CFT derivation of entanglement phase transition in pseudo entropy

This paper utilizes boundary conformal field theory methods to demonstrate that pseudo entropy between distinct boundary states undergoes a phase transition dependent on the conformal weight of the boundary condition changing operators, a result that is confirmed to match holographic AdS calculations.

The Gist

Here is an explanation of the paper "CFT derivation of entanglement phase transition in pseudo entropy," translated into simple language with creative analogies.

The Big Picture: Measuring "What If"

Imagine you are a physicist trying to understand how two quantum systems are connected. Usually, we measure Entanglement Entropy. Think of this as measuring how much two friends (System A and System B) are "in sync" with each other right now. If they are highly entangled, they share a deep, invisible bond.

But this paper introduces a new, slightly more magical concept called Pseudo Entropy.

The Analogy: The "What If" Movie
Imagine you are watching a movie.

• Standard Entanglement: You watch the movie from start to finish. The characters are in a specific state at the beginning and a specific state at the end. You measure how connected they are throughout the whole story.
• Pseudo Entropy: You take the opening scene (the Initial State) and the ending scene (the Final State), but you force them to be different from each other. You ask: "If the universe started like this but ended up like that, how much 'connection' or 'information' was shared in the middle?"

It's like asking, "If I start a journey in New York but my destination is actually Paris (even though I started in New York), how much of the road did we travel together?"

The Experiment: Changing the Rules

The researchers wanted to see what happens to this "Pseudo Entropy" when they change the rules of the game. Specifically, they looked at a system where the "Initial State" and "Final State" are defined by different boundary conditions.

The Metaphor: The Two Sides of a Wall
Imagine a long hallway (the quantum system).

• Boundary Condition: This is the rule for how the hallway ends.
  • Condition A: The walls are made of mirrors (Neumann).
  • Condition B: The walls are made of black velvet (Dirichlet).
• The Setup: The researchers prepared the system with "Mirror Walls" at the start and "Velvet Walls" at the end. They then watched how the "connection" (Pseudo Entropy) grew over time.

They tested two very different types of universes to see if the result changed.

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Experiment 1: The "Chaotic" Universe (Holographic CFT)

This is a complex, messy universe (like a real black hole or a chaotic fluid). In physics terms, this is a Holographic CFT.

The Discovery: A Phase Transition
The researchers found that the behavior of the "connection" depended entirely on how different the Mirror and Velvet walls were. They used a dial to control the difference (let's call it the "Deformation").

1. Small Change (The "Linear Growth" Phase):

   • Scenario: The walls are slightly different (e.g., Mirrors with a tiny scratch).
   • Result: The connection grows linearly over time. It's like a steady stream of water filling a bucket. The more time passes, the more "shared history" the system accumulates.
   • Metaphor: It's like two people starting a conversation with a slight accent difference. They keep talking, and the conversation gets longer and deeper the more time passes.

2. Critical Point (The "Logarithmic" Phase):

   • Scenario: The walls are different at a very specific, critical threshold.
   • Result: The connection grows very slowly, like a logarithmic curve. It's a "tipping point."

3. Huge Change (The "Frozen" Phase):

   • Scenario: The walls are completely different (e.g., Mirrors vs. a bottomless pit).
   • Result: The connection stops growing. It hits a ceiling and stays constant.
   • Metaphor: Imagine trying to have a conversation with someone who speaks a completely different language. No matter how long you talk, you never build a deeper connection. The "information" gets stuck.

Why is this cool?
This looks exactly like a "Measurement Induced Phase Transition" seen in quantum computers. It suggests that if you change the rules of the universe too drastically, the system stops learning or evolving its connections. It's a sudden switch from "growing" to "stagnating."

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Experiment 2: The "Simple" Universe (Free Dirac Fermions)

To make sure this wasn't just a fluke, they tested a Free Dirac Fermion CFT. Think of this as a "toy universe"—it's simple, predictable, and doesn't have chaos (it's "integrable").

The Discovery: Nothing Happens
They tried the exact same experiment: Start with Mirrors, end with Velvet.

• Result: The connection grew exactly the same whether the walls were different or the same.
• Metaphor: In this simple universe, it doesn't matter if you start with a red car and end with a blue car. The engine runs the same way. The "chaos" required to create the "phase transition" is missing.

The Lesson: The dramatic "switch" from growing to freezing only happens in complex, chaotic systems (like the Holographic ones). Simple, orderly systems don't care about the difference in boundary conditions.

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The "Gravity" Connection (The Holographic Trick)

The paper also uses a famous idea called Holography (AdS/CFT).

• The Idea: A complex 2D quantum system (the CFT) is mathematically equivalent to a 3D universe with gravity (AdS space).
• The Visualization:
  • In the "Small Change" phase, the gravity universe looks like a smooth, expanding shape.
  • In the "Huge Change" phase, the gravity universe develops a "defect" or a sharp corner (a conical singularity).
  • The math shows that the "surface" used to measure the connection (the geodesic) has to stretch into imaginary numbers (complex coordinates) to find the shortest path in this weird geometry.

Summary: What Does It All Mean?

1. Pseudo Entropy is a tool to measure connections between a "Start" and a "Finish" that are different.
2. In Complex/Chaotic Universes, changing the rules too much causes the system to stop growing its connections (a Phase Transition). It's like hitting a wall.
3. In Simple/Orderly Universes, changing the rules does nothing special. The system just keeps chugging along.
4. This suggests that this "freezing" behavior is a signature of chaos and complexity. It might help us understand how black holes process information or how quantum computers behave when we try to force them into specific states.

In a nutshell: The universe has a "tipping point." If you try to force a chaotic system to start and end in too-different ways, it gives up on building connections. But if the system is simple, it doesn't even notice the difference.

Technical Summary

Here is a detailed technical summary of the paper "CFT derivation of entanglement phase transition in pseudo entropy" by H. Kanda, T. Takayanagi, and Z. Wei.

1. Problem Statement

The paper investigates the entanglement phase transition of pseudo entropy within Conformal Field Theories (CFTs).

• Context: Pseudo entropy ($S_A$) generalizes entanglement entropy to scenarios involving a post-selection process. It is defined via a transition matrix $\tau_A = \text{Tr}_B \left( \frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle} \right)$, where $|\psi\rangle$ is the initial state and $|\phi\rangle$ is the final (post-selected) state.
• Motivation: Previous holographic studies (using AdS/BCFT models) suggested that pseudo entropy exhibits a phase transition depending on the "distance" between the initial and final boundary states. Specifically, for small differences, entropy grows linearly (thermalization), while for large differences, it becomes time-independent. However, these results relied on bottom-up gravity models where the microscopic CFT dual was not explicitly defined.
• Goal: To provide a rigorous microscopic CFT derivation of this phase transition in a holographic CFT (large central charge $c$) and to contrast this behavior with a non-holographic, integrable CFT (free Dirac fermions).

2. Methodology

The authors employ two distinct approaches to compute pseudo entropy:

A. Holographic CFT (Large $c$ Limit)

• Setup: The system is defined on a strip of width $\beta/2$ with different boundary conditions ($a$ and $b$) on the two edges. The states are regularized boundary states $|B_a\rangle$ and $|B_b\rangle$.
• Operators: The difference between boundary conditions is mediated by Boundary Condition Changing (b.c.c.) operators ($\Psi_{ab}$ and $\Psi_{ba}$) with conformal weight $h_\Psi$.
• Calculation Technique:
  1. Replica Method: The pseudo entropy is computed via the one-point function of twist operators $\sigma_n$ on the strip.
  2. Conformal Mapping: The strip is mapped to the Upper Half Plane (UHP) via $z = i e^{2\pi w/\beta}$.
  3. Mirror Trick (Doubling): The boundary 3-point function $\langle \Psi_{ba}(\infty) \sigma_n(z) \Psi_{ab}(0) \rangle$ is mapped to a chiral 4-point function on the complex plane.
  4. Heavy-Heavy-Light-Light (HHL) Approximation: Assuming the b.c.c. operators are heavy ($h_\Psi = O(c)$), the 4-point function is approximated by the dominance of the vacuum conformal block.
  5. Analytic Continuation: The Euclidean result is analytically continued to Lorentzian time to study time evolution.

B. Free Dirac Fermion CFT (Integrable Case)

• Setup: A $c=1$ massless free Dirac fermion on a cylinder.
• States: The initial and final states correspond to Neumann ($|N\rangle$) and Dirichlet ($|D\rangle$) boundary conditions.
• Calculation Technique:
  1. Replica Method: Using $n$ copies of the fermion fields.
  2. Bosonization: The fermions are bosonized into free scalar fields to handle the twist operators.
  3. Boundary State Construction: Explicit construction of boundary states for the replicated system and calculation of the partition function ratio.

3. Key Results

A. Holographic CFT: Phase Transition

The behavior of the pseudo entropy $S_A(t)$ depends critically on the conformal weight $h_\Psi$ of the b.c.c. operator relative to the central charge $c$. The transition is governed by the parameter $\alpha_\Psi = \sqrt{1 - 24h_\Psi/c}$.

1. Small Deformation Phase ($0 < h_\Psi < c/24$):

   • Here $\alpha_\Psi$ is real and $0 < \alpha_\Psi < 1$.
   • Result: The pseudo entropy grows linearly with time at late times.
   • Formula: $S_A \sim \frac{c}{6} \frac{2\pi \alpha_\Psi}{\beta} t$.
   • Physical Interpretation: This mimics the linear growth of entanglement entropy in a quantum quench, indicating thermalization. The dual geometry is a BTZ black hole with a deficit angle.

2. Critical Phase ($h_\Psi = c/24$):

   • Here $\alpha_\Psi = 0$.
   • Result: The pseudo entropy grows logarithmically.
   • Formula: $S_A \sim \frac{c}{6} \log t$.
   • Physical Interpretation: This corresponds to a critical point where the geometry transitions.

3. Large Deformation Phase ($h_\Psi > c/24$):

   • Here $\alpha_\Psi$ becomes purely imaginary ($|\alpha_\Psi| > 0$).
   • Result: The pseudo entropy becomes time-independent (saturates to a constant).
   • Formula: $S_A \approx \text{const}$.
   • Physical Interpretation: The dual geometry corresponds to a "thermal AdS" (or a naked singularity from the open string perspective) where the geodesic connecting the boundary points does not probe the deep interior in a way that generates growth.

• Holographic Consistency: The CFT results perfectly match the holographic calculations performed in AdS space using the AdS/BCFT model, confirming the existence of the phase transition. The authors note a coefficient difference in the logarithmic growth ($c/6$ here vs. $c/3$ in previous AdS/BCFT work) due to different microscopic descriptions of the boundary deformation (heavy primary operator vs. marginal deformation on the brane).

B. Free Dirac Fermion CFT: No Phase Transition

• Result: When calculating pseudo entropy for the transition between Neumann and Dirichlet boundary states, the result is identical to the standard entanglement entropy for a single boundary condition.
• Implication: The pseudo entropy does not exhibit any phase transition or non-trivial time-dependence change based on the boundary state difference.
• Reasoning: This is attributed to the integrable structure of the free CFT. Unlike holographic (chaotic/irrational) CFTs, free theories lack the complex scrambling dynamics required to generate the phase transition.

4. Significance and Contributions

1. Microscopic Validation: This paper provides the first rigorous CFT-level proof of the pseudo entropy phase transition, moving beyond the effective gravity dual (AdS/BCFT) to a concrete CFT calculation using heavy operators and conformal blocks.
2. Universality vs. Integrability: By contrasting the holographic CFT with the free Dirac fermion, the authors demonstrate that this phase transition is likely a feature of chaotic/irrational CFTs (those with holographic duals) rather than a universal property of all quantum systems. Integrable systems do not exhibit this transition.
3. Connection to Measurement-Induced Transitions: The results strongly resemble the "measurement-induced phase transition" observed in quantum many-body systems. The paper suggests a deep link between post-selection (pseudo entropy) and non-unitary evolution (measurements), where the "strength" of the boundary condition change acts analogously to the measurement rate.
4. Geometric Interpretation: The work clarifies the geometric nature of the transition in the bulk. The transition from linear growth to saturation corresponds to a change in the bulk geometry from a BTZ black hole (with a deficit angle) to a thermal AdS space with a naked singularity, mediated by the conformal weight of the boundary operator.

Conclusion

The paper establishes that pseudo entropy in holographic CFTs undergoes a phase transition driven by the conformal weight of the operators connecting initial and final boundary states. This transition mirrors the measurement-induced phase transition and is absent in integrable free theories, highlighting the role of quantum chaos and holography in the dynamics of post-selected quantum states.