# The Implications of Quantum Entanglement on Space-Time: A Focus on the Time Direction

## About the "entangled" Microscopic and Macroscopic Realities

Through a detailed investigations on both the __ Conformal Field Theory__ (

__) and__

**CFT****Gravity**(

**Space - Time**) sides, it’s possible to find the profound connections between initial

**correlations**in

**quantum systems**and the

**geometric structure**of

__s. Recent developments in__

**Dual Space-Time**__(__

**Anti - de - Sitter**__)/__

**AdS****CFT**

**correspondence**[

__1__] have uncovered intriguing links between

**Quantum Information Theory**and Gravity, specifically focusing on the structure of

**Quantum Entanglement**[

__2__] (

__Figure 1__) in conformal field theories (CFTs) and its impact on the dual spacetime. The

**entanglement structure**of quantum subsystems is argued to be a key determinant of classically connected spacetimes.

## Let's start with the Quantum Entanglement!

In the **standard approach**, we commence by examining **two independent conformal field theories** (CFTs) on the **sphere** [math]\normalsize{S^d}[/math] (**x time**) (__Figure 2__). These CFTs correspond to **subsystems**, **Left** (**L**) and **Right **(**R**), with their __ Hilbert Space__s decomposed as:

**Equation 1**. Decomposed Hilbert Space of LR Entangled State

[math]\large{H_{LR}}[/math] is the Hilbert Space for **LR Entangled State**; [math]\large{H_{L}}[/math] is the Hilbert Space for **L Subsystem** and [math]\large{H_{R}}[/math] is the Hilbert Space for **R Subsystem** ([math]\large{\otimes}[/math] is the **Product Operator**).

Initially uncorrelated, the joint state is a **product state**, [math]\normalsize{\rho_{LR}}[/math]

**Equation 2**. LR Entangled State Density Matrix

where [math]\normalsize{\rho_{L}}[/math] and [math]\normalsize{\rho_{R}}[/math] are the __ Density Matrix__(ces) for the left and right subsystems, representing

__s [__

**Thermal State**__3__]; [math]\normalsize{\Psi_{\beta}}[/math] is the

**LR Entangled State Wavefunction**. For initially entangled states, such as the thermofield double state, the joint state [math]\normalsize{\rho_{LR}}[/math] involves entangled pure states for subsystems L and R.

**Figure 1**. An abstract illustration of the two-Particles Quantum Entanglement

In the **AdS/CFT framework**, this uncorrelated state corresponds to disconnected AdS spacetimes. We quantify correlations using mutual information as follows.

**Equation 3**. The Quantum Information equation for the LR State

[math]\large{S(\rho_{LR})}[/math], [math]\large{S(\rho_{L})}[/math] and [math]\large{S(\rho_{R})}[/math] are the **LR State**, **L** and **R Subsystems Entropies**, respectively.

**Equation 4**. Low-Entropy Conditions Quantum Information

According to the __ Second Law of Thermodinamics__ (

__Equation 5__), the Entropy of the composite system must increase as individual entropies evolve, giving:

**Equation 5**. The Second Law of Thermodynamics for the LR State

[math]\large{\Delta{S}(\rho)}[/math] is the **Entropy Variation** for each subsystem and state involved.

In the absence of **initial correlations**, **the dual Space - Time** is composed of **disconnected AdS regions**, while initial entanglement leads to classical connectivity. The degree of entanglement is shown to dynamically influence the connectivity of the dual spacetime. Disentangling __ Degrees of Freedom__ decreases

**mutual information**and Entropy.

## Traveling the Space-Time aboard the Thermodynamic Arrow of Time

Building upon recent debates on the __ Thermodynamic Arrow of Time__ [

__4__], it has been established a connection between the initial conditions of quantum correlations and the emergence of a

**preferred direction**for the arrow of time. If there are

**no initial correlations**, the arrow of time is directed toward

**increasing Entropy**. However, in contrast to the uncorrelated case, initial correlations alter the entropy evolution. The thermodynamic arrow can now

**reverse**, allowing for

**both orientations**.

**Figure 2**. A 3D Representation of a [math]\small{S^d}[/math] Sphere

## And on Gravity … Side?

Furthermore, the concepts of **Space - Time Sidedness** [__5__] and **Time - Orientability** have to be discussed. **Initial entanglement** in the composite quantum system is argued to lead to a **time-unoriented**, **one-sided Space - Time**, while **decreasing entanglement** results in a **time-oriented**, **two-sided Space - Time**. In the latter condition, the dual spacetime features disconnected components with **opposing time orientations**, reflecting the reversed arrows of time in the individual CFTs.

## The Fluctuations between Entanglement States

The effects of varying the **degree of entanglement** between the dual CFTs affect the Space - Time. High
correlations are associated with a connected one-sided spacetime, while disentangling the degrees of
freedom leads to a disconnected two-sided Space - Time. The **maximal entanglement** is interpreted as
building a connection between the **two sides** of Space - Time.

**Figure 3**. Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group

## Just a Multi-Effect Dynamics

As shown, the insights into the **relationship** between **quantum entanglement**, **Space - Time sidedness**,
and the **thermodynamic arrow of time**, within the **AdS/CFT correspondence framework**, highlight the
**crucial play** of dynamic between **initial correlations** and the **geometric dual structure**, in understanding
the **emergence** and **orientation** of the **thermodynamic arrow of time**.

- nLab.org. "AdS-CFT correspondence in nLab"
__https://ncatlab.org/nlab/show/AdS-CFT+correspondence__ - IOPscience. "Quantum Entanglement and Its Application in Quantum
Communication"
__https://iopscience.iop.org/article/10.1088/1742-6596/1827/1/012120__ - Astronomy & Astrophysics. "The thermal state of molecular clouds in the Galactic center: evidence for non-photon-driven heating"
__https://www.aanda.org/articles/aa/full_html/2013/02/aa20096-12/aa20096-12.html__ - Forbes. "No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time"
__https://www.forbes.com/sites/startswithabang/2019/11/22/no-thermodynamics-does-not-explain-our-perceived-arrow-of-time/?sh=4694b68c3109__ - vXra.org. "The Placement of Two-sided Time in Physics"
__https://vixra.org/pdf/1906.0353v2.pdf__