Pure states for subregions in gravity and their entanglement entropy

This paper proposes a framework where spatial subregions in quantum gravity are assigned pure states via a partially frozen gravitational path integral, leading to a new holographic prescription for entanglement entropy that satisfies key consistency conditions and introduces an observer-dependent entanglement wedge.

The Gist

The Big Idea: A Universe Where "Parts" Can Be Whole

Imagine you have a giant, complex jigsaw puzzle representing the entire universe. In normal quantum physics (the rules that govern tiny particles), if you look at just one small piece of the puzzle, it usually looks messy and incomplete. You can't describe that single piece perfectly on its own because it's "entangled" with the rest of the puzzle. To describe it, you have to use a "mixed state," which is like a blurry, statistical guess because you are missing information about the other pieces.

This paper proposes a radical new idea for gravity: If you look at a specific region of the universe through the lens of quantum gravity, that region might actually be able to be described as a perfect, pure, complete picture on its own.

The author, Zixia Wei, suggests that we can treat a piece of the universe as a self-contained "pure state" rather than a blurry, incomplete one.

How Do We "Freeze" a Piece of the Universe?

To understand how this works, the author uses a mathematical tool called a Gravitational Path Integral. Think of this as a giant simulation that tries to calculate every possible way the universe could have formed.

Usually, this simulation sums up everything. But Wei proposes a "partially frozen" version:

1. The Frozen Region: Imagine you take a specific chunk of the universe (a spatial subregion) and "freeze" it in place. You fix its shape and its internal rules. You treat this chunk like a solid, unchangeable box.
2. The Rest of the Universe: Everything outside this box is allowed to wiggle, change, and fluctuate. The simulation sums up all the possibilities for the outside world, but it must respect the boundaries of your frozen box.

The Analogy: Imagine you are in a room (the frozen region) while a chaotic storm rages outside (the rest of the universe). You can't control the storm, but the walls of your room are solid and fixed. The paper argues that by fixing the room, you can define a perfect, pure state for what happens inside, even though the outside is chaotic.

The "Holographic" Recipe for Entanglement

Once we have this "pure state" for a region, the next question is: How much is this region "entangled" with itself? (In quantum physics, entanglement is like a deep, invisible connection between parts of a system).

The author proposes a new recipe to calculate this, similar to a famous formula called Ryu-Takayanagi.

• The Old Recipe: To measure the connection between two parts of a hologram, you draw a surface (like a soap film) connecting them. The size of that surface tells you the amount of entanglement.
• The New Recipe: Because we have a "frozen" region, the rules change slightly. You can still draw that surface, but it has to obey a new rule: It must hug the boundary of your frozen region. It can explore the "wiggly" outside world, but it can't cross the frozen boundary in a way that breaks the rules.

This creates a new kind of "entanglement wedge" (a region of space that is connected to your subregion). The paper shows that this new recipe works perfectly: it follows all the logical rules of quantum mechanics (like "strong subadditivity," which is a fancy way of saying the math doesn't break when you combine regions).

Why Does This Matter? The "Observer" Twist

The most surprising part of the paper is what this means for observers.

In the old view, the universe is one big thing, and we just look at pieces of it. In this new view, the description of the universe depends on who is looking and where they are standing.

• The Metaphor: Imagine a giant, shifting landscape.
  • Observer A decides to freeze a mountain. For them, the mountain is a solid, pure object, and the rest of the world is a fluctuating sea.
  • Observer B decides to freeze a valley. For them, the valley is the solid, pure object, and the mountains are the fluctuating sea.

The paper suggests that the same underlying spacetime can be described as two completely different "pure states" depending on which region you choose to freeze.

• Observer A sees a specific quantum state.
• Observer B sees a different quantum state.
• Both are correct, but they are looking at the universe through different "lenses."

Summary of Key Claims

1. Pure States for Parts: Unlike normal physics where parts of a system are messy (mixed), a region in quantum gravity can be described as a perfect, pure state if you fix its boundary conditions.
2. The Frozen Path Integral: This state is created by a mathematical calculation where a specific region is held fixed ("frozen") while the rest of the universe is summed over.
3. New Entanglement Rules: The author provides a new formula to calculate how connected different parts of this frozen region are. This formula works consistently and matches known physics in special cases (like black holes or holographic universes).
4. Observer Dependence: The "entanglement wedge" (the region of space connected to your observation) changes depending on which region you choose to freeze. This implies that the quantum description of the universe is relative to the observer's location and choices.

What the paper does NOT claim:

• It does not claim this solves the mystery of black hole information loss (though it relates to it).
• It does not claim we can build a machine to "freeze" regions of space.
• It does not claim this applies to everyday objects like chairs or apples (it is strictly about the fundamental quantum nature of spacetime).

In short, the paper suggests that in the quantum world of gravity, how you define a "piece" of the universe determines the reality of that piece. By freezing a region, you turn a messy, incomplete picture into a perfect, pure one.

Technical Summary

Technical Summary: Pure States for Subregions in Gravity and Their Entanglement Entropy

Problem Statement
In standard quantum many-body systems, a closed system is described by a pure state, while a proper subsystem is described by a mixed reduced density matrix due to the tracing out of exterior degrees of freedom. The paper addresses the corresponding question for spatial subregions in quantum gravity: does a subregion admit a pure-state description? While semiclassical expectations suggest a mixed state upon tracing over exterior degrees of freedom, the holographic principle implies that information appearing to lie outside a subregion may be encoded on its boundary. This raises the possibility that a spatial subregion in quantum gravity can be associated with a pure state, distinct from the mixed states typical of open subsystems in non-gravitational theories.

Methodology
The author proposes a framework based on a "partially frozen" gravitational path integral (GPI).

1. Gravitationally Prepared States: The construction generalizes the concept of gravitationally prepared states defined on a full Cauchy slice [6]. Instead of freezing the entire spacetime tube $\Sigma \times I$, the author considers a general spacetime subregion $M_F$ (a $D$-dimensional manifold) that covers only a spatial subregion $F$.
2. Path Integral Formulation: The state is defined by a GPI where the geometry and field configurations within $M_F$ are fixed (frozen), while the ambient geometry and fields in the complementary region $M_G$ are summed over. The partition function is given by:
   $$Z_{\text{grav}}[M_F] = \int_{\partial M_G = \partial M_F} \mathcal{D}g_{\mu\nu} \int_M \mathcal{D}\phi \exp(-I_E)$$
   This setup imposes Dirichlet boundary conditions (DBC) at the interface between the frozen region $F$ and the gravitating region $G$, while allowing quantum fields to traverse the entire spacetime.
3. Purity Verification: The author employs the replica trick to determine the nature of the state. By constructing $n$-replica geometries and comparing the trace of the $n$-th power of the density matrix to the $n$-th power of the trace, it is demonstrated that $(\text{Tr}(\rho))^n = \text{Tr}(\rho^n)$ holds for the ensemble. This equality implies that the rank of the density matrix $\rho_{F^\circ \cup \partial F}$ is 1, confirming the state is pure.
4. Holographic Prescription: In the semiclassical regime, assuming a single dominant saddle geometry, a Ryu-Takayanagi (RT)-like prescription is proposed for the entanglement entropy (EE) of a bipartition $A \subset F$. The formula minimizes over regions $e_A$ satisfying a "frozen condition" $e_A \cap F = A$:
   $$S^F_A = \min_{e_A \cap F = A} \left( \frac{\text{Area}(\gamma^F(e_A))}{4G_N} + S_{\text{QFT}}(e_A) \right)$$
   Here, $\gamma^F(e_A)$ is the codimension-2 surface defined by the closure of the interface between $e_A$ and the complement within the gravitating region $G$.

Key Contributions and Results

• Pure State Assignment: The paper establishes that spatial subregions in quantum gravity can be assigned pure states $|\Psi\rangle_{F^\circ \cup \partial F}$, where the interior $F^\circ$ is purified by holographic degrees of freedom living on the boundary $\partial F$. This contrasts with the mixed states of open subsystems in standard quantum mechanics.
• Observer-Dependent Entanglement Wedge: The construction introduces an entanglement wedge $E^F_A$ and a quantum RT surface $\Gamma^F_A$ that depend explicitly on the choice of the frozen subregion $F$. For a fixed subregion $A$, the inclusion of a point $p$ in the entanglement wedge depends on how $F$ is chosen.
• Consistency Conditions: The proposed holographic prescription (Eq. 16) is shown to satisfy nontrivial self-consistency conditions required for a valid quantum entropy definition:
  • Strong Subadditivity (SSA): Proven via the properties of the area term and QFT entropy.
  • No-Cloning: Proven by showing that disjoint regions have disjoint entanglement wedges.
  • Entanglement Wedge Nesting: If $A \subset B$, then $E^F_A \subset E^F_B$.
  • Complementarity: For $A \cup B = F$, the entanglement wedges cover the entire slice $\Sigma$ and are disjoint in their interiors.
• Recovery of Known Formulas: The prescription reproduces several established entropy formulas as special cases:
  • The standard Ryu-Takayanagi formula when $F$ is the asymptotic boundary.
  • The island formula when $F$ represents a heat bath region.
  • Holographic EE in the presence of End-of-the-World (EOW) branes (AdS/BCFT).
  • Wedge holography formulas for higher-codimension defects.
  • The Bousso-Penington generalized entanglement wedge for bulk subregions.

Significance and Claims
The paper claims to provide a precise realization of the intuition that subregions in quantum gravity admit pure-state descriptions. The primary significance lies in the identification of an observer-dependent entanglement wedge. The author argues that a single semiclassical spacetime can be encoded in different quantum states $|\Psi\rangle_F$ and $|\Psi'\rangle_{F'}$ living in different spatial subregions, corresponding to the perspectives of different observers localized at $F$ and $F'$. Each observer trades knowledge of local geometric data for a quantum state describing the entire spacetime.

The work suggests that the entanglement entropy and entanglement wedge are not absolute properties of a subregion but are relative to the "frozen" data defining the observer's frame. The paper concludes by noting that isolating the observer-independent content shared among these descriptions and formulating precise maps between different viewpoints remains a challenge to be addressed in future work [19].