Far down the Quantum Chemistry: how to Intoduce an “Entangled” Electron Correlation


Far down the Quantum Chemistry: how to Introduce an “Entangled” Electron Correlation

Exceeding the Old Methods

In Quantum Chemistry, the Hartree–Fock (HF) Method a, which describes interacting Fermion systems using an effective Single-Particle Model [1], is widely used to approximate the Electronic Structure of atoms and molecules.

However, this method neglects Electron Correlation, which results from the inherent interactions between Electrons in quantum systems. The Correlation Energy [2], defined as the energy difference between the HF limit and the exact solution (Equation 1) of the nonrelativistic Schrödinger Equation (further information in here, Section 2), is one measure of this correlation.

[math]\LARGE{E_{c} = |E^{Exact} - E^{HF}| }[/math]

[math]\normalsize{E_{c} = |E^{Exact} - E^{HF}| }[/math]

Equation 1.   The Electronic Energy in HF Method

Nevertheless, other metrics, such as Statistical Correlation Coefficients [3] and Shannon Entropy, have been proposed to quantify electron correlation. Despite these methods, electron correlation remains challenging to calculate accurately for Large Systems [4]. 

Quantum Entanglement (further information in here, Ref. 2), a fundamental concept in quantum mechanics, offers a promising alternative for measuring electron correlation. Unlike traditional measures, entanglement is directly observable and represents a non-classical correlation between quantum systems.

A New Definition of Quantum Entanglement

To quantify entanglement, one can consider a pure two-electron state in a 2m-dimensional Spin-Orbital space, represented by fermionic Annihilation and Creation Operators (further information in here, Section 2), [math]\normalsize{c_{a}}[/math] and [math]\normalsize{{c^{\dagger}}_{a}}[/math], with [math]\small{|0 \rangle}[/math] as the vacuum state. The general form of a two-electron state, [math]\small{| \Psi \rangle}[/math] can be written as:

[math]\LARGE{| \Psi \rangle = \sum_{a, b \in \{1, 2, \ldots, 2m\}} \omega_{a, b} c^\dagger_a c^\dagger_b |0 \rangle}[/math]

[math]\normalsize{| \Psi \rangle = \sum_{a, b \in \{1, 2, \ldots, 2m\}} \omega_{a, b} c^\dagger_a c^\dagger_b |0 \rangle}[/math]

Equation 2.   The form of the Electronic Wavefunction for a two-electron state

where [math]\normalsize{x_{a, b}}[/math] represents the Antisymmetric Expansion Coefficient Matrix [5], satisfying [math]\normalsize{x_{a, b} = - x_{b, a}}[/math]. Using this representation, we can derive a Reduced Density Matrix (further information in here, Section 2), [math]\normalsize{\rho}[/math] by tracing out all but one Spatial Orbital, resulting in a [math]\normalsize{4 \times 4}[/math] matrix. Then one can define the Von Neumann Entropy, like below.

[math]\LARGE{S(\rho) = - \operatorname{Tr}(\rho \log_2 \rho)}[/math]

[math]\normalsize{S(\rho) = - \operatorname{Tr}(\rho \log_2 \rho)}[/math]

Equation 3.   The Von Neumann Entropy equation

With [math]\normalsize{\rho}[/math] of the following form:

[math]\LARGE{\rho = \operatorname{Tr}|\Psi \rangle \langle \Psi|}[/math]

[math]\normalsize{\rho = \operatorname{Tr}|\Psi \rangle \langle \Psi|}[/math]

Equation 4.   The expansion of the Reduced Density Matrix

This approach provides a measure of the entanglement for atomic and molecular systems, focusing on the von Neumann entropy of the reduced density matrix.

Is it an Efficient Model?

Using the derived reduced density matrix, the entanglement for various systems, focusing on the Hydrogen Molecule ([math]\small{H_{2}}[/math]) [6], as an example, can be calculated. The entanglement is evaluated as a function of the Interatomic Distance, R.

These calculations show that entanglement (Equation 6) and electron correlation exhibit similar trends, with maximum entanglement occurring at specific interatomic distances. This behavior aligns with previous findings, indicating that entanglement can be an effective metric for measuring electron correlation.

Additionally, if one explores a model system of two spin-1/2 electrons with an Exchange Coupling Constant a, [math]\small{J}[/math] and a transverse Magnetic Field strength, B. The general Hamiltonian for this system is given by:

[math]\LARGE{H = - \frac{J}{2} (1 + \gamma) \sigma^x_1 \sigma^x_2 - \frac{J}{2} (1 - \gamma) \sigma^y_1 \sigma^y_2 - B \sigma^z_1 \otimes I_2 - B I_1 \otimes \sigma^z_2}[/math]

[math]\small{H = - \frac{J}{2} (1 + \gamma) \sigma^x_1 \sigma^x_2 - \frac{J}{2} (1 - \gamma) \sigma^y_1 \sigma^y_2 - B \sigma^z_1 \otimes I_2 - B I_1 \otimes \sigma^z_2}[/math]

Equation 5.   The Hamiltonian form for a two-electron state

where the subscripts, [math]\small{1}[/math] and [math]\small{2}[/math], are the two electrons, respectively; [math]\normalsize{\sigma^{x}}[/math], [math]\normalsize{\sigma^{y}}[/math] and [math]\normalsize{\sigma^{z}}[/math] are Pauli Matrices, and [math]\normalsize{\gamma}[/math] the Degree of Anisotropy ([math]\small{I}[/math]). This model provides a simplified framework for examining entanglement in a quantum system.

[math]\LARGE{S = \frac{1}{2} \log_2\left( \frac{1}{4} + \frac{1}{4 + {\lambda^2}} + \right) + \frac{1}{\sqrt{4 + \lambda^{2}}} \log_2 \frac{( \sqrt{4 + \lambda^{2}} - 2} {\sqrt{4 + \lambda^{2}} + 2}}[/math]

[math]\normalsize{S = \frac{1}{2} \log_2\left( \frac{1}{4 + {\lambda^2}} + \frac{1}{4} + k^2 \right) + \frac{1}{\sqrt{4 + \lambda^{2}}} \log_2 \frac{( \sqrt{4 + \lambda^{2}} - 2} {\sqrt{4 + \lambda^{2}} + 2}}[/math]

Equation 6.   The Quantum Entanglement form expressed via Von Neumann Entropy equation

[math]\large{\lambda}[/math] represents the Eigenvalue for the Hamiltonian of the two spin system (further information in here, Section 3).

Figure 1.   A Pictorial Representation of a Wormhole Interior

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

Figure 2.   Illustration of the Hologram for a Sphere

A Cross-cutting Solution with no Traces of Complexity...

The results demonstrate that quantum entanglement can serve as an effective measure of electron correlation in quantum chemistry. The use of von Neumann entropy allows for a more observable and intuitive understanding of electron correlation, without relying on traditional methods that require complex wave function calculations.

This approach has implications for larger atomic and molecular systems and can be extended to other quantum systems, offering a robust alrternative for evaluating electronic structures in atoms and molecules. Future work will explore the application of this method to more complex systems and its potential for advancing quantum chemistry calculations.

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  2. Inspire HEP. "Factorization and Non-Factorization of In-Medium Four-Quark Condensates" https://inspirehep.net/literature/676262

  3. arXiv.org. "Editorial: New frontiers in holographic duality" https://arxiv.org/abs/2210.03315

  4. Big Think. "Are we living in a baby universe that looks like a black hole to outsiders?" https://bigthink.com/hard-science/baby-universes-black-holes-dark-matter/

  5. Wiley Online Library. "Energy-Efficient Memristive Euclidean Distance Engine for Brain-Inspired Competitive Learning" https://onlinelibrary.wiley.com/doi/full/10.1002/aisy.202100114

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  7. ResearchForLife7 (revisited from arXiv). "Discussion on Consistent Truncations: Uplifting the GPPZ Solutions" https://httpsresearchforlife7.com/wp-content/uploads/2024/05/Discussion_on_Consistent_Truncations__Uplifting_the_GPPZ_Solutions.pdf

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How Can Teleportation involve Time Travels?

Albert Einstein's Theory of General Relativity permits the existence of Closed-Timelike Curves (CTCs) [1], which are paths within Space - Time that, if traversed, would enable a traveler to interact with their own Past self, whether that traveler be human or elemental particle. Kurt Gödel was among the first to highlight the possibility of CTCs, and subsequent research has proposed various Space - Time configurations accommodating these curves.

However, such scenarios of Time Travel inevitably introduce paradoxes, such as the infamous Grandfather Paradox [2], wherein the time traveler inadvertently alters the past in a way that prevents their own existence. This concept troubled even Einstein, who was close friends with Gödel. The reconciliation of CTCs with quantum mechanics poses a formidable challenge, tackled through various approaches, including Path-Integral Techniques [3].

The Removed “Memories” Approach...

Any theory aiming to unify quantum mechanics and gravity must address the complexities inherent in closed timelike curves, which introduce nonlinearities that challenge the Linearity [4] of conventional quantum mechanics. Deutsch proposed a resolution in his influential work, suggesting a Self-Consistency Condition (Equation 1) concerning the states within CTCs. This condition demands equivalence between measurements at the CTC's entrance and exit. However, this formulation necessitates the assumption of Factorization, implying invalidation of Future [5] "memories". However, Deutsch's theory has faced criticism for apparent inconsistencies.

[math]\LARGE{\rho_{CTC} = Tr_{A}[U(\rho_{A} \otimes \rho_{CTC})U^{\dagger}}][/math]

[math]\large{\rho_{CTC} = Tr_{A}[U(\rho_{A} \otimes \rho_{CTC})U^{\dagger}}][/math]

Equation 1.   Deutsch Self-Consistency Condition form

where [math]\small{\rho_{CTC}}[/math] is the density matrix (further information in here, Section 1) of the system state, [math]\small{A}[/math] inside the CTC; [math]\small{Tr_{A}}[/math] is the Trace of [math]\small{A}[/math]; [math]\small{U}[/math] is the Unitary Matrix; [math]\small{\rho_{A}}[/math] is the density matrix of [math]\small{A}[/math] and [math]\small{U^{\dagger}}[/math] is the transpose unitary matrix.

A New View: Overcoming the CTCs

In contrast, while acknowledging the strangeness of time travel quantum mechanics, P-CTCs, based on the Novikov Principle, appear to offer a less problematic framework. The concept of Probabilistic Closed Timelike Curves (P-CTCs) [6] was initially conceived to tackle the enigma posed by the integration of quantum mechanics into the framework of General Relativity, particularly concerning closed-timelike curves. However, its implications extend beyond this specific domain, offering insights into the potentiality of time travel in alternative scenarios.

N   [math]\LARGE{[\rho]\propto {Tr_{E}[U_{AE}] = C_{A}\rho C_{A}^{\dagger}}}[/math]

N  [math]\large{[\rho]\propto{Tr_{E}[U_{AE}] = C_{A}\rho C_{A}^{\dagger}}}[/math]

Equation 2.   P-CTC External System Time Evolution Equation

N [math]\small{[\rho]}[/math] is the Time Evolution of External System; [math]\small{\rho}[/math] is the density matrix of external system; [math]\small{Tr_{E}}[/math] is the trace of the Hilbert space (further information in here, Section 2), [math]\small{E}[/math] over the system into the CTC; [math]\small{C_{A}}[/math] is the partial trace of [math]\small{A}[/math]; [math]\small{U_{AE}}[/math] is the unitary matrix coupling the internal and external systems; [math]\small{{C_{A}}^{\dagger}}[/math] is the transpose partial trace of [math]\small{A}[/math].

Fundamentally, any quantum theory that permits non-linear processes like the Projection onto specific states, such as the Entangled States (further information in here, references) associated with P-CTCs, inherently allows for the prospect of time travel, even in the absence of Space - Time configurations supporting closed-timelike curves. The P-CTS mechanism translates mathematically to the time evolution of the external system being , with the absence of evolution enforced if certain conditions (Equation 2) are met. The paradigm of non-general relativistic P-CTCs can be instantiated through the generation and projection onto entangled pairs of particle-antiparticle. This method mirrors renowned Wheeler's Thought Experiment [7] of a telephone call through time.


Figure 1.   A typical Macroscopic Example of Quantum Tunneling: a Ball (Subatomic Particle) which overcomes a Wall (Potential Energy Barrier)

Although the process of projection is inherently nonlinear, defying deterministic implementation within conventional quantum mechanics, it can be executed in a probabilistic manner. Consequently, experimental validation of P-CTCs is achievable through Quantum Teleportation experiments, where outcomes corresponding to the desired entangled-state output are selectively post-processed. Should it transpire that the linearity of quantum mechanics is merely an approximation, and projection onto specific states indeed manifests, such occurrences could potentially be witnessed at the singularities (further information in here) of black holes.

In such a scenario, even in the absence of general relativistic closed-timelike curves, the realization of time travel might still be feasible. The theoretical framework of P-CTCs elucidates that quantum time travel can be conceived as a form of retrograde Quantum Tunneling (Figure 1), permitting temporal traversal devoid of a classical trajectory from future to past. P-CTCs rely on Destructive Interference (Figure 2) to prevent self-contradictory events, emphasizing a different self-consistency condition from Deutsch's approach.

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

Two Different Perspectives

Illustrating the link between P-CTCs and teleportation provides further insights, showcasing their behavior through Qubits. This demonstration underscores the compatibility of P-CTCs with Higher-Dimensional Systems and the extension to infinite-dimensional scenarios. Presently, no definitive conclusion favors either approach (Deutsch or P-CTCs), given their respective foundations and consistency with different theoretical frameworks. The aspiration in elaborating on the theory of P-CTCs is that it may furnish valuable insights for formulating a Quantum Theory of Gravity. By shedding light on one of the most enigmatic ramifications of general relativity—the prospect of time travel—this theory may contribute significantly to our understanding of Gravity at the quantum level.


Figure 2.   A classic Experiment of Waves Interference: Ripples in the Water

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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.

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