**Title:** *Temporal Entanglement: A New Frontier in Quantum Systems*

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**Supervisor:** Jay Wisdom

**Abstract:**

Quantum entanglement has traditionally been studied as a spatial phenomenon, where correlations between particles are examined at specific points in space. However, the role of time in entanglement is often treated as secondary or merely a parameter in dynamic evolution. This thesis proposes a novel approach to quantum entanglement by focusing on **temporal entanglement**—the correlations between quantum states across different moments in time. By shifting the focus from spatial to temporal entanglement, we aim to uncover previously hidden aspects of quantum behavior, challenge the universality of existing entanglement measures, and propose new methods to quantify these temporal correlations.

**Research Objectives:**

1. **Defining Temporal Entanglement:**

- Develop a rigorous mathematical framework to define and quantify temporal entanglement in quantum systems.

- Investigate the conditions under which temporal entanglement manifests most prominently, distinguishing it from spatial entanglement.

2. **Challenging Spatial Entanglement Assumptions:**

- Critically analyze the assumptions underlying current entanglement measures, particularly their focus on spatial correlations.

- Explore how temporal entanglement could provide alternative explanations for phenomena traditionally attributed to spatial entanglement.

3. **Entanglement in Excited States:**

- Extend the concept of temporal entanglement to excited states of quantum field theory, contrasting it with spatial entanglement in these states.

- Examine whether temporal entanglement exhibits universal features across different quantum systems, or if it reveals the non-universality of excited states.

4. **New Measures of Entanglement:**

- Propose and develop new measures that can accurately capture both temporal and spatial entanglement.

- Validate these measures against known quantum systems, both in ground and excited states, to assess their efficacy.

5. **Implications for Quantum Computing and Information:**

- Explore the potential applications of temporal entanglement in quantum computing, particularly in the context of time-based quantum algorithms and protocols.

- Investigate how temporal entanglement could enhance our understanding of quantum information theory, potentially leading to new paradigms in quantum communication and encryption.

**Methodology:**

- **Mathematical Modeling:** Develop and apply advanced mathematical techniques to define and quantify temporal entanglement. This will involve extending existing models and possibly creating new ones to capture the dynamics of temporal correlations.

- **Numerical Simulations:** Utilize computational methods to simulate quantum systems and analyze temporal entanglement under various conditions. These simulations will help validate theoretical models and provide insights into the behavior of temporal entanglement in both ground and excited states.

- **Comparative Analysis:** Perform a detailed comparison between temporal and spatial entanglement in different quantum systems. This will include examining known phenomena through the lens of temporal entanglement to see if new insights can be gained.

**Expected Outcomes:**

1. A comprehensive framework for understanding and quantifying temporal entanglement.

2. A re-evaluation of the universality of current entanglement measures, with potential challenges to existing paradigms.

3. New insights into the nature of excited states in quantum field theory, informed by the study of temporal entanglement.

4. Novel applications of temporal entanglement in quantum computing and information theory.

**Conclusion:**

This thesis aims to pioneer a new approach to understanding quantum entanglement by shifting the focus from spatial correlations to temporal ones. By exploring this uncharted territory, we hope to uncover new aspects of quantum behavior, challenge existing assumptions, and open up new avenues for research and application in quantum theory.

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This thesis proposal sets the stage for innovative research that could potentially reshape our understanding of quantum entanglement and its applications in modern physics.

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