⚛️ high-energy theory

Entanglement of excited states after measurements in conformal field theory

This paper investigates the entanglement of low-energy excited states in (1+1)-dimensional conformal field theories following projective measurements on spatial intervals, deriving a framework based on boundary CFT correlation functions to analyze specific excitations in the compact free boson model and proposing numerical methods to verify these predictions in critical spin chains.

Original authors: Hui-Huang Chen

Published 2026-07-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hui-Huang Chen

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: Measuring a Quantum Symphony

Imagine a quantum system (like a chain of atoms) as a giant, complex orchestra playing a piece of music. In physics, this "music" is called entanglement. It's a special kind of connection where the notes played by one section of the orchestra instantly affect the notes played by another, even if they are far apart.

Usually, scientists study the "ground state"—the quiet, calm music the orchestra plays when no one is conducting. But what happens if you suddenly stop the music in a specific section of the hall (a measurement) and force that section to play a single, fixed note?

This paper asks: How does forcing one part of the orchestra to play a specific note change the "connection" (entanglement) between the remaining musicians? Furthermore, what happens if the orchestra was already playing a slightly more complex tune (an excited state) before you stopped them?

The Main Characters

The Orchestra (The System): A one-dimensional chain of particles (specifically, a "critical XX chain" in the lab, modeled by a "compact free boson" in the math).
The Conductor's Cut (The Measurement): The researchers imagine "cutting" the orchestra in the middle and forcing that cut section to hold a specific pose (a "conformal boundary condition"). In the math world, this looks like a slit in a piece of paper.
The Soloists (Excited States): Before the cut, the orchestra might be playing a simple melody (ground state) or a melody with a specific soloist added (an excited state, like a "chiral current" or a "vertex operator").
The Audience (The Entanglement): The researchers want to know how connected the musicians on the left of the cut are to the musicians on the right, after the cut has been made.

The Core Discovery: The "Phase" Secret

The most interesting finding of this paper is about superpositions.

Imagine a musician who is in a "superposition." They aren't just playing a high note or a low note; they are playing both at the same time, like a ghostly echo of two notes overlapping. This overlap has a phase—a timing difference that determines how the two notes interfere (do they boost each other or cancel each other out?).

Before the measurement (The Unmeasured Case): If you listen to the whole orchestra without stopping anyone, the "phase" of this ghostly double-note is invisible. The connection between the left and right sides of the orchestra looks the same regardless of the timing difference. It's like trying to hear a specific echo in a noisy room; the echo gets lost in the general sound.
After the measurement (The Measured Case): When the researchers "cut" the orchestra and force a specific note, they create a new environment (a "slit geometry"). In this new environment, the phase suddenly becomes visible. The connection between the left and right sides now depends heavily on that timing difference.

The Analogy:
Think of the unmeasured system as a foggy day. You can't see the details of a person's face (the phase). But when you shine a specific flashlight (the measurement) through a narrow slit, the shadows cast by the person's features suddenly become sharp and distinct. The measurement didn't just "look" at the system; it revealed hidden information (the phase) that was previously invisible.

How They Did It (The Math Magic)

The authors used a branch of math called Conformal Field Theory (CFT).

The Map: They took the complex shape of the "slit" (the cut in the orchestra) and mathematically stretched and bent it into a simple circle (a disk). This is like taking a crumpled piece of paper with a hole in it and ironing it flat so you can draw on it easily.
The Formula: Once the shape was simple, they used standard rules of quantum mechanics to calculate the "connection strength" (entanglement entropy).
- For simple "current" excitations, the math involved something called hafnians (a complex cousin of determinants, like a specialized recipe for mixing ingredients).
- For the "superposition" cases, they had to add up the contributions of different "ghostly" paths, which created interference patterns (the phase sensitivity).

The Lab Check (The XX Chain)

Theory is great, but does it work in the real world? The authors tested their math using a critical XX chain, which is a specific type of magnetic chain of atoms that behaves like free electrons.

The Challenge: When you have a superposition (two states mixed together), the math gets messy because it's no longer a simple "Gaussian" (smooth bell-curve) shape. It's like trying to predict the weather when two different storm systems are colliding.
The Solution: They invented a new computer method called the "Multi-Slater Determinant" technique. Imagine trying to calculate the shape of a cloud made of two different types of water vapor. Instead of treating it as one blob, they treated it as two overlapping clouds and calculated how they interact.
The Result: They ran simulations on a computer with 500 to 800 atoms. The results matched their CFT predictions perfectly.
- They confirmed that for simple excitations, the math holds up.
- Crucially, they confirmed that for the superposition states, the "phase" really does change the entanglement after the measurement, exactly as their "slit" theory predicted.

Summary of Claims

Measurement Reveals Hidden Phases: A fixed measurement on a quantum system can make the "relative phase" of a superposition state visible in the entanglement of the remaining parts. This phase is invisible if you don't measure.
Mathematical Framework: They developed a general formula to calculate this entanglement by mapping the "slit" geometry to a disk and using correlation functions.
Specific Formulas: They provided exact mathematical formulas (involving hafnians and interference terms) for specific types of excitations (currents and vertex operators) in a free boson system.
Numerical Verification: They proved these formulas work by simulating a real quantum chain (the XX chain) using a new method for handling superpositions, showing that the theory matches the computer data.

In short: The paper shows that if you "pinch" a quantum system in a specific way, you can suddenly "see" the hidden timing relationships (phases) between different quantum states, turning a previously invisible feature into a measurable one.

Technical Summary: Entanglement of Excited States After Measurements in Conformal Field Theory

Problem Statement
This paper investigates the entanglement structure of low-energy excited states in (1+1)-dimensional conformal field theories (CFTs) following a fixed-outcome projective measurement on a spatial interval. While the entanglement of ground states after such measurements is understood via boundary conformal field theory (BCFT) and slit geometries, the behavior of excited states—specifically how operator insertions modify the post-measurement entanglement—remains less explored. The authors address how a fixed measurement outcome, which flows to a conformal boundary condition in the infrared, alters the Rényi entropy of excited states created by scaling operators or their coherent superpositions.

Methodology
The authors employ a combination of analytical CFT techniques and lattice numerical simulations:

Analytical Framework (BCFT and Replicas):
- Geometry: The measurement on an interval $A$ is modeled as a slit in the Euclidean path integral carrying a conformal boundary condition $a$ . The unmeasured system resides on a slit cylinder.
- Excited States: Excitations are introduced via operator insertions ( $\Upsilon$ ) at the infinite Euclidean past and future.
- Replica Trick: To compute the $n$ -th Rényi entropy, the slit cylinder is replicated $n$ times and cyclically glued along the entanglement cut.
- Uniformization: The replicated slit geometry is mapped to the upper half-plane and subsequently uniformized to a single disk. This transformation reduces the calculation of the excited-state contribution to a normalized correlation function of the operators on the disk with boundary condition $a$ .
- Specific Models: The framework is applied to the compact free boson CFT. The authors analyze:
  - Chiral current excitations ( $J = i\partial\phi$ ).
  - Coherent superpositions of left and right currents ( $J$ and $\bar{J}$ ).
  - Coherent superpositions of conjugate vertex operators ( $V_{\alpha, \bar{\alpha}}$ and $V_{-\alpha, -\bar{\alpha}}$ ).
- Calculations: For current excitations, the correlation functions are evaluated using hafnians. For superposition states, the authors derive general formulas involving sums over charge sectors and interference terms dependent on the relative phase of the superposition.
Lattice Numerics (XX Chain):
- The critical XX spin chain is used as the lattice realization of the compact free boson CFT.
- Current Excitation: Since the ground state and single current excitation are Slater determinants, the post-measurement state remains Gaussian. The authors compute the entanglement using conditional free-fermion correlation matrices derived from the projection of fixed measurement outcomes (alternating occupation patterns).
- Superposition States: Coherent superpositions of Slater determinants are non-Gaussian. To handle these, the authors develop a multi-Slater determinant method. They utilize a "transition-Gaussian determinant formula" to compute the purity (and thus the second Rényi entropy) of the reduced density matrix for the superposed state after projection.

Key Contributions and Results

General Formula for Excited States: The paper derives a universal expression for the post-measurement Rényi entropy of a primary excited state:
$S^{(\Upsilon, M)}_n = S^{(0, M)}_n + \frac{1}{1-n} \log F^{(n)}_{\Upsilon, a}$
where $S^{(0, M)}_n$ is the ground-state entropy on the slit geometry, and $F^{(n)}_{\Upsilon, a}$ is a normalized correlation function on the uniformized disk.
Chiral Current Excitation: For the chiral current $J$ , the authors obtain closed-form expressions for the second and third Rényi ratios in terms of hafnians. They explicitly demonstrate that these results recover the standard unmeasured cylinder results in the limit where the slit length approaches the UV cutoff.
Phase Sensitivity in Superpositions:
- $J/\bar{J}$ Superposition: The authors show that the measurement-induced boundary mixes holomorphic and antiholomorphic sectors. Consequently, the post-measurement Rényi ratio depends on the relative phase $\theta$ between the $J$ and $\bar{J}$ components. This phase dependence arises from the relative Jacobian factors of the conformal map for operators with different chiral weights.
- Conjugate Vertex Superposition: For a superposition of conjugate vertex operators (e.g., $V_{1,-1}$ and $V_{-1,1}$ ), the authors find a striking contrast: the ordinary-cylinder (unmeasured) second Rényi ratio is independent of the relative phase. However, the finite-slit post-measurement ratio contains explicit interference terms ( $\cos \theta$ and $\cos 2\theta$ ). This indicates that the fixed-outcome measurement activates phase-sensitive interference sectors that are invisible in the unmeasured case.
Numerical Verification:
- The analytical predictions for the current excitation are verified against XX chain simulations using conditional correlation matrices, showing excellent agreement for both $n=2$ and $n=3$ .
- The predictions for superposition states are verified using the multi-Slater determinant method. The numerical data confirms the spatial dependence and, crucially, the phase-sensitive interference terms predicted by the CFT formulas.

Significance
The paper establishes a rigorous framework for calculating the entanglement of excited states after projective measurements, bridging the gap between operator content in CFT and the entanglement properties of post-measurement lattice states.

The primary significance lies in the demonstration that fixed-outcome measurements can reveal coherent phase information that is otherwise hidden. Specifically, for conjugate vertex superpositions, the measurement boundary renders the relative phase observable in the Rényi entropy, whereas the unmeasured system's entropy remains phase-insensitive. This provides a concrete example of how measurement-induced boundaries can alter the universality class of entanglement, activating interference sectors between different conformal families.

Furthermore, the development of the multi-Slater determinant method provides a practical numerical tool for studying non-Gaussian states in critical systems, enabling direct tests of CFT predictions for coherent superpositions that were previously inaccessible via standard correlation-matrix techniques. The work suggests that such measurement-induced phase sensitivity could be a probe for wavefunction structure and quantum criticality in monitored quantum systems.