Relative Information, Relative Facts

This paper proposes a new way to understand the relational interpretation of quantum mechanics by defining "relative facts" through quantifiable information and associating perspectives with commutative observables rather than entire systems, thereby resolving previous theoretical issues and describing measurement as a continuous process.

The Gist

Imagine you are at a massive, crowded music festival. You are standing near the main stage, and you can hear the heavy bass of a techno track. To you, the "fact" of the festival is that it is a techno concert.

But your friend, who is standing a mile away near the acoustic tent, hears nothing but soft guitar strumming. To them, the "fact" of the festival is a folk concert.

Who is right? In classical physics, we would say one of you is wrong, or there is a "true" soundtrack playing that you are both just perceiving differently. But this paper, written by Andrea Di Biagio and Carlo Rovelli, suggests a radical new way of looking at the universe: There is no "true" soundtrack. There are only facts relative to where you are standing.

Here is a breakdown of their big ideas using everyday language.

1. Facts are like "Information Packages"

In our normal lives, we think of a "fact" as something solid and unchangeable, like a rock sitting on a table. But the authors argue that a fact is actually just a measure of information.

Think of a "fact" as a perfectly clear, high-definition photograph. If you have a blurry, pixelated image of a cat, you don't really "know" it's a cat; you just have a vague idea. But if the image is crystal clear, you have a "fact."

The authors say that in the quantum world, things aren't always "clear photos." Instead, information is constantly flowing between objects. A "fact" only exists when one part of the universe has enough information about another part to be certain.

2. The "Perspective" vs. The "Observer"

Usually, when people talk about quantum mechanics, they talk about "Observers"—which sounds like it requires a human being with a brain. This makes people uncomfortable, as if the universe needs us to look at it to exist.

The authors fix this by replacing "Observers" with "Perspectives."

Imagine a smartphone. A smartphone isn't a "person," but it can "know" things. It has a sensor that detects light. When the light hits the sensor, the phone has a "perspective" on the brightness of the room. It doesn't need a human to look at the phone for the phone to have recorded that information.

In this paper, a "perspective" is just any collection of things that can be correlated. A single electron can have a "perspective" on another electron. The universe isn't a stage where humans watch a play; it’s a massive web of objects constantly "noticing" each other.

3. The "Merging" of Truths (How we agree on reality)

If everyone has their own "relative facts," you might ask: "If my facts are different from yours, how do we ever agree on anything? Why does the world feel so solid and shared?"

The authors use the idea of Merging.

Imagine you and I are both looking at a closed box. I think there is a red ball inside; you think there is a blue ball inside. Our perspectives differ. But if we both walk over to the box, open it, and look at the same ball, our perspectives merge. We have interacted with the same object, and now we share a fact.

In the real world, this happens through something called decoherence. The environment (air molecules, light, dust) is constantly "poking" everything. This constant poking forces different parts of the universe to interact and "sync up" their information. This is why, even though quantum mechanics is weird and relative, the world we see feels stable, objective, and shared.

4. The "No-Go" Zone: Why we shouldn't ask "What's really happening?"

The most profound part of the paper is a warning against a specific type of question.

When we see something strange in a lab, we often ask: "But what is really happening behind the scenes? What is the 'true' state of the particle when no one is looking?"

The authors say this question is a trap. It’s like asking, "What is the 'true' color of a sound?" The question itself doesn't make sense because it's trying to find a property (color) in a domain where it doesn't exist (sound).

They argue that searching for a "view from nowhere"—a single, absolute truth that exists outside of all relationships—is a waste of time. Physics isn't about finding the "ultimate object"; it's about mapping the relationships between things.

Summary: The Universe as a Conversation

If the old way of thinking about the world was like a Library (a collection of fixed, unchangeable books sitting on shelves), this paper proposes that the universe is more like a Conversation.

In a conversation, the "truth" of what is being said depends on who is talking, who is listening, and what was said a moment ago. There is no single "truth" of the conversation floating in the air; there is only the information being exchanged between the participants.

The universe isn't a collection of things; it is a web of interactions.

Technical Summary

Technical Summary: Relative Information, Relative Facts

Authors: Andrea Di Biagio and Carlo Rovelli
Date: February 10, 2026

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1. The Problem: Ambiguity in Relational Interpretations

The Relational Interpretation of Quantum Mechanics (RQM) posits that physical properties are not absolute but exist only as relations between systems. However, previous formulations of RQM faced significant technical and conceptual hurdles:

• Vagueness: A lack of precise, quantitative definitions for what constitutes a "fact" or "information" within a relational context.
• The "System vs. Variable" Problem: Previous attempts tried to define facts relative to entire quantum systems. This led to mathematical ambiguities, such as the possibility of complementary (non-commuting) observables being considered "facts" simultaneously, which contradicts the structure of quantum mechanics.
• The Observer Problem: The need to clarify whether "observers" or "agents" are fundamental requirements for the emergence of facts.

2. Methodology: Information-Theoretic Reformulation

The authors propose a fresh perspective by grounding RQM in Shannon’s Information Theory and Probability Theory. Instead of treating "perspectives" as belonging to entire quantum systems, they associate perspectives with commutative subalgebras of observables (classical subsystems).

Key Mathematical Framework:

• Information ($I_A$): Quantified as the difference between the maximum possible information and the Shannon entropy ($H_A$) of a variable $A$.
• Mutual Information ($I_{A:B}$): Quantifies the correlation between two variables, representing how much information about $A$ can be gained by knowing $B$.
• Relative Information ($I_{A|B}$): Defined as the expectation value of the information about $A$ conditional on the value of $B$.
• Relative Fact: A value of a variable $A$ is defined as a fact relative to $B$ if the relative information is maximal ($I_{A|B} = I_{max}^A$).

3. Key Contributions

• Definition of Relative Facts: By using information theory, the authors provide a rigorous mathematical criterion for a "fact." A fact is no longer a metaphysical "state of being" but a state of maximal information/certainty relative to another system.
• Commutative Subalgebras as Perspectives: The authors resolve the ambiguity of non-commuting observables by asserting that a "perspective" is defined by a set of commuting variables. This ensures that only variables that can be simultaneously known (in a classical sense) can be treated as facts within a single perspective.
• Continuous Measurement Process: The paper demonstrates that quantum measurement is not a discontinuous "collapse" but a continuous growth of mutual information ($I_{A:B}$) between the measured system and the apparatus, as governed by the Schrödinger equation.
• Naturalistic Standpoint: The authors decouple "observers" from "perspectives." While observers (humans) are physical systems with complex perspectives, any set of commuting observables constitutes a perspective, allowing for a purely physical, non-anthropocentric description of the world.

4. Results and Implications

• Merging of Perspectives: The authors show through mathematical examples (e.g., entanglement and interaction) how two different perspectives can "merge." When two systems interact and become correlated, they can come to agree on the values of certain variables, providing a mechanism for intersubjectivity.
• Resolution of the Combination Problem: The "combination problem" (how individual subsystem perspectives form a global one) is addressed by showing that if variables couple correctly, their perspectives naturally align through correlation.
• Explanation of Paradoxes: The framework provides a way to understand the Wigner’s Friend and Frauchiger-Renner paradoxes. These paradoxes arise because different observers (e.g., a friend in a lab vs. Wigner outside) are operating from different perspectives with different correlation structures. The "contradictions" are simply the result of trying to apply one perspective's facts to another's.
• The End of Absolute Facts: The primary result is the rejection of "absolute facts." The authors argue that asking what "really" happens beyond the information available to any physical system is a metaphysical question without empirical content.

5. Significance

This paper provides a mathematically robust foundation for Relational Quantum Mechanics, moving it from a qualitative philosophical stance to a quantitative physical theory. By aligning RQM with standard quantum formalism and information theory, the authors:

1. Eliminate the need for "collapse" postulates or special ontological entities.
2. Bridge the gap between quantum mechanics and classical experience via decoherence and information spreading.
3. Offer a coherent worldview where the "objective" world is an emergent phenomenon arising from the robust, intersubjective agreement between various physical perspectives.