⚛️ quantum physics

The Locality Gap: A Thermodynamic Law for Objective Facts

This paper establishes that the redundancy of objective facts in quantum Darwinism generates a specific thermodynamic resource called the "thermodynamic factuality charge," which quantifies the exact work asymmetry and erasure penalty incurred by controllers with restricted access compared to a global controller, without modifying the fundamental laws of quantum mechanics.

Original authors: Maxim V. Churilov

Published 2026-08-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Maxim V. Churilov

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

The Hidden Cost of Knowing: Why Facts Feel Different When Everyone Knows Them

Imagine you are trying to understand how the universe turns a fleeting quantum event into a solid, unchangeable "fact." In the world of quantum physics, things are usually fuzzy and exist in many states at once. But in our daily life, facts are clear: a broken vase is broken, not half-broken and half-whole. Scientists have long believed that this shift happens through a process called decoherence, where a tiny system gets entangled with its environment, like a secret whispered into a crowded room. If enough people hear the whisper, it becomes a "public fact" that everyone agrees on. This idea is known as Quantum Darwinism: the environment "selects" certain facts by copying them over and over again into many different fragments, like a viral meme spreading across the internet.

But here is the puzzle: if you have one bit of information (a single yes/no answer) and you copy it a million times, you still only have one bit of logical information. You haven't created new data. So, what physical difference does it make that the information is now everywhere? Does it cost anything to have a fact that is "objective" (known by many) versus a fact that is "private" (known by only one)? This is the question that sits at the intersection of thermodynamics (the study of heat and work) and information theory. While we know that erasing information costs energy (a rule called Landauer's Principle), we didn't know if the way information is shared—redundantly across many people versus stored in one place—changes the energy bill.

The Paper's Big Idea: The "Locality Gap"

In this paper, researcher Maxim V. Churilov proposes a new thermodynamic law that acts like a "fact meter." The core finding is that objective facts are expensive to erase if you can't talk to each other.

To understand this, imagine a group of friends trying to delete a shared secret.

Scenario A (Global Controller): One super-smart friend has access to all the notes everyone else wrote down. They can look at the whole pile, see the patterns, and delete the secret with minimal effort.
Scenario B (Partition-Local Controller): The friends are in separate rooms and cannot talk to each other. Each friend has to delete their own note independently, without knowing what the others are doing.

The paper proves that Scenario B always costs more energy than Scenario A. This extra energy cost is called the Locality Gap. It is the "tax" you pay for having a fact that is distributed across many independent places but cannot be accessed all at once.

The "Fact Charge" Formula

The author introduces a specific quantity called the Thermodynamic Factuality Charge ( $F^T_P$ ). Think of this as a score that tells you how "factual" a piece of information is, based on how much extra work it takes to erase it when the controllers are separated.

Perfect Broadcasting (The Objective Fact): If the information is copied perfectly into every single fragment (like a viral video everyone has), and you try to erase it without talking, the energy cost is maximal. The "Fact Charge" is high and positive. This is what makes a fact "objective"—it is robust and public, but thermodynamically expensive to undo if you are isolated.
Unique Storage (The Private File): If the information is stored in only one place and the rest are blank, there is no extra cost. The "Fact Charge" is zero.
Secret Sharing (The Private Key): This is the most fascinating case. Imagine a secret that is split into pieces (like a puzzle) where no single piece tells you anything, but if you put them all together, the secret is revealed. In this case, erasing the pieces separately actually costs less energy than erasing them together. The "Fact Charge" becomes negative. This represents "privacy" or "synergy," where the whole is greater than the sum of its parts, but the parts are useless on their own.

The "Thermodynamic Record Number"

The paper also defines a "Record Number" to quantify this.

If you have perfect secret sharing, the number is 0.
If you have a unique storage (one copy), the number is 1.
If you have perfect broadcasting across $m$ blocks, the number is $m$ .

This means that the more redundant the record is (the more "objective" the fact), the higher the thermodynamic record number, and the more energy you must spend to erase it if you are restricted to local control.

What This Paper Rules Out and Confirms

It is important to note what this paper does not do. It does not claim to change the laws of quantum mechanics, nor does it suggest that "objective reality" is an illusion. It does not prove the "Born rule" (how probabilities work in quantum mechanics) or introduce a new "collapse" of the wave function.

Instead, the paper argues that objectivity is a thermodynamic resource. It confirms that:

Redundancy creates a work asymmetry: The same information requires different amounts of energy to erase depending on whether you can access it globally or locally.
Classicality is required for maximum objectivity: The paper shows that for a fact to reach the "perfect broadcast" state (maximum positive charge), the information must be classical (like a standard bit). You cannot perfectly broadcast a quantum state to multiple independent observers without destroying the quantum nature of it (a concept related to the "No-Broadcasting Theorem").
The math is exact: The formulas provided are not just guesses; they are derived as exact equalities for reversible processes. The author has also stress-tested these formulas with thousands of random simulations, and they hold up perfectly.

Why It Matters

This research gives us a new way to measure "truth" in the physical world. It suggests that when a quantum event becomes a "fact" that everyone can agree on, it leaves a specific, measurable scar on the universe's energy budget. It's as if the universe charges a "redundancy fee" for making things public.

The paper also suggests a way to test this in the lab. By using quantum computers or trapped ions, scientists could set up a scenario where they try to erase a shared record either globally or locally. If the theory is right, the local erasure should always cost more energy, and the difference should match the "Fact Charge" formula exactly.

In short, this paper tells us that facts are not just ideas; they are physical structures with a price tag. The more people who know a fact independently, the more energy it takes to make them forget it, unless they can all sit down and talk to each other first. This "Locality Gap" is the thermodynamic signature of objectivity itself.

Technical Summary: The Locality Gap and Thermodynamic Factuality

Problem Statement
In the framework of quantum Darwinism, objective facts are defined as values of a system imprinted redundantly into many independently accessible fragments of the environment. While redundancy explains how observers reach a consensus without direct system access, it does not increase the Shannon information of the underlying event; $R$ copies of a bit still contain only one bit of logical uncertainty. Consequently, the thermodynamic meaning of this redundancy has remained unclear. The paper addresses the question: What physical resource is accumulated when an event acquires a large cloud of records, and is there a universal thermodynamic signature distinguishing objective broadcast records from unique storage or secret sharing?

Methodology
The author proposes a comparative thermodynamic framework based on reversible isothermal erasure (Landauer erasure) under restricted control. The methodology involves:

Defining Architectures: Establishing a record cloud $F$ of an event $X$ and a partition $\mathcal{P}$ of this cloud into blocks that cannot be jointly controlled.
Four-Work Protocol: Comparing the minimum work required to reset the record cloud under four distinct conditions:
- Global reset without source information ( $W^\emptyset_{glob}$ ).
- Partition-local reset without source information ( $W^\emptyset_{\mathcal{P}}$ ).
- Global reset with the event $X$ supplied as catalytic classical side information ( $W^X_{glob}$ ).
- Partition-local reset with the event $X$ supplied as catalytic classical side information ( $W^X_{\mathcal{P}}$ ).
Double Difference: Calculating the "Locality Gap" ( $G^\emptyset_{\mathcal{P}}$ ) as the work difference between local and global erasure without side information. Then, calculating the "Source-Law" gap ( $G^X_{\mathcal{P}}$ ) with side information. The core quantity, the Thermodynamic Factuality Charge ( $F^T_{\mathcal{P}}(X)$ ), is derived as the double difference: $F^T_{\mathcal{P}}(X) = G^\emptyset_{\mathcal{P}} - G^X_{\mathcal{P}}$ .
Modeling: The theory is tested against exact solvable models, including perfect classical broadcasts, independent binary-symmetric records, pure-state quantum collision models, and parity secret sharing.

Key Contributions and Results

The Locality Gap Law: The paper establishes that the work asymmetry between a global controller and a partition-local controller is exactly the correlation free energy inaccessible to the local controller. Specifically, $G^\emptyset_{\mathcal{P}} = k_B T T_{\mathcal{P}}(F)$ , where $T_{\mathcal{P}}(F)$ is the partition total correlation.
The Source-Loss Law of Factuality: By subtracting the source-assisted gap, the author isolates correlations specifically attributable to the event $X$ . The resulting quantity is:
$F^T_{\mathcal{P}}(X) = k_B T \left[ \sum_{B \in \mathcal{P}} I(X:F_B) - I(X:F) \right]$
This quantity represents the temperature times the event-specific increase in entropy production caused by the loss of the source key under restricted control.
Thermodynamic Record Number: The author defines a normalized coordinate $N^{\mathcal{P}}_X = 1 + F^T_{\mathcal{P}}(X) / (k_B T H(X))$ $N_{X}^{P} = 1 + F_{P}^{T} (X) / (k_{B} T H (X))$ , which ranges from 0 to $|\mathcal{P}|$ $∣ P ∣$ .
- Perfect Broadcast: $N^{\mathcal{P}}_X = |\mathcal{P}|$ (Maximum positive charge). Every block independently determines $X$ .
- Unique Storage: $N^{\mathcal{P}}_X = 1$ (Zero charge). Only one block carries $X$ ; others are uncorrelated.
- Perfect Secret Sharing: $N^{\mathcal{P}}_X = 0$ (Maximum negative charge). No block reveals $X$ , but the union determines it.
Landauer–Darwin Bounds: The paper derives lower bounds for the factuality charge based on the number of independently decodable fragments. If $R$ disjoint blocks allow inference of $X$ with deficit $\delta$ , then $F^T_{\mathcal{P}} \geq [R(1-\delta) - 1] k_B T H(X)$ .
Endpoint Rigidity: The bounds are "rigid." A charge close to the broadcast ceiling certifies that every block is an almost complete record. A charge close to the secret-sharing floor certifies that every block is almost ignorant of $X$ .
Interaction Corrections: For Hamiltonians with cross-block interactions, the entropy-only law acquires an exact interaction-energy correction bounded by $2\|H_{int}\|$ . The paper provides a Pinsker-refined bound that scales with the observed correlation, offering a sharper positivity criterion.

Significance and Claims
The paper claims to provide a precise thermodynamic signature for objective facts without modifying quantum mechanics or positing objective collapse.

Resource Identification: It identifies the thermodynamic resource generated by redundant records as a "work asymmetry" between control architectures, rather than an additional state variable.
Operational Objectivity: It reframes objectivity from a statement about accessible information to a quantitative work asymmetry under local control. A "thermodynamically public fact" is defined as a classical variable independently decodable from multiple blocks where source loss creates a positive local-global erasure double difference.
Classicality: The maximal positive factuality endpoint requires the record blocks to contain perfectly distinguishable conditional states, implying the saturating variable must be classical (commuting pointer algebra). This connects the operational baseline to the no-broadcasting theorem.
Causal Sealing: The author proposes (as a conjecture, not a theorem) that the factuality charge can define a "causal sealing" barrier, where an event becomes sealed for an agent if no future admissible controller can reunite enough records to remove the positive locality penalty.

Limitations and Scope
The author explicitly states that the theory does not derive the Born rule, modify the Schrödinger equation, or introduce physical collapse. The results are exact within stated operational assumptions:

Reversible, quasistatic erasure (asymptotic regime).
Strictly partition-local protocols (no cross-block communication or shared memory).
Classical side information treated catalytically.
Additive Hamiltonians (with a specific correction term provided for non-additive cases).

The paper concludes that the finite-system law is closed, exact, numerically verified, and directly testable via four-work double difference measurements in controlled quantum systems.