🔬 condensed matter

The Principle of Minimum Justified Correlation

This paper establishes that Shannon entropy, Fisher information, and quantum kinetic energy are all measures of correlation that increase under a fundamental map to independent marginals, thereby supporting a "principle of minimum justified correlation" which posits that physical systems should be described by the least correlated distributions consistent with given constraints.

Original authors: John H Van Drie

Published 2026-09-30

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: John H Van Drie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the physical world, things are rarely isolated. A gas molecule bumps into its neighbor, a planet feels the pull of a distant star, and two electrons in an atom repel one another. These interactions create links, or correlations, between the parts of a system. For decades, physicists have used two different languages to describe these links. One language comes from information theory, a field that measures how much knowing the state of one thing tells you about another. The other comes from quantum mechanics, the rules that govern the behavior of the smallest particles, where energy is constantly in motion. While these fields often talk past each other, a new analysis suggests they are actually describing the same underlying reality from different angles. The core idea is simple: when two things are linked, they share information, and that link costs energy. If you could magically sever that link while keeping the individual properties of each part the same, the system would lose a specific amount of energy, and that loss is a direct measure of how strongly they were connected.

This connection between information and energy is the focus of a new paper by John H. Van Drie, a researcher from North Andover, Massachusetts. The work brings together three concepts that are usually studied separately: a measure of how much two variables depend on each other, a measure of how sharply that dependence changes across space, and the kinetic energy of a quantum system. Kinetic energy, in this context, is simply the energy of motion. The paper demonstrates that for a quantum system, the extra energy required to keep two particles correlated is exactly equal to a specific mathematical measure of how much those particles are linked. It is a precise accounting identity, not an approximation. The author shows that if you take a quantum system and mathematically remove the correlation between its parts—keeping the individual behavior of each part exactly as it was—the system's energy drops. The amount of energy lost is not random; it is a perfect match for the "Fisher information" of the correlation, a concept that describes how rapidly the relationship between the particles varies from one point in space to another.

The study begins with a thought experiment involving a fundamental transformation. Imagine you have a detailed map showing the joint behavior of two variables, such as the positions of two particles. This map shows not just where each particle is likely to be, but how their locations are tied together. The researcher applies a simple operation to this map: he replaces the joint picture with two separate pictures, one for each particle, and multiplies them together. This new, combined picture has the exact same individual probabilities for each particle as the original, but it completely removes the link between them. In the world of information, this act of "decorrelation" always increases the total entropy, or disorder, of the system. The paper confirms that this same act, when applied to continuous distributions like those found in physics, reveals a hidden structure. The increase in entropy is exactly equal to the amount of mutual information that was removed. But the paper goes further, showing that this process also affects the energy of the system.

When the researcher applies this same correlation-removing map to a quantum wavefunction, a striking result emerges. The kinetic energy of the system decreases. The paper proves that the difference between the energy of the original, correlated system and the new, uncorrelated system is directly proportional to the relative Fisher information. This is a measure of how "sharp" or concentrated the correlation is in space. If the particles are linked in a way that changes rapidly over small distances, the Fisher information is high, and the energy cost of maintaining that link is high. If the link is smooth and broad, the cost is lower. The author shows that this relationship holds true even when the wavefunction is complex, involving phases and signs, provided the comparison is made correctly. The energy lost by removing the correlation is exactly the energy stored in the spatial structure of that correlation.

This finding supports a principle the author calls the "principle of minimum justified correlation." In the language of information theory, developed by the late physicist E. T. Jaynes, the best guess for a system's state, given only certain constraints, is the one with the maximum entropy. The paper argues that this is equivalent to saying the system should have the least amount of correlation that is strictly required by the physical laws or constraints at hand. If a physical law, such as the Pauli exclusion principle for electrons, demands that two particles be correlated, then that correlation must exist. But if there is no physical reason for them to be linked, the system will naturally settle into a state where they are independent. The paper suggests that nature avoids adding unnecessary links. Any correlation that exists is there because the physics of the situation demands it, and the energy of the system reflects exactly that necessity.

The author illustrates this with a concrete example involving two coupled quantum harmonic oscillators, a standard model in physics. In this system, the strength of the link between the two oscillators is controlled by a single number. The paper calculates the exact amount of entropy gained and energy lost when the link is removed. The results show that for weak links, the energy loss is proportional to the square of the link strength, while for very strong links, the energy loss grows much faster. This confirms that the two measures of correlation—the total amount of information shared and the spatial sharpness of that sharing—are related but distinct. One tells you how much information is shared, while the other tells you how tightly packed that information is in space. The paper notes that this distinction is crucial because the energy cost is tied to the spatial sharpness, not just the total amount.

The work also clarifies the relationship between this new finding and existing ideas in quantum chemistry. In that field, "correlation energy" usually refers to the difference between the exact energy of a system and the energy calculated using a simplified model called the Hartree-Fock method. The author points out that this traditional definition is different from the one proposed here. The traditional measure includes changes in both kinetic and potential energy and is often negative, meaning the simplified model overestimates the energy. In contrast, the new measure looks only at the kinetic energy and compares the real system to a state where the particles are completely independent. This new "kinetic Fisher correlation" is always positive, representing the extra energy required to maintain the link. The paper argues that while the traditional view focuses on the total energy gap, this new view isolates the specific kinetic cost of the correlation itself.

The author is careful to state what this work does not do. It does not derive the fundamental laws of quantum mechanics from scratch, nor does it claim that all kinetic energy is simply correlation energy. The relationship is exact for the specific case of removing correlation while holding the individual particle distributions fixed. The paper also notes that for identical particles like electrons, the simple uncorrelated state used in the comparison does not respect the quantum rule that identical particles must be antisymmetric. Therefore, the comparison is a diagnostic tool to measure the kinetic structure of the dependence, rather than a proposal for a new physical wavefunction. The work is presented as a first part of a larger investigation, with a second paper planned to address more complex questions, such as how these ideas apply to time-dependent systems, the role of particle spin, and how to generate the Schrödinger equation itself from these principles.

Ultimately, the paper offers a unified way of seeing the physical world. It suggests that the energy we see in a quantum system is not just a random property but a direct reflection of the information structure holding the system together. When we look at a system and see particles moving with a certain amount of kinetic energy, we are seeing the cost of the links between them. If those links are removed, the energy drops, and the system becomes more disordered in terms of information. The principle of minimum justified correlation provides a clear rule for understanding why systems behave the way they do: they carry only as much correlation as the physical constraints force them to carry. This perspective bridges the gap between the abstract world of information and the tangible world of energy, showing that in the quantum realm, to know is to pay.

Technical Summary: The Principle of Minimum Justified Correlation (Part I)

Problem Statement
The paper addresses the conceptual unification of three distinct measures in physics and information theory: Shannon entropy, Fisher information, and quantum-mechanical kinetic energy. While these quantities are often treated separately, the author posits that they can all be viewed as measures of correlation. The core problem is to demonstrate that a single, fundamental operation—the removal of statistical dependence between variables—produces consistent, exact changes in all three quantities. Specifically, the paper seeks to clarify the relationship between the "amount" of correlation (measured by entropy) and the "spatial sharpness" of that correlation (measured by Fisher information), and how the latter manifests as a change in quantum kinetic energy.

Methodology
The methodology centers on a "fundamental correlation-destroying map," denoted as $C$ . This map transforms a joint probability distribution (discrete or continuous) into the product of its marginal distributions, effectively removing all dependence between the variables while preserving the marginals.

Discrete Case: For discrete variables $X$ and $Y$ with joint probability $p_{ij}$ , the map $C$ replaces $p_{ij}$ with $\pi_{ij} = P_i Q_j$ . The change in Shannon entropy under this map is shown to be exactly the mutual information $I(X;Y)$ , which is non-negative.
Continuous Case: The map is extended to continuous densities $\rho(x,y) \to \rho_x(x)\rho_y(y)$ . The author constructs a coordinate-invariant measure of correlation, the continuous mutual information $I[\rho]$ , defined as the difference between the sum of marginal differential entropies and the joint differential entropy.
Fisher Information: The paper introduces the translational Fisher information $J[\rho]$ and defines a "Fisher correlation" $J_{corr}[\rho]$ as the excess Fisher information of the joint distribution over the sum of the Fisher informations of its marginals. This is shown to be equivalent to the relative Fisher information $J(\rho \parallel \rho_x \rho_y)$ .
Quantum Mechanical Application: The author applies this framework to quantum mechanics by considering a normalized wavefunction $\psi$ with Born density $\rho = |\psi|^2$ . A reference state $\Phi$ is constructed as the product of the square roots of the marginal densities (with a phase-matched extension for complex wavefunctions). The paper compares the expectation value of the kinetic energy $\langle T \rangle$ for the correlated state $\psi$ against the decorrelated state $\Phi$ .

Key Contributions and Results

Unification of Entropy and Fisher Information: The paper establishes that both Shannon entropy and Fisher information possess a "decorrelation property." Under the map $C$ , entropy increases (or stays constant), while Fisher information decreases (or stays constant). The magnitude of these changes corresponds exactly to the mutual information $I[\rho]$ and the relative Fisher information $J_{corr}[\rho]$ , respectively.
The Kinetic-Energy Identity: A central result is the derivation of an exact identity linking the decrease in mean kinetic energy to the relative Fisher information. For a real wavefunction $\psi = \sqrt{\rho}$ , the difference in kinetic energy between the correlated state and the decorrelated product state is:
$\langle T \rangle_\psi - \langle T \rangle_\Phi = \frac{\hbar^2}{8m} J(\rho \parallel \rho_x \rho_y)$
This identity holds exactly, not as an approximation. It demonstrates that the von Weizsäcker kinetic energy functional is superadditive under marginal factorization.
Complex Wavefunctions: The analysis is extended to complex wavefunctions $\psi = \sqrt{\rho}e^{i\theta}$ . By choosing a reference state $\Phi$ that retains the original phase field $\theta$ but uses the decorrelated amplitude, the phase contribution to kinetic energy cancels out. The result remains that the reduction in kinetic energy is determined solely by the amplitude's correlation, quantified by the relative Fisher information.
Exact Solvable Example: The paper validates these identities using the ground state of two coupled quantum harmonic oscillators (a bivariate Gaussian). It explicitly calculates $I[\rho]$ and $J_{corr}[\rho]$ in terms of the correlation coefficient $R$ , showing that while both diverge as $|R| \to 1$ , they do so at different rates (logarithmically vs. as a simple pole), highlighting their distinct sensitivities to strong correlation.
Principle of Minimum Justified Correlation: The paper synthesizes these findings into a physical principle. It argues that Jaynes's Maximum Entropy Principle can be reinterpreted as a "Principle of Minimum Justified Correlation." Given physical constraints (such as conservation laws, interactions, or boundary conditions), the physical state should be the one that exhibits the least amount of correlation not strictly demanded by those constraints.

Significance and Claims
The paper claims to provide a unified geometric and information-theoretic perspective on quantum kinetic energy. By identifying the kinetic energy difference between a correlated state and its decorrelated product counterpart with the relative Fisher information, the author suggests that quantum kinetic energy (specifically the von Weizsäcker term) is a direct measure of the "spatial sharpness" of statistical dependence.

The significance lies in:

Exactness: Unlike many applications of Fisher information in density functional theory which serve as approximations, this derivation provides an exact identity for the kinetic energy difference.
Conceptual Clarity: It reframes the Maximum Entropy Principle not just as a method for selecting the least biased distribution, but as a principle of minimizing unjustified correlation. Correlation is only "justified" if required by physical constraints; otherwise, the system tends toward the product state (maximum entropy, minimum kinetic energy).
Foundational Scope: The author explicitly states that this is Part I of a two-part paper. Part I establishes the identities and the principle. The paper modestly notes that it does not derive the Schrödinger equation from scratch in this section, nor does it claim to derive quantum mechanics solely from Shannon entropy. Instead, it establishes a diagnostic framework: given a state, one can measure its correlation content via these functionals. Part II is reserved for addressing time dependence, spin, and a generative route to the Schrödinger equation.

The work distinguishes itself from B. Roy Frieden's "Extreme Physical Information" (EPI) by focusing on the correlation aspect of Fisher information (relative to marginals) rather than using Fisher information as a variational basis for physical laws, and from L"owdin's correlation energy by focusing strictly on the kinetic energy component of the correlation rather than the total energy gap.

More like this