued Quasiprobability Representations

• Complex-valued quasiprobability representations are advanced frameworks that assign complex distributions to quantum states, measurements, and transformations, capturing phase and interference.
• They utilize invertible frame maps and dual frames to express quantum operations in generalized probabilistic theories, unifying classical and quantum probabilities.
• Such representations provide actionable insights into quantum contextuality, computational speedup, and experimental tomography, bridging theory with practical simulation.

A complex-valued quasiprobability representation is a mathematical and operational framework that assigns to the elements of a physical theory—most notably, quantum states, measurements, and transformations—distributions that take values in the complex numbers. These representations extend traditional (real-valued) quasiprobability frameworks, such as the Wigner function and its finite-dimensional analogs, to settings where phases and coherence are intrinsic and cannot be completely encoded by real numbers alone. Complex-valued quasiprobability representations have become essential in quantum information science and foundational studies for capturing phenomena such as contextuality, quantum interference, and the operational impossibility of classically simulating genuinely quantum computation.

1. Formal Structure and Definitions

In both finite-dimensional quantum theory and more general classes of generalized probabilistic theories (GPTs), a complex-valued quasiprobability representation is formulated as a collection of maps from the convex cones of states and effects to (typically finite-dimensional) complex vector spaces, such that:

• For each system, there exists an invertible linear map (a "frame" map) $\chi_A$, and the representation of any process $T: A \to B$ can be written as

$$\mathsf{Q}(T) = \chi_B \circ T \circ \chi_A^{-1}\,,$$

where $\chi_A$ is determined uniquely by its action on states and identity effects (Wagner et al., 13 Sep 2025).

• If the representation is not strictly functorial but only semi-functorial (i.e., does not necessarily preserve identities), a generalized form applies with a corestriction involving the idempotent $D_A = Q(\text{id}_A)$.
• In quantum theory, this framework encompasses standard real-valued representations, such as the Wigner function, as well as quintessentially complex-valued ones, such as the Kirkwood–Dirac (KD) distribution:

$$\mu_{\mathrm{KD}}(a, b | \rho) = \langle a|\rho|b\rangle \langle b|a\rangle\,,$$

with $|a\rangle, |b\rangle$ orthonormal bases—typically noncommuting—to probe contextuality and coherence (Wagner et al., 13 Sep 2025, Halpern et al., 2017).

• Every such representation is specified by its choice of frames (and dual frames), which allows arbitrary "coordinates" (possibly overcomplete and nonorthogonal) for expressing operators as complex-valued functions.

2. Mathematical Framework: Frames and Complexification

The construction is grounded in the mathematical theory of frames for (real or complex) Hilbert spaces, and in category-theoretic notions within process theories:

• The real vector space of Hermitian operators $\mathrm{Herm}$ is complexified: $$\mathbb{C}(\mathrm{Herm}) \cong \mathrm{Herm} \oplus i\,\mathrm{Herm}$$, allowing for full flexibility in representing complex structure (Wagner et al., 13 Sep 2025).
• Frame representations consist of a collection of operator-valued maps $F(\alpha)$ over a set $\Gamma$ and corresponding dual maps $\tilde{F}(\alpha)$, such that every operator $A$ can be reconstructed as

$$A = \int_{\Gamma} d\mu(\alpha)\, \langle A, F(\alpha) \rangle\, \tilde{F}(\alpha) \,,$$

with inner product $\langle A,B \rangle = \mathrm{tr}(A B)$.

• For each process in a GPT, the induced representation is then completely determined by the structure theorem for linearity-preserving, empirically-adequate (semi-)functors. In this formalism, diagrams (compositions and tensors) in the process theory are preserved under the functorial map into complex vector spaces (Wagner et al., 13 Sep 2025).

3. Comparison With Real-Valued and Classical Representations

Complex-valued quasiprobability representations generalize and unify both real-valued (but possibly negative) representations and the structure of classical probability:

• In classical theories, states and measurements are described by nonnegative real-valued functions, and probabilities are computed via the standard law of total probability.
• In quantum theory, any real-valued quasiprobability representation must involve negativity either in states or measurements or employ a deformed (nonclassical) probability calculus (0711.2658, Ferrie, 2010). Complex-valued representations are necessary for fully capturing features such as phase and noncommutativity.
• The KD representation captures not only negative but also imaginary contributions, which are crucial for expressing contextuality and interference; these quantities cannot generally be encoded using real-valued functions (Halpern et al., 2017, Wagner et al., 13 Sep 2025).
• The structure theorem demonstrates that for any tomographically local, finite-dimensional GPT, all (complex-valued) representations are characterized by a system-wise choice of invertible maps $\chi_A$, subsuming all real-valued, negative-valued, and complex-valued instances as special cases (Wagner et al., 13 Sep 2025).

4. Examples and Special Cases

Complex-valued quasiprobability representations naturally arise in quantum information science:

• The Kirkwood–Dirac (KD) distribution, used extensively for analyzing weak values, out-of-time-ordered correlators (OTOCs), and state retrodiction:

$$\mu_{\mathrm{KD}}(a,b|\rho) = \langle a | \rho | b \rangle \langle b | a\rangle$$

appears as an explicit instance of a (generally non-self-adjoint) frame representation. For noncommuting measurement bases, $\mu_{\mathrm{KD}}$ is generically complex (Halpern et al., 2017, Wagner et al., 13 Sep 2025).

• The general structure accommodates all families of Weyl–Heisenberg or symmetric informationally complete (SIC) frame representations, which may be real or complex, depending on the construction (Zhu, 2016, DeBrota et al., 2017).
• Weak value quasiprobabilities, as introduced via Aharonov's formulation and parameterized by a complex number $\alpha$, interpolate between forward and time-reversed processes and are used to assign consistent values to noncommuting observables (Fukuda et al., 2016).
• Deforming frame operators by complex parameters (e.g., via fractional Fourier transforms or phase-space displacements) leads to families of quasiprobability representations with tunable phase and interference properties, including those suited for quantum optics and infinite-dimensional (continuous-variable) systems (Anaya-Contreras et al., 2019, Berra-Montiel et al., 2020).

5. Operational and Foundational Implications

The adoption of complex-valued quasiprobability representations brings several advantages and implications:

• Resource Theory: The presence of complex negativity in these representations is tightly linked to resources for quantum computational speedup, contextuality, and nonclassicality. In certain computational models, negativity in the complex quasiprobability is necessary for quantum advantage, whereas positive or real representations correspond to classically simulable regimes (Zhu, 2016, Ferrie, 2010, Wagner et al., 13 Sep 2025).
• Process Tomography and Channel Representation: Quantum processes can be represented by matrices or maps in the quasiprobability framework, and the complex structure allows more general forms, such as non-Hermitian process tensors, relevant for open-system dynamics and non-unitary evolution (Wagner et al., 13 Sep 2025, Aw et al., 2023).
• Compatibility With Quantum Logical Inference: Complex-valued representations enable a precise and transparent formulation of quantum Bayesian inference (e.g., via the Petz recovery map), in which the manipulation of the "reference prior" (expressed in the structure of the representation, e.g., via M-matrices) accommodates quantum complementarity and noncommutativity (Aw et al., 2023).
• Foundational and Contextuality Tests: The availability of representations such as KD distributions and extended frame constructions provides tools for experimentally accessible tests of quantum contextuality and for probing the limits of classical-quantum interface (Halpern et al., 2017).

6. Extensions and Generalized Probabilistic Theories

The structure theorem applies far beyond quantum theory:

• For any tomographically local, finite-dimensional GPT, the theorem guarantees that every empirically adequate, linearity-preserving (semi-)functorial complex-valued quasiprobability representation arises from system-wise choices of invertible complex-linear maps (Wagner et al., 13 Sep 2025).
• This applies even when the representations map into infinite-dimensional spaces or when the target theory comprises infinite-dimensional complex vector spaces, provided the necessary structure is preserved.
• The notion of complexification functor $\mathbb{C}$ (which assigns $W \mapsto W \oplus W \cong W + iW$) generalizes the construction to arbitrary real vector spaces, making the theoretical apparatus suitable for a wide class of probabilistic physical theories.

7. Outlook and Significance

Complex-valued quasiprobability representations provide a mathematically and physically unified language for expressing and analyzing quantum and generalized probabilistic theories. They:

• Capture phase, interference, and contextuality in a manner inaccessible to real or strictly positive representations.
• Are essential for the theoretical analysis and practical simulation of quantum computation, especially in settings where quantum effects are operationally significant.
• Establish a canonical structure for all representations via system-wise invertible mappings, clarifying the role of frames in quantum and GPT state spaces.
• Offer flexibility for generalization to higher dimensions, infinite-dimensional systems, and future extensions involving nonstandard forms of probability.

The structure theorem (Wagner et al., 13 Sep 2025) consolidates this perspective, demonstrating that the full class of complex-valued quasiprobability representations is both rich and tightly constrained, with quantum theory serving as a central and paradigmatic example. This framework is now foundational in quantum information science, underpinning research in quantum computation, simulation, tomography, error correction, metrology, and the broader mathematical structure of physical theories.