Decoherent histories with(out) objectivity in a (broken) apparatus

This paper demonstrates that while approximate decoherent histories emerge in both measurement and scrambling phases of a solvable quantum model, true classicality characterized by environment-induced decoherence and preferred pointer states is uniquely distinguished by the non-ergodicity and correlation with the measured qubit found only in the measurement apparatus phase.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Question: How Does the Quantum World Become "Real"?

Imagine you are watching a magic show. The magician (the quantum world) can be in two places at once, or spin in two directions simultaneously. But when you look at the result, you only see one thing: a rabbit, or a coin landing on heads.

Physicists have two main stories to explain how this magic trick turns into a solid reality. This paper asks: Are these two stories actually telling the same thing, or are they different?

1. Story A (The "Objective" Story): Reality becomes real because the information about the magic trick gets copied and spread around the room. If 100 people all see the rabbit, it's "objective." Everyone agrees on what happened.
2. Story B (The "History" Story): Reality becomes real because the different possible paths the magic trick could have taken stop interfering with each other. It's like a river splitting into streams; eventually, the streams stop mixing, and you can trace a single, clear path.

The authors of this paper built a digital "toy universe" to test these stories. They wanted to see if you can have a clear history (Story B) without having objective reality (Story A).

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The Experiment: The "Tree of Copies"

To test this, the scientists created a model that looks like a growing tree.

• The Seed: At the bottom, there is a single "system" (a tiny quantum coin) that is in a superposition (both heads and tails).
• The Growth: As time passes, this coin interacts with new coins, which interact with more coins, creating a massive, expanding tree of entangled particles.
• The Twist: They have a "dial" (called $\theta$) that controls how the tree grows.

By turning this dial, they created two different worlds:

World 1: The "Perfect Photocopier" (The Apparatus Phase)

When the dial is set one way, the tree acts like a perfect photocopier.

• What happens: The state of the original coin is copied perfectly onto every branch of the tree.
• The Result: If you look at just a small twig, you can tell what the original coin was. If you look at a whole branch, you get the same answer.
• The Analogy: Imagine a rumor starting in a small town. Everyone hears the exact same version. If you ask 10 random people, they all tell you the same story. This is Objectivity (Quantum Darwinism). The information is redundant and accessible.

World 2: The "Scrambler" (The Encoding Phase)

When the dial is turned the other way, the tree acts like a chaotic blender.

• What happens: The information about the original coin gets mixed up so thoroughly that it is hidden inside the complex relationships between the coins.
• The Result: If you look at a small twig, you see nothing but noise. Even if you look at a whole branch, you can't easily tell what the original coin was. The information is still there, but it's "scrambled" like a jigsaw puzzle where the pieces are glued together in a weird way.
• The Analogy: Imagine a rumor that gets twisted every time someone repeats it. By the time it reaches the end of the line, no one knows what the original story was. This is Scrambling.

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The Surprise: Both Worlds Have "Clear Histories"

Here is the big discovery. The scientists looked at the "history" of the tree in both worlds. They asked: "If we watch the tree grow, do we see a clear, single path, or a blurry mess of possibilities?"

They found that in both worlds, the history becomes clear.

• The Analogy: Imagine watching a foggy forest.
  • In the Photocopier World, the fog clears up because everyone is shouting the same thing. You see a clear path.
  • In the Scrambler World, the fog also clears up, but for a different reason. The chaos of the scrambling process naturally averages out the noise if you look at the "big picture" (coarse-graining).

The Key Insight: You don't need "Objectivity" (everyone agreeing) to get a "Classical History" (a clear path). You just need to look at the system with "fuzzy glasses" (coarse-graining). If you try to look too closely (fine-grained), the quantum weirdness comes back.

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The Difference: "Frozen" vs. "Wandering"

Even though both worlds have clear histories, the nature of those histories is totally different.

In the Photocopier World (Apparatus Phase)

• The Behavior: The history is non-ergodic. This is a fancy way of saying the tree "freezes" into a specific state.
• The Analogy: Imagine a compass needle. Once it points North, it stays pointing North. No matter how long you watch it, it doesn't wander.
• Why it matters: The tree "remembers" the original coin. The history is correlated with the past. The system has selected a "preferred state" (like Heads or Tails) and stuck with it. This is how a real measurement works.

In the Scrambler World (Encoding Phase)

• The Behavior: The history is ergodic. It wanders around randomly.
• The Analogy: Imagine a drunk person walking in a field. They might go North for a bit, then South, then East. If you watch them for a long time, they cover the whole field.
• Why it matters: The tree has forgotten the original coin. The history is just random noise. There is no "pointer state" selected. The system has scrambled the information so well that the past is inaccessible.

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The Conclusion: Two Types of "Classical"

The paper concludes that we need to stop using the word "Classical" to mean just one thing. There are actually two different flavors:

1. Classical as "Objective Reality" (Quantum Darwinism): This is when the world is stable, and everyone agrees on what happened. This requires a specific kind of interaction (like a measurement apparatus) that copies information redundantly.
2. Classical as "Decoherent Histories": This is just when the quantum interference stops, and you can describe the system as a single path. This happens almost everywhere if you just look at the system "fuzzily."

The Takeaway:
Just because a system looks "classical" (has a clear history) doesn't mean it's a "measurement apparatus" that has created objective reality.

• A measurement device is a special, rare object that freezes the past and makes it public.
• A scrambler (like a chaotic gas or a black hole) might look classical if you squint, but it has lost the connection to the past.

The authors showed that you can have the appearance of a classical world without the substance of an objective reality. It's the difference between a photograph that everyone agrees on, and a blurry painting that just happens to look like a landscape if you stand far enough away.

Technical Summary

Here is a detailed technical summary of the paper "Decoherent histories with(out) objectivity in a (broken) apparatus" by Ferté, Farci, and Cao.

1. Problem Statement and Motivation

The paper addresses the fundamental question of how classical reality emerges from quantum mechanics, specifically aiming to distinguish between two influential but often conflated notions of "classicality":

• Environment-Induced Decoherence (Quantum Darwinism): Defines classicality as objectivity, where information about a system is redundantly encoded in the environment, allowing multiple observers to access the same "pointer states" without disturbing the system.
• Decoherent Histories: Defines classicality intrinsically via consistent histories, where the quantum evolution of an isolated system can be approximated by a stochastic sum over non-interfering trajectories (histories) in configuration space.

The Core Conflict: There is a prevailing belief that decoherent histories require objectivity (i.e., that a system must have redundant records to exhibit consistent histories). Recent works suggested that approximate decoherent histories can emerge in chaotic, scrambling systems where information is not objective (inaccessible). However, the relationship and distinction between these two mechanisms remained unclear.

Goal: The authors aim to demonstrate a solvable model where these two notions can be turned on and off independently, clarifying the mechanisms behind each.

2. Methodology and Model

The authors utilize a solvable unitary model based on an "inflationary" quantum circuit (a dynamically expanding tree), which acts as a bridge between microscopic and macroscopic realms.

• Model Structure:
  • System ($S$): A single qubit initially entangled with a "ready" apparatus qubit ($A$).
  • Dynamics: At each time step $t$, every existing qubit interacts with a new qubit via an isometry $v$. This creates a binary tree structure where the number of qubits grows as $N_t = 2^t$.
  • Interaction: The isometry $v$ is parameterized by a tuning angle $\theta \in (0, \pi/2)$.
    • At $\theta = 0$, the model acts as a repetition code (perfect copying).
    • As $\theta$ increases, the code is perturbed, leading to scrambling.
• Phases:
  • Apparatus Phase ($\theta < \theta_c$): Information about $S$ is redundantly stored in the apparatus qubits (Objectivity/Quantum Darwinism).
  • Encoding/Scrambler Phase ($\theta > \theta_c$): Information about $S$ is scrambled and becomes inaccessible to local or extensive observables (No Objectivity).
• Threshold: The phase transition occurs at $\theta_c = \pi/4$, determined by the scaling dimension $x_1 = -\log_2(\cos \theta)$. The transition happens when $x_1 = 1/2$.
• Probe for Classicality:
  • Instead of projective measurements, the authors use weak, coarse-grained measurements of an extensive observable (total magnetization $Q_t = \sum Z_j$) described by smeared Kraus operators.
  • They test the Decoherent Histories Condition (DHC) by checking if the outcome distribution of measurements at a subset of times is independent of measurements performed at other times (non-disturbance). This is quantified by the $L_1$ norm of the difference $\|\Delta\|_1$.

3. Key Contributions and Results

A. Emergence of Decoherent Histories in Both Phases

The most significant finding is that approximate decoherent histories emerge in both the apparatus phase and the scrambling phase, provided the measurements are coarse-grained (measuring extensive quantities with finite relative precision).

• Mechanism: The emergence relies on the coarse-graining parameter $\Gamma$. If measurements are too fine-grained (high precision), quantum coherence is revealed, and histories do not decohere.
• Result: For macroscopic times ($\tau \gg 1$), the interference between different histories vanishes ($\|\Delta\|_1 \to 0$) in both phases. This confirms that decoherent histories do not strictly require objectivity.

B. Distinction via Ergodicity and Pointer States

While decoherent histories exist in both phases, their statistical properties differ fundamentally, allowing the authors to distinguish the "Apparatus" from the "Scrambler":

1. Apparatus Phase ($\theta < \theta_c$):

   • Non-Ergodicity: The history outcomes $m_t$ are non-ergodic. In a single experimental run, the outcome freezes to a random constant value $M$ (plus noise), which varies between runs but is constant in time.
   • Correlation: The histories are strongly correlated with the measured qubit $S$.
   • Pointer States: The conditional state of $S$ (the "pointer state ensemble") collapses towards a specific basis (e.g., $|0\rangle$ or $|1\rangle$), though imperfectly. The apparatus successfully selects a preferred ensemble of states.

2. Encoding/Scrambler Phase ($\theta > \theta_c$):

   • Ergodicity: The history outcomes behave as a Gaussian process with finite temporal correlation (Ornstein-Uhlenbeck like). The system "forgets" its initial state rapidly.
   • No Correlation: The histories are uncorrelated with the initial qubit $S$.
   • No Pointer States: The conditional state of $S$ remains maximally mixed ($I/2$). No information about the system is inferred from the history.

C. Violation of Leggett-Garg Inequality

The authors demonstrate that the Leggett-Garg inequality (a test for macro-realism) can be violated at arbitrarily late times in both phases using specific global observables.

• This highlights that while coarse-grained observables yield classical-looking histories, the underlying dynamics retain quantum coherence accessible only to fine-grained, non-local measurements.

4. Significance and Implications

• Decoupling Classicality Concepts: The paper rigorously proves that Decoherent Histories and Objectivity (Quantum Darwinism) are distinct concepts.
  • Decoherent Histories are a generic feature of macroscopic systems under coarse-grained observation, arising from the suppression of interference due to averaging.
  • Objectivity is a specific property of measurement apparatuses where information is redundantly stored and accessible.
• Definition of an Apparatus: The authors conclude that "an apparatus is a macroscopic object, but not every macroscopic object is an apparatus." A macroscopic object can exhibit classical trajectories (histories) without actually measuring or recording information about a specific system (scrambler phase).
• Role of Coarse-Graining: The study emphasizes that the "classicality" of decoherent histories is an artifact of the observer's limited resolution (coarse-graining). Without this limitation, the system remains fully quantum.
• Theoretical Framework: By using a solvable tree model (related to MERA tensor networks), the authors provide exact analytical and numerical evidence for these transitions, offering a rare rigorous testbed for foundational quantum mechanics questions.

Conclusion

The paper resolves the ambiguity between two major approaches to quantum classicality. It shows that while coarse-grained monitoring is sufficient to generate decoherent histories (stochastic trajectories) in both ordered and chaotic regimes, only the apparatus phase (characterized by non-ergodicity and system-history correlation) supports objectivity and the emergence of a preferred pointer basis. This distinction clarifies that the "classical world" of trajectories is broader than the "classical world" of objective measurement records.