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 (CFT) and Gravity (Space - Time) sides, it’s possible to find the profound connections between initial correlations in quantum systems and the geometric structure of Dual Space-Times. Recent developments in 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 Spaces decomposed as:

[math]\LARGE{{H_{LR}} = {H_{L}} \otimes {H_{R}}}[/math]

[math]\Large{H_{LR} = H_{L} \otimes H_{R}}[/math]

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]

[math]\LARGE{\rho_{LR} = \rho_{L} \otimes \rho_{R} = |\Psi_{\beta}\Psi_{\beta}|} [/math]

[math]\Large{\rho_{LR} = \rho_{L} \otimes \rho_{R} = |\Psi_{\beta}\Psi_{\beta}|} [/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 Thermal States [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.

Due particelle interagenti con un fascio di energia

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.

[math]\LARGE{{I({\rho_{LR}})} = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR}})}}[/math]

[math]\large{{I({\rho_{LR}})} = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR}})}}[/math]

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.

In a low - Entropy (Figure 3) environment,

[math]\boxed{\LARGE{{I({\rho_{LR}})}} = 0} \hspace{0.5cm} \LARGE{\Rightarrow}[/math]     [math]\LARGE{0 = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR})}}}[/math];

[math]\LARGE{{S({\rho_{LR}})} = {S({\rho_{L}})} + {S{(\rho_{R})}}}[/math]

[math]\boxed{\large{{I({\rho_{LR}})} = 0}}[/math]


[math]\large{0 = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR})}}}[/math];

[math]\large{{S({\rho_{LR}})} = {S({\rho_{L}})} + {S{(\rho_{R})}}}[/math]

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:

[math]\LARGE{{\Delta{S}({\rho_{L}})} + {\Delta{S}({\rho_{R}})} \geq 0}[/math]

[math]\Large{{\Delta{S}({\rho_{L}})} + {\Delta{S}({\rho_{R}})} \geq 0}[/math]

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.

  1. nLab.org. "AdS-CFT correspondence in nLab"https://ncatlab.org/nlab/show/AdS-CFT+correspondence

  2. IOPscience. "Quantum Entanglement and Its Application in Quantum Communication" https://iopscience.iop.org/article/10.1088/1742-6596/1827/1/012120

  3. 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

  4. 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

  5. vXra.org. "The Placement of Two-sided Time in Physics" https://vixra.org/pdf/1906.0353v2.pdf

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