⚛️ quantum physics

The spontaneous disentanglement hypothesis and causality

This paper proposes a maximum entropy-based formulation of the spontaneous disentanglement hypothesis, utilizing Lagrange multipliers to resolve conflicts with the causality principle in finite-dimensional quantum systems.

Original authors: Eyal Buks

Published 2026-04-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Eyal Buks

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 Big Picture: A Quantum Mystery

Imagine the universe as a giant, perfectly predictable machine running on a set of rules called Quantum Mechanics. Usually, this machine works in a very specific way: things evolve smoothly and predictably (like a movie playing forward).

However, physicists have always been bothered by two "glitches" in this movie:

The Measurement Glitch: When we look at something, the smooth movie suddenly jumps to a single, definite scene. This is called "collapse," and nobody knows exactly how or why it happens.
The Time Glitch: The laws of physics usually work the same forward and backward in time. But when things get hot and messy (thermalization), time seems to only move forward.

The Author's Idea:
Eyal Buks suggests a new way to fix these glitches. Instead of a magical "collapse," he proposes that quantum systems have a natural tendency to spontaneously untangle themselves.

Think of two dancers (quantum particles) who are holding hands and moving in perfect sync (entangled). Buks suggests that over time, they naturally let go of each other and start dancing solo (disentangled). This "letting go" happens automatically, without anyone needing to measure them.

The Problem: The "Faster-Than-Light" Loophole

Here is where things get tricky. In the quantum world, when two dancers are holding hands, they are connected in a spooky way. If you change the move of one dancer, the other instantly knows, no matter how far apart they are.

Buks explains that if this "untangling" happens spontaneously and randomly, it creates a dangerous loophole.

The Analogy: Imagine Alice and Bob are on opposite sides of the galaxy, holding a magical, invisible rope (entanglement).
If Alice could force her end of the rope to snap (disentangle) at a specific moment, and Bob could instantly feel that snap, Alice could send a message to Bob faster than light.
Why is this bad? Einstein's rule of Causality says nothing can travel faster than light. If Alice can send a message instantly, she could theoretically send a message back in time, creating a paradox (like killing your own grandfather before you were born).

The paper shows that if we just let quantum systems "untangle" however they want, we break the rules of the universe and allow time travel paradoxes.

The Solution: The "Maximum Entropy" Rule

So, how do we keep the idea of spontaneous untangling without breaking the speed-of-light rule?

Buks proposes a new set of rules based on Maximum Entropy.

The Analogy: Imagine you have a messy room (high entropy). Nature loves messiness. It wants the room to be as messy as possible.
Buks suggests that when the quantum dancers let go of each other, they don't just let go randomly. They let go in a very specific way that maximizes the "messiness" (entropy) of the whole system, but strictly keeps the individual dancers' states exactly the same as they were before.

The "Lagrange Multiplier" Trick:
To make this work mathematically, the author uses a tool called "Lagrange multipliers."

The Metaphor: Think of this as a strict bouncer at a club. The bouncer's only job is to ensure that while the party gets wilder (entropy increases and entanglement breaks), no one's personal identity changes.
Alice's side of the rope must look exactly the same to her, and Bob's side must look exactly the same to him.
Because neither Alice nor Bob sees any change in their local reality, neither of them can tell if the rope snapped or not. Therefore, no message can be sent.

What This Means for Physics

No More Magic Collapses: We don't need a mysterious "collapse" when we measure something. The system naturally untangles itself, and because we forced the rules to keep local properties unchanged, it looks exactly like a standard measurement to us.
Safety First: By using this "Maximum Entropy" rule, we can have a universe where things naturally get messy and untangle, but we don't accidentally break the speed of light or allow time travel.
It's Testable: The author notes that this isn't just math; it makes different predictions than standard quantum mechanics. Scientists could potentially test this in a lab (like with spinning magnets) to see if particles really do "untangle" in this specific, rule-abiding way.

Summary

The paper is like a mechanic trying to fix a car engine that makes a weird noise (the measurement problem).

Old Idea: The engine just stops and restarts magically (Collapse).
Buks' Idea: The engine naturally settles down (Spontaneous Disentanglement).
The Catch: If the engine settles down too fast, it might explode the car (Break Causality/Speed of Light).
The Fix: Install a governor (Maximum Entropy/Lagrange Multipliers) that lets the engine settle down smoothly but ensures the car never speeds up too fast.

This allows the universe to be a bit more "messy" and natural, while still obeying the strict laws of cause and effect.

1. Problem Statement

The paper addresses a fundamental conflict between two concepts in quantum foundations:

The Spontaneous Disentanglement Hypothesis: A proposal suggesting that quantum systems naturally evolve toward disentangled states (a process analogous to wavefunction collapse and thermalization) without external measurement. This hypothesis relies on nonlinear extensions to the Schrödinger equation to generate disentanglement, as the set of disentangled states is not a linear subspace.
The Causality Principle: The requirement that no information can be transmitted faster than light (superluminal signaling).

The Core Conflict: Previous work (notably by Gisin) demonstrated that nonlinear quantum dynamics, when combined with entanglement, generally allow for superluminal signaling. If Alice performs a local operation on her part of an entangled system, the nonlinearity could theoretically alter the reduced density matrix of Bob's distant system instantaneously, violating causality. The paper seeks to formulate a version of the spontaneous disentanglement hypothesis that avoids this violation.

2. Methodology

The author employs a combination of theoretical derivation, information-theoretic constraints, and numerical simulation to resolve the conflict.

Nonlinear Master Equation: The study utilizes a modified master equation for the density operator $\rho$ :
$\frac{d\rho}{dt} = i\hbar^{-1}[\rho, H] + \Omega(\Theta)$
where $\Omega(\Theta) = -\Theta\rho - \rho\Theta + 2\langle\Theta\rangle\rho$ . This equation describes stochastic evolution (equivalent to a Langevin-Schrödinger equation) that preserves the norm and positivity of the density matrix. The operator $\Theta$ is chosen to drive specific physical processes (e.g., thermalization or disentanglement).
Maximum Entropy Principle & Constraints: To prevent superluminal signaling, the author proposes that the disentanglement process must not alter the local properties of the subsystems.
- The hypothesis is constrained such that the reduced density operators of the subsystems ( $\rho_a$ and $\rho_b$ ) remain fixed during the evolution.
- This is achieved by maximizing the total entropy $\sigma = -\text{Tr}(\rho \log \rho)$ subject to the constraints that $\rho_a$ and $\rho_b$ are constant.
- Lagrange Multipliers: The method of Lagrange multipliers is implemented to enforce these constraints. The operator $\Theta$ is modified to $\Theta'$ by adding terms proportional to the generators of the local algebras (generalized Bloch vectors), ensuring that the evolution of the global state does not change the local reduced states.
Bloch Matrix Formalism: The system is analyzed using generalized Gell-Mann matrices to define a Bloch matrix $B$ . The constraints are mathematically enforced by ensuring that the "single subsystem Bloch vectors" (extracted from the first row and column of $B$ ) remain invariant under the nonlinear evolution.
Numerical Simulation: The author simulates the time evolution of a two-spin-1/2 system under two conditions:
1. Unconstrained: Standard nonlinear disentanglement without fixing local states.
2. Constrained: Nonlinear disentanglement with Lagrange multipliers enforcing fixed local Bloch vectors.

3. Key Contributions

Causality-Compliant Formulation: The paper proposes a specific formulation of the spontaneous disentanglement hypothesis where the nonlinearity is constrained to preserve the reduced density matrices of spatially separated subsystems. This effectively mimics the Nakajima-Zwanzig projection ( $\rho \to \rho_a \otimes \rho_b$ ) without requiring an instantaneous collapse that alters local statistics.
Derivation of Stochasticity and Born Rule: The author argues that the requirement to exclude superluminal signaling (via fixed reduced density matrices) naturally leads to the stochastic nature of quantum measurement and implies the Born rule.
Unification of Processes: The framework unifies unitary evolution, thermalization (entropy maximization), and disentanglement (collapse) under a single nonlinear master equation, provided the causality constraints are met.
Falsifiability: The paper emphasizes that this hypothesis is experimentally distinguishable from standard Quantum Mechanics (QM) and notes recent experimental tests using spin resonators.

4. Results

Mitigation of Superluminal Signaling: Numerical results (Figures 1 and 2) demonstrate that when constraints are applied:
- The local Bloch vectors ( $P_a$ and $P_b$ ) remain constant over time, regardless of the disentanglement process occurring in the global system.
- Consequently, an observer (Bob) cannot detect any change in his local subsystem resulting from operations or evolution on a distant subsystem (Alice), thereby preventing superluminal signaling.
Efficiency of Disentanglement: The simulations show that applying the causality constraints does not significantly hinder the efficiency of the disentanglement process. The system still evolves from a pure entangled state to a mixed disentangled state (increasing entropy and relative entropy) at a rate comparable to the unconstrained case.
Thermalization Mimicry: The constrained evolution successfully mimics thermalization and measurement-like behavior. For example, in the presence of dipolar coupling, the constrained evolution drives the system toward a state that looks like a measurement of the spin's z-component, while keeping local properties fixed.
Limitations: The author acknowledges that while the conflict is mitigated, the formulation is based on non-relativistic QM. Therefore, issues like superluminal tunneling (which exist even in standard QM) are not fully resolved, but the specific conflict arising from nonlinear signaling is addressed.

5. Significance

Foundational Resolution: The paper offers a potential pathway to reconcile nonlinear modifications of quantum mechanics (which solve the measurement problem and explain thermalization) with the principle of causality. It suggests that the "weirdness" of quantum measurement (collapse) might be a result of a constrained nonlinear evolution rather than a fundamental postulate.
Theoretical Framework: By linking the exclusion of superluminal signaling to the preservation of reduced density matrices, the work provides a rigorous mathematical justification for why quantum measurements appear stochastic and why the Born rule might emerge from causality requirements.
Experimental Relevance: The hypothesis is presented as falsifiable. The author cites recent experimental tests, suggesting that deviations from standard QM predictions could be observed in specific quantum systems (like spin resonators) if the spontaneous disentanglement hypothesis is correct.

In summary, Buks proposes that spontaneous disentanglement can occur without violating causality if the nonlinear dynamics are constrained to leave local subsystem properties unchanged. This approach preserves the utility of nonlinear extensions for explaining collapse and thermalization while adhering to the no-signaling principle.