Day: December 18, 2025

Kirkwood-Dirac Quasiprobability Distribution

Recently, I have been working on several topics related to the Kirkwood–Dirac quasiprobability distribution (often abbreviated as the KD distribution), which has inspired me to write down some informal notes.

The KD distribution is a fascinating conceptual tool that sits at the boundary between classical probability theory and the strange, counterintuitive structure of quantum mechanics. It provides a way to represent quantum states that goes beyond classical statistics, allowing for features such as negativity and even complex values. These properties have no classical analogue, yet they encode genuinely quantum aspects of measurement, interference, and contextuality.

In this sense, the KD distribution offers a particularly transparent window into the nonclassical nature of quantum theory, making it an invaluable framework for both foundational studies and modern applications in quantum information.

What Are Quasiprobability Distributions in Quantum Mechanics?

Before turning to the KD distribution itself, it is helpful to place it within a broader conceptual landscape.

In classical physics, the state of a system can be described directly in phase space (concept developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs). A point (x,p) specifies the position and momentum of a particle, and our uncertainty about the system is encoded in a genuine probability density \omega(x,p). Such distributions are always non-negative real functions and satisfy a normalization condition, reflecting the familiar rules of classical probability theory. All classical observables are functions on phase space, F(x,p), and their expectation values are given by
\langle F \rangle = \int F(x,p)\,\omega(x,p)\,dx\,dp.

Quantum mechanics, however, adopts a very different language. The fundamental object is the wavefunction \psi(x) (or, more generally, a density operator), which does not represent a probability distribution in phase space. While |\psi(x)|^2 gives the probability density for position and |\psi(p)|^2 gives the probability density for momentum, there is no single function that simultaneously assigns probabilities to both position and momentum. This obstruction is not merely technical but fundamental, rooted in the noncommutativity of quantum observables and formalized by the Heisenberg uncertainty principle.

Nevertheless, it is natural to ask whether quantum states can still be represented in a phase-space-like manner, even if some classical features must be sacrificed. This motivation led to the development of quasiprobability distributions. These objects resemble classical probability distributions in form, but they relax one crucial requirement: they need not be non-negative, or even real-valued. Allowing for negative or complex values makes it possible to encode quantum interference, coherence, and other genuinely nonclassical effects within a phase-space framework.

Well-known examples include:

The Kirkwood–Dirac (KD) quasiprobability distribution belongs to this same family, but with an important distinction. It can be defined for arbitrary pairs of observables (in fact the Wigner function can also be extended in this way), not just position and momentum, and it is generically complex-valued. These features make the KD distribution especially powerful for studying quantum measurements, temporal correlations, and applications in quantum information processing.

Historical Background: From Kirkwood to Dirac

The KD distribution has its roots in the early days of quantum mechanics, when physicists were grappling with how to reconcile quantum phenomena with classical intuition.

  • John G. Kirkwood’s Contribution (1933): American physicist John Gamble Kirkwood first introduced the distribution in his paper “Quantum Statistics of Almost Classical Assemblies“. Kirkwood aimed to describe quantum systems that are nearly classical, using a phase-space representation. His version was essentially a complex-valued joint distribution for position and momentum.
  • Paul Dirac’s Independent Discovery (1945): The legendary physicist Paul A. M. Dirac independently rediscovered the distribution in his paper “On the Analogy Between Classical and Quantum Mechanics“. Dirac highlighted its role in illustrating analogies between classical and quantum theories, particularly for noncommuting operators.

The real part of KD distribution is called Margenau-Hill quasiprobability distribution, which was proposed by by H. Margenau and R. N. Hill in 1961 in their paper “Correlation between Measurements in Quantum Theory“.

For decades, the KD distribution remained relatively obscure, overshadowed by the Wigner function. However, in recent years—especially since the 2010s—it has gained renewed attention due to its applications in quantum information theory, where it serves as a powerful tool to quantify nonclassical resources such as coherence and contextuality. See the review paper “Properties and applications of the Kirkwood–Dirac distribution” by Arvidsson-Shukur et al for more details.

Mathematical Formulation: Defining the KD Distribution

Let’s dive into the mathematics. We’ll assume a basic familiarity with quantum mechanics notation (e.g., density operators \hat{\rho}, bras \langle \cdot |, kets | \cdot \rangle), but I will explain as we go.

Basic Definition for Two Observables

Consider a quantum system in a state described by the density operator \hat{\rho} (which may be pure or mixed). Let \hat{A} and \hat{B} be two Hermitian observables with eigenbases {|a_i\rangle} and {|b_j\rangle}, respectively, assuming a finite-dimensional Hilbert space of dimension d for simplicity.

The Kirkwood–Dirac (KD) quasiprobability distribution is defined as:

Q_{i,j}(\hat{\rho}) = \langle b_j | a_i \rangle \langle a_i | \hat{\rho} | b_j\rangle

Here:

  • \langle b_j | a_i \rangle is the overlap between the eigenvectors.
  • This expression is generally complex because the overlaps can introduce phases.

For projectors \hat{\Pi}_a = |a\rangle\langle a| and \hat{\Pi}_b =|b\rangle\langle b|, it can also be written as:

Q(a,b; \hat{\rho}) = \mathrm{Tr}(\hat{\Pi}_b \hat{\Pi}_a \hat{\rho})

In continuous variables (e.g., position x and momentum p):

Q(x, p; \hat{\rho}) = \langle p | x \rangle \langle x | \hat{\rho} | p \rangle

The real part of KD distribution is called Margenau-Hill quasiprobability distribution, which by definition is a real-valued quasiprobability distribution.

Generalizations

The KD distribution extends naturally to more general scenarios:

  • Multiple observables (k observables): Q_{i_1, \dots, i_k}(\hat{\rho}) = \mathrm{Tr} (\hat{\Pi}_{i_k}^{(k)} \cdots \hat{\Pi}^{(1)}_{i_1} \hat{\rho} )
  • Positive-operator-valued measures (POVMs) {\hat{M}^{(l)}_{i_l}}: Q_{i_1, \dots, i_k}(\hat{\rho}) = \mathrm{Tr} ( \hat{M}_{i_k}^{(k)} \cdots \hat{M}^{(1)}_{i_1} \hat{\rho} )

Key Mathematical Properties

  • Normalization:
    \sum_{i,j} Q_{i,j}(\hat{\rho}) = 1
  • Marginals: The sums over one index reproduce the standard quantum probabilities (Born rule):
    \sum_j Q_{i,j}(\hat{\rho}) = \langle a_i | \hat{\rho} | a_i \rangle, \quad \sum_i Q_{i,j}(\hat{\rho}) = \langle b_j | \hat{\rho} | b_j \rangle
    For multiple observable case, this property becomes Kolmogorov consistency condition.
  • State Reconstruction: Assuming all overlaps \langle a_i | b_j \rangle are non-zero, the density operator can be recovered from the KD distribution: \hat{\rho} = \sum_{i,j} Q_{i,j}(\hat{\rho}) \frac{|b_j\rangle\langle a_i|}{\langle a_i | b_j \rangle^*}

Properties: What Makes the KD Distribution Nonclassical?

The KD distribution is particularly powerful because it captures features of quantum states that have no classical analogue:

  • Complex Values and Negativity: Entries of the KD distribution can be negative (in their real part) or contain imaginary components. Negativity is a direct signature of quantum interference, while the imaginary parts reflect the disturbance caused by measurement.
  • Quantifying Nonclassicality: One common measure is the total non-positivity:
    N(Q) = \sum_{i,j} |Q_{i,j}| - 1 \ge 0,
    where N(Q) = 0 for states that behave classically with respect to the chosen observables.

Connections to Quantum Foundations:

  • Negativity in the KD distribution can serve as a witness of contextuality, showing that measurement outcomes cannot be explained by noncontextual hidden-variable models.
  • It signals the incompatibility of observables, reflecting the fundamentally noncommutative structure of quantum mechanics.
  • It is also related to violations of macrorealism, for example as quantified by Leggett–Garg inequalities.

Applications in Modern Quantum Research

  • Quantum Metrology: The imaginary part enables ultra-sensitive parameter estimation.
  • Weak Measurements: Negativity gives rise to anomalous weak values, allowing signal amplification.
  • Quantum Thermodynamics: Defines coherent work distributions and fluctuation theorems.
  • Quantum Information: Quantifies coherence, entanglement, and information scrambling via out-of-time-order correlators (OTOCs). It can also be used for direct quantum state measurement, requiring fewer resources than full state tomography.
  • Quantum Computing: The nonclassicality of the KD distribution is closely related to the computational power of magic-state injection schemes in quantum computation. (This connection has been extensively studied within the Wigner distribution framework, as it has been recognized that the advantage of quantum computation is intimately tied to quantum contextuality; both the KD distribution and the Wigner distribution are closely related to quantum contextuality.)