⚛️ quantum physics

Unobservables and Decoherence from Complexity

This paper argues that the apparent classicality of macroscopic systems arises from computational complexity constraints, which render certain formally valid quantum measurements unperformable and prevent the observation of coherence in specific superpositions, thereby linking the quantum-classical transition to the inability of quantum systems to efficiently solve NP-complete problems.

Original authors: Michael Epping, Jochen Szangolies

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael Epping, Jochen Szangolies

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

Imagine the universe as a giant, incredibly complex video game. In this game, the rules of the "micro-world" (quantum mechanics) are very different from the "macro-world" (our everyday experience).

In the micro-world, things can be in two places at once (superposition) and can interfere with each other like waves in a pond. In our macro-world, we never see a cat that is both dead and alive, or a coin that is both heads and tails simultaneously. Usually, scientists explain this difference by saying that the environment (air, light, heat) constantly "measures" big objects, washing out these weird quantum effects. This is called decoherence.

However, this paper proposes a different, surprising reason why the world looks classical to us: It's too hard to compute.

Here is the breakdown of their argument using simple analogies:

1. The "Undo Button" Trap

In quantum mechanics, if you measure something and get a result you don't like, there are theoretical ways to "undo" that measurement and try again, effectively erasing the outcome.

The authors imagined a clever quantum algorithm that uses this "undo button" to solve a very difficult type of logic puzzle (called 3SAT). Think of this puzzle as trying to find the one specific combination of switches that turns on a massive light panel with millions of switches.

The Trick: The algorithm tries a switch combination. If it fails, it uses the "undo" button to reset the universe to before the attempt, then tries a different path.
The Problem: If nature allowed us to build a machine that could perform these "undo" measurements easily, we could solve these massive logic puzzles almost instantly.

2. The Complexity Wall

We know from computer science that some logic puzzles are "NP-complete." This means that as the puzzle gets bigger, the time it takes to solve it grows exponentially. Even the fastest supercomputers would take longer than the age of the universe to solve a large enough version.

The authors argue that if quantum mechanics allowed us to perform these specific "undo" measurements, we would be able to solve these impossible puzzles efficiently. Since we believe nature doesn't allow us to solve these puzzles instantly (because that would break the laws of complexity), nature must forbid these specific measurements.

3. The "Unobservable"

Because nature forbids these measurements, there are certain things in the quantum world that are mathematically valid but physically impossible to see. The authors call these "unobservables."

Analogy: Imagine a library with a perfect catalog system. The catalog says a book exists on a specific shelf. However, the shelf is made of a material so dense and complex that no human hand could ever reach it or pull the book out. The book exists in the catalog (math), but it is "unobservable" to us (physics).
In this paper, certain quantum measurements (like specific rotations of particles) are like that unreachable book. They are valid in the equations, but the computational effort required to perform them is so high that no observer with finite resources can ever do it.

4. The "Time Travel" Barrier

The paper also discusses how quantum states change over time. Usually, we think you can evolve from any quantum state to any other if you wait long enough.

But if a measurement is "unobservable" because it's too complex, then the time evolution (the process of changing from one state to another) that would lead to that measurement is also impossible.

Analogy: Imagine two islands in an ocean. Math says there is a bridge between them. But the bridge is made of a material so heavy that no ship could ever build it or cross it. Therefore, for all practical purposes, the two islands are disconnected. You can never travel from Island A to Island B.

5. The Result: "Decoherence by Complexity"

This leads to the paper's biggest conclusion: Superpositions become invisible.

If you have two quantum states that are "disconnected" (you can't evolve one into the other because the path is too complex), you can never prove they are in a "superposition" (a mix of both).

The Illusion: To an observer, a quantum superposition of these two states looks exactly the same as a random mixture (like flipping a coin and getting heads or tails).
The Shift: Because we cannot distinguish the "weird quantum mix" from a "normal random mix," the system acts like it has lost its quantum nature. It behaves classically.

Summary

The paper suggests that the reason the macroscopic world looks boring and classical isn't just because the environment is noisy. It might be because the universe is computationally lazy.

The "weird" quantum behaviors (like being in two places at once) are still there in the math, but observing them requires solving puzzles that are too hard for any observer to compute. Because we can't compute the solution, we can't see the quantum effect. The complexity of the universe itself acts as a filter, hiding the quantum weirdness and leaving us with the classical world we experience.

Technical Summary: Unobservables and Decoherence from Complexity

Problem Statement
The interface between quantum mechanics and classical reality remains a subject of intense research. While the quantum regime is defined by interference and superposition, macroscopic systems appear classical, lacking these phenomena. Standard explanations often rely on environmental decoherence. This paper investigates an alternative hypothesis: that the "unobservability" of certain quantum features in macroscopic systems arises not from environmental interaction, but from fundamental limitations imposed by computational complexity. Specifically, the authors explore whether the assumption that quantum systems cannot efficiently solve NP-complete problems implies that certain formally valid quantum measurements are physically unperformable.

Methodology
The authors construct a hypothetical quantum algorithm designed to solve the 3SAT problem (an NP-complete problem) in polynomial time by exploiting the ability to "undo" quantum measurements.

Measurement Undoing: The algorithm utilizes a "sandwich-POVM" (Positive Operator-Valued Measure), denoted as $\mathcal{N}(F)$ , for a propositional logic formula $F$ . This measurement involves a sequence of projective measurements on clauses of the formula, followed by a "reset" measurement that undoes unwanted outcomes (specifically, the projection onto the subspace violating a clause).
Dilution Technique: To address the issue of exponentially low success probabilities in sequential measurements, the authors employ a "dilution" strategy. They transform the original formula $F$ into a diluted formula $D(F)$ by introducing new variables and random XOR clauses. This modification ensures that the probability of measuring a satisfying clause outcome remains high (greater than $1/2$ on average) throughout the algorithm.
Complexity Assumption: The core argument relies on two assumptions:
- (INC): NP $\not\subseteq$ BQP (Quantum systems cannot efficiently solve NP-hard problems).
- (OBS): Every formally valid quantum measurement can be experimentally realized.
  The authors demonstrate that if (OBS) holds, the proposed algorithm would solve 3SAT efficiently, contradicting (INC).

Key Contributions and Results
The paper establishes that if NP-complete problems are hard for quantum systems, then specific formally valid quantum measurements must be physically unperformable. These quantities are termed unobservables.

Existence of Unobservables: The sandwich-POVM $\mathcal{N}(F)$ is shown to be a valid mathematical object in quantum mechanics but is physically unperformable for sufficiently large problem sizes (e.g., hundreds of variables) because its execution would violate the assumption that NP-hard problems are intractable.
Unobservable Pauli Matrices: Through Naimark's dilation theorem, the unobservable POVM $\mathcal{N}(F)$ is linked to unobservable Projector-Valued Measures (PVMs). The authors show that in a sufficiently large Hilbert space (specifically a $2^{M+N}$ -dimensional qudit), there exist bases where the Pauli operators ( $\tilde{X}, \tilde{Y}, \tilde{Z}$ ) are unobservables. While the canonical Pauli operators are observable, the transformed versions required to map between specific states are not.
Disconnected Time-Evolution Orbits: The authors define the set $\mathcal{T}$ of physically realizable time evolutions using finite resources. They prove that there exist unitary operators (such as the one mapping an unobservable to an observable) that are not in $\mathcal{T}$ . Consequently, there are pairs of pure states $|\psi_1\rangle$ and $|\psi_2\rangle$ that cannot be connected by any physically realizable time evolution. The transition probability between such states decays exponentially with system size ( $\exp(-\Omega(N))$ ).
Complexity-Induced Decoherence: Because no physically realizable unitary can connect $|\psi_1\rangle$ and $|\psi_2\rangle$ , the expectation values of any admissible observable $A$ for the coherent superposition $\rho_{coh} = \frac{1}{2}(|\psi_1\rangle + |\psi_2\rangle)(\langle\psi_1| + \langle\psi_2|)$ are indistinguishable from the incoherent mixture $\rho_{incoh} = \frac{1}{2}(|\psi_1\rangle\langle\psi_1| + |\psi_2\rangle\langle\psi_2|)$ . The difference in expectation values scales as $\exp(-\Omega(N))$ .
Effective Superselection Rules (SSR): This indistinguishability implies the emergence of an effective superselection rule. The coherence between such states cannot be observed, effectively turning a pure superposition into a classical mixture. This provides a mechanism for classicality based on computational complexity rather than environmental interaction.

Significance and Claims
The paper claims to be the first to directly link the observability of quantum coherence to a computationally hard problem. The authors propose that the apparent classicality of macroscopic systems may be partly due to the fact that observing certain quantum phenomena (like specific superpositions or performing specific measurements) requires resources that scale superpolynomially, rendering them impossible for any observer with finite resources.

The authors are modest in their scope, stating that they have only demonstrated the existence of this mechanism. They do not claim it is the sole source of classicality or decoherence, nor do they provide a quantitative estimate of its contribution relative to environmental decoherence. They suggest that future work could quantify how much of the "classical world" is a result of these complexity-theoretic limitations. The results imply that the set of physically realizable operations is a strict subset of the mathematically valid operations in quantum mechanics, constrained by the hardness of NP-complete problems.