Left-Right Relative Entropy

This paper introduces the "left-right relative entropy" as a universal measure of distinguishability for boundary states in two-dimensional conformal field theories, revealing that certain orthogonal global states become indistinguishable under this metric and thereby defining "relative entanglement sectors" that link quantum information theory, modular data, and 't Hooft anomalies.

The Gist

Imagine you are standing in a vast, magical library where every book represents a different version of the universe. Some books describe a world where everything is frozen, others where everything is chaotic. In the world of physics, these "books" are called Quantum States.

Usually, if you want to know how different two books are, you read them from cover to cover. If the stories are completely different, the books are "orthogonal" (totally distinct). If they are the same, they are identical.

But what if you could only read the left-hand pages of the books? Or only the right-hand pages?

This is the core idea of the paper "Left-Right Relative Entropy" by Mostafa Ghasemi. The author introduces a new way to measure how different two quantum worlds are, but with a twist: we only look at half the picture.

The Big Idea: The "One-Eyed" Comparison

In quantum physics, systems often have two sides that dance together: a "left-moving" side and a "right-moving" side. Usually, to tell two states apart, you need to see the whole dance.

The author asks: What happens if we cover our eyes and only look at the left dancer?

He developed a mathematical tool called Left-Right Relative Entropy (LRRE). Think of this as a "distinguishability score."

• High Score: The left sides of the two books look very different.
• Zero Score: The left sides look exactly the same, even if the full books are totally different.

The Magic Trick: When "Different" Looks "Same"

Here is the most surprising part of the paper.

Imagine two people, Alice and Bob.

• Alice is wearing a red hat and a blue shirt.
• Bob is wearing a blue hat and a red shirt.

If you look at them fully, they are clearly different people. In physics terms, their "Global States" are orthogonal (completely distinct).

However, imagine you are a detective who can only see their shirts (the "left-moving" part).

• Alice's shirt is blue.
• Bob's shirt is blue.

To your "shirt-only" eyes, Alice and Bob are identical. Your "distinguishability score" is zero.

The paper proves that in certain quantum systems (like the famous Ising Model, which describes how magnets work), this happens naturally. You can have two completely different quantum universes that look exactly the same if you only observe one specific part of them.

The "Secret Club": Relative Entanglement Sectors

Because of this phenomenon, the author invents a new concept called Relative Entanglement Sectors.

Think of this as a VIP club.

• Normally, every quantum state is its own unique individual.
• But in this club, if two states look the same when you only check their "left side," they are considered members of the same group.

It doesn't matter that they are different people in the grand scheme of things; for the purpose of this specific measurement, they are twins.

The paper shows that these groups aren't random. They follow strict rules based on symmetry.

• In the Ising Model (a simple model of magnetism), there is a symmetry that flips "Up" spins to "Down" spins.
• The "Up" state and the "Down" state are totally different globally.
• But when you look at just the "left side," they look identical.
• So, they belong to the same Relative Entanglement Sector.

Why Does This Matter?

You might ask, "So what? Why do we care if two things look the same from one angle?"

1. It's a New Compass: This tool helps physicists navigate the "landscape" of quantum matter. It tells us which states are truly unique and which are just different faces of the same coin.
2. It Connects to Anomalies: The paper finds that these groups of "look-alike" states behave in a way that mirrors deep, mysterious rules of nature called Anomalies (specifically 't Hooft anomalies). It's like finding that the way these groups form is a hidden fingerprint of the universe's underlying laws.
3. It's Universal: The math works for almost any system, from simple magnets to complex theories about strings and black holes. The formula is surprisingly clean, relying on a "modular S-matrix" (which is just a fancy table of numbers that describes how the system's parts mix and match).

The Takeaway

This paper is like discovering a new pair of glasses. When you put them on, you realize that two things you thought were completely different are actually indistinguishable from a specific viewpoint.

It teaches us that distinguishability depends on what you are allowed to see. In the quantum world, two universes can be totally different, yet if you only look at the "left side" of the story, they are one and the same. This insight helps us understand the hidden symmetries and structures that hold our universe together.

Technical Summary

1. Problem Statement

The paper addresses the challenge of quantifying the distinguishability between quantum states in the context of two-dimensional Conformal Field Theories (CFTs), specifically focusing on boundary states.

• Context: While Entanglement Entropy (EE) is a standard measure of quantum correlations, it suffers from ultraviolet (UV) divergences in relativistic field theories and ambiguities in gauge theories.
• Gap: Relative entropy is known to be UV-finite and a robust measure of distinguishability, but its application to boundary states (specifically those arising from D-branes, quantum impurities, or critical quenches) in CFTs requires a specialized formulation.
• Specific Question: How can one define a universal, finite measure of distinguishability between two boundary states by tracing over either the left-moving or right-moving sectors? Furthermore, what are the implications when this measure vanishes for states that are globally orthogonal?

2. Methodology

The author develops a formalism to calculate the Left-Right Relative Entropy (LRRE) for arbitrary regularized boundary states on a circle.

• Regularization: Boundary states $|B\rangle$ in CFT are generally non-normalizable. The author introduces a regularization scheme using Euclidean time evolution $e^{-\epsilon H}$, where $H$ is the Hamiltonian and $\epsilon$ is a UV cutoff parameter. The regularized state is defined as $|\tilde{B}\rangle = (Z_B)^{-1/2} e^{-\epsilon H} |B\rangle$.
• Factorization: The Hilbert space of a diagonal Rational CFT (RCFT) is factorized into left ($V_h$) and right ($\bar{V}_{\bar{h}}$) moving sectors. The boundary state is expanded in terms of Ishibashi states $|h_a\rangle\rangle$.
• Reduced Density Matrices: By tracing out the right-moving sector, the author derives the reduced density matrix $\rho_L$ for the left-moving sector.
• Replica Trick: To compute the relative entropy $D(\rho_L \| \sigma_L)$, the author employs the replica trick in the thermodynamic limit ($\ell/\epsilon \gg 1$). This involves calculating traces of powers of the density matrices, $\text{tr}(\rho^n)$ and $\text{tr}(\rho \sigma^{n-1})$, and taking the derivative with respect to $n$ at $n=1$.
• Modular Transformation: The calculation utilizes the modular $S$-matrix transformation properties of the Virasoro characters to evaluate the traces in the thermodynamic limit, eliminating dependence on the UV cutoff $\epsilon$.

3. Key Contributions

A. Universal Formula for LRRE

The paper derives a universal formula for the Left-Right Relative Entropy between two regularized boundary states $|\tilde{B}_\alpha\rangle$ and $|\tilde{B}_\beta\rangle$.

• The result reduces to the Kullback–Leibler (KL) divergence:
  $$D(\rho_L^{(\alpha)} \| \sigma_L^{(\beta)}) = \sum_a p_a^{(\alpha)} \log \left( \frac{p_a^{(\alpha)}}{p_a^{(\beta)}} \right)$$
• Here, the probability distribution $p_a$ is determined entirely by the modular $S$-matrix ($S_{1a}$) and the boundary coefficients ($\psi_a$) of the state:
  $$p_a^{(\alpha)} \equiv \frac{S_{1a} |\psi_a^{(\alpha)}|^2}{\sum_i S_{1a} |\psi_i^{(\alpha)}|^2}$$
• Crucially, this quantity is UV-finite and independent of the regularization parameter $\epsilon$.

B. Extension to Rényi Entropy and Fidelity

The framework is extended to define:

• Left-Right Sandwiched Rényi Relative Entropy ($D_n$): A one-parameter generalization.
• Left-Right Quantum Fidelity ($F$): Derived from the $n=1/2$ limit, providing a measure of overlap between reduced density matrices.

C. Discovery of "Relative Entanglement Sectors" (RES)

The most significant finding is the identification of a phenomenon where the LRRE between two globally orthogonal boundary states is zero.

• Observation: For specific boundary states (e.g., Cardy states in diagonal CFTs), $D(\rho_L^{(\alpha)} \| \sigma_L^{(\beta)}) = 0$ even if the global states $|\tilde{B}_\alpha\rangle$ and $|\tilde{B}_\beta\rangle$ are orthogonal.
• Definition: The author defines Relative Entanglement Sectors (RES) as equivalence classes of boundary states that are indistinguishable with respect to LRRE (i.e., states with zero mutual LRRE).
• Physical Implication: This implies that while the global states are distinct, their local (left-moving) physics is identical.

4. Results and Examples

The author applies the formalism to three specific models to demonstrate the theory:

1. Ising Model ($c=1/2$):

   • The RES consists of two sectors: $E_1 = \{|e_0\rangle, |e_{1/2}\rangle\}$ and $E_2 = \{|f_{1/16}\rangle\}$.
   • The states $|e_0\rangle$ (fixed up) and $|e_{1/2}\rangle$ (fixed down) have zero LRRE between them, despite being orthogonal globally.
   • These sectors transform under the global $Z_2$ symmetry as a NIM-representation (Non-negative Integer Matrix representation).

2. Tricritical Ising Model ($c=7/10$):

   • Similar structures are found, with specific pairs of Cardy states forming RESs.

3. $\widehat{su(2)}_k$ WZW Model:

   • The RESs are defined by the equivalence $j \sim k/2 - j$.
   • Anomaly Connection: The structure of these sectors depends on whether the level $k$ is even or odd.
     • Even $k$: Contains a unique fixed point (singlet) and doublets, corresponding to a non-anomalous $Z_2$ symmetry.
     • Odd $k$: All sectors are doublets with no fixed point, mirroring the presence of a $Z_2$ 't Hooft anomaly in the boundary theory.

5. Significance and Implications

• Bridge Between Information and Symmetry: The work establishes a direct link between quantum information measures (distinguishability) and the algebraic structure of boundary CFTs (symmetries and fusion rules).
• Anomaly Constraints: The level-dependent structure of RESs mirrors 't Hooft anomalies, suggesting that information-theoretic measures can detect and characterize quantum anomalies in boundary states.
• Topological Phases: The results offer insights into topological phases of matter. Since boundary states in CFTs describe edge modes of topological systems (e.g., $p+ip$ superconductors), the vanishing of LRRE suggests that different bulk excitations (quasiparticles) can lead to indistinguishable edge physics under specific symmetry transformations.
• New Framework: The introduction of RES provides a new tool for classifying exotic phases of matter constrained by quantum anomalies, moving beyond traditional order parameters.
• Future Directions: The paper suggests applications to D-branes in string theory and potential gravity duals via AdS/CFT, proposing that LRRE could serve as a probe for distinguishability in quantum gravity.

In summary, the paper introduces a rigorous, UV-finite measure of distinguishability for CFT boundary states, revealing that global orthogonality does not imply local distinguishability. This leads to the discovery of "Relative Entanglement Sectors," which are deeply tied to global symmetries and quantum anomalies, offering a novel perspective on the interplay between quantum information and conformal symmetry.