Bit Threads: From Entanglement to Geometric Entropies (2508.18941v1)

Abstract: In this work, we attempt to construct bit threads configurations for various backgrounds using expressions from CPS formalism. We find that when the Ryu-Takayanagi surface is same as horizon, such expressions are sufficient. In other cases, it differs by gradient of a harmonic function. We explore its relation to Wald and differential entropy, and re-express the first law of entanglement entropy in terms of bit threads. Inclusion of quantum effects imposes some constraints on the bulk entanglement via the dominant energy condition. We also apply our method to ascertain bit thread configuration in certain dynamical spacetime.

• The paper demonstrates a novel bit thread construction leveraging the covariant phase space formalism to unify the computation of entanglement and geometric entropies.
• It employs harmonic gauge corrections to adjust divergenceless vector fields, ensuring saturation of norm bounds on RT surfaces across various backgrounds.
• It extends the framework to cover Wald, cosmological, and differential entropies while reformulating first law expressions and incorporating quantum corrections.

Bit Threads: From Entanglement to Geometric Entropies

Introduction and Motivation

The bit thread formalism provides a geometric, flow-based alternative to the Ryu-Takayanagi (RT) prescription for holographic entanglement entropy (HEE) in AdS/CFT. Instead of minimising the area of a codimension-2 surface, the bit thread approach maximises the flux of a divergenceless vector field subject to a norm bound, with the maximal flux equating to the entanglement entropy. This formalism is deeply connected to quantum information, as the flow lines ("bit threads") can be interpreted as representing Bell pairs shared between boundary regions.

The present work systematically investigates the construction of bit thread configurations using the covariant phase space (CPS) formalism, extending the approach to a variety of backgrounds, including static and dynamical spacetimes, and generalising the flow prescription to other geometric entropy concepts such as Wald entropy and differential entropy. The authors also reformulate the first law of entanglement entropy in terms of bit threads and discuss quantum corrections, including constraints from energy conditions.

Canonical Bit Threads via Covariant Phase Space

The CPS formalism, particularly in the Iyer-Wald framework, provides a systematic method for constructing conserved currents associated with symmetries of the gravitational theory. The key object is the codimension-2 current $\mathbf{k}_\xi$, which is closed for exact symmetries (Killing vectors), and can be mapped to a divergenceless vector field $v^a_{cps}$ via the Hodge dual on a Cauchy surface:

$v_{cps}^a = g^{ab} (\star {\mathit{j}_\xi})_b$

where ${\mathit{j}_\xi}$ is related to the Noether charge and symplectic potential.

While $v^a_{cps}$ is always divergenceless, it does not generically satisfy the norm bound $|v| \leq 1/4G_N$ or the saturation condition $v^a|_{RT} = n^a/4G_N$ required for bit threads. The authors show that the general solution is given by an equivalence class:

$$[v^a] = \{ v^a_{cps} + \nabla^a \varphi \mid \nabla^2 \varphi = 0 \}$$

where the harmonic function $\varphi$ encodes a gauge freedom that can be used to enforce the bit thread constraints.

Example: Poincaré AdS$_3$

In Poincaré AdS$_3$, the naive $v^a_{cps}$ derived from CPS fails to saturate the norm bound on the RT surface. By solving the Laplace equation for $\varphi$ with appropriate boundary conditions, the authors construct a corrected vector field $v^a = v^a_{cps} + \nabla^a \varphi$ that satisfies all bit thread requirements and reproduces the HEE for a boundary interval.

Figure 1: Flow lines (black curves with arrows) for bit threads obtained from transforming a valid thread configuration in AdS-Rindler to Poincaré AdS. The gray horizontal line denotes the boundary and the red semicircle is the RT surface corresponding to the boundary subregion $A \in [-2,2]$.

Example: Rindler AdS$_3$

For Rindler AdS$_3$, where the RT surface coincides with the horizon, the CPS-derived vector field $v^a_{cps}$ automatically satisfies all bit thread constraints without the need for gauge corrections. This demonstrates that the necessity of the harmonic correction is background-dependent.

Extensions to Other Geometric Entropies

The CPS formalism is not limited to HEE. The authors extend the bit thread prescription to other entropy concepts:

• Wald Entropy (Black Hole Horizons): The bit thread vector field constructed from the CPS current, with norm bound imposed on the horizon, yields the Wald entropy upon integration.
• Cosmological Horizons: The same construction applies to de Sitter horizons, reproducing the Gibbons-Hawking entropy.
• Differential Entropy: For spacetimes with "holes" (e.g., spherical Rindler), the bit thread formalism computes the differential entropy as the flux through the boundary of the hole.

First Laws in the Bit Thread Language

The authors reformulate the first law of entanglement entropy, black hole thermodynamics, and cosmological horizon thermodynamics in terms of bit thread flows. For Killing symmetries, the associated bit thread vector field is divergenceless, and the first law is expressed as a local conservation equation:

$\nabla_a v^a_{th} = 0$

For differential entropy, where the relevant symmetry is conformal rather than Killing, the symplectic current does not vanish, and the first law acquires a volume term, relating Lorentzian and Riemannian bit threads.

Quantum Corrections and Energy Conditions

Quantum corrections to HEE introduce a bulk entanglement entropy term. The quantum bit thread vector field $v^a = v^a_0 + v^a_q$ satisfies a modified divergence relation:

$$\nabla_a v^a_q = s_q(x) = \langle T^{\text{bulk}}_{ab}(x) \rangle \xi^a n^b$$

where $s_q(x)$ is the bulk entanglement density. The positivity of relative entropy imposes an energy condition equivalent to the dominant energy condition (DEC) on the bulk stress tensor:

$$\langle \delta T^{\text{bulk}}_{ab}(x) \rangle \xi^a n^b \geq 0$$

Bit Threads in Dynamical Spacetimes

The construction is extended to dynamical backgrounds such as AdS-Vaidya, where the presence of a conformal Killing vector allows for a perturbative bit thread flow. However, the full non-perturbative construction in general time-dependent spacetimes remains challenging due to the lack of symmetries.

PDE Approach to Bit Threads

The authors also discuss a PDE-based approach, where the bit thread field is constructed by solving the Laplace equation for a potential, subject to non-linear norm constraints and boundary conditions. This method is particularly effective in simple geometries.

Figure 2: Flow lines for bit threads obtained from solving the PDE: $\nabla_a v_{PDE}^a=0$ and imposing the conditions $v_{PDE}^a|_{RT} = n^a$ and $|v_{PDE}| \le 1$. The blue horizontal line denotes the boundary subregion and the red semicircle is the corresponding RT surface.

Implications and Future Directions

The canonical construction of bit threads via CPS formalism provides a unified framework for geometric entropies in gravitational theories. The equivalence class structure, with gauge freedom encoded in harmonic functions, clarifies the non-uniqueness of bit thread configurations and their dependence on background geometry.

The extension to higher curvature gravities, covariant bit threads, quantum energy conditions, and generalised entanglement wedges are promising directions. The connection to geometric modular flows in CFTs and the challenge of constructing bit threads in fully dynamical spacetimes are highlighted as open problems.

Conclusion

This work demonstrates that the CPS formalism offers a systematic and canonical approach to constructing bit thread configurations for a wide class of backgrounds and entropy concepts. The necessity of gauge corrections is shown to be background-dependent, and the formalism naturally generalises to black hole and cosmological horizon entropy, as well as differential entropy. The reformulation of first laws and the incorporation of quantum corrections via energy conditions further solidify the utility of the bit thread perspective. The results provide a foundation for future investigations into the geometric and information-theoretic structure of spacetime in holography and beyond.