Quantum Entanglement Basics

Table of Contents

1. Introduction
2. What Is Quantum Entanglement?
3. Historical Background and EPR Paradox
4. Entangled States: Definition and Examples
5. Tensor Product and Composite Systems
6. Bell States and Maximally Entangled Qubits
7. Nonlocal Correlations and Bell’s Theorem
8. Separability and Mixed State Entanglement
9. Entanglement Measures
10. Entanglement Swapping and Monogamy
11. Decoherence and Loss of Entanglement
12. Experimental Realizations
13. Applications in Quantum Technologies
14. Interpretational Implications
15. Conclusion

---

1. Introduction

Quantum entanglement is a fundamental feature of quantum mechanics, describing the deep, non-classical correlations between particles. Two or more systems are entangled if the state of one cannot be described independently of the state of the others, even when separated by large distances. It is central to quantum computing, cryptography, teleportation, and foundational debates in physics.

---

2. What Is Quantum Entanglement?

Entanglement occurs when quantum systems interact in such a way that their individual states become inseparably linked. A measurement on one subsystem instantaneously affects the state of the other, regardless of the distance between them. This nonlocal behavior defies classical expectations.

---

3. Historical Background and EPR Paradox

In 1935, Einstein, Podolsky, and Rosen (EPR) challenged the completeness of quantum mechanics. They argued that if quantum mechanics were complete, then it would allow spooky action at a distance, violating locality. This paradox prompted decades of theoretical and experimental work, culminating in Bell’s theorem.

---

4. Entangled States: Definition and Examples

A bipartite pure state \( |\Psi\rangle_{AB} \) is entangled if it cannot be written as a product of states:

\[
|\Psi\rangle_{AB} \ne |\psi\rangle_A \otimes |\phi\rangle_B
\]

Example (Bell State):

\[
|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
\]

This state is maximally entangled—neither qubit has an independent state.

---

5. Tensor Product and Composite Systems

Composite quantum systems are described by tensor products of Hilbert spaces:

\[
\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B
\]

States in this joint space can exhibit entanglement, where individual subsystems have no well-defined state alone.

---

6. Bell States and Maximally Entangled Qubits

There are four Bell states for two qubits:

• \( |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \)
• \( |\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle – |11\rangle) \)
• \( |\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle) \)
• \( |\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle – |10\rangle) \)

These form a complete orthonormal basis for entangled two-qubit systems.

---

7. Nonlocal Correlations and Bell’s Theorem

Bell’s theorem (1964) proves that no local hidden variable theory can reproduce all the predictions of quantum mechanics. Experiments testing Bell inequalities (e.g., CHSH) confirm the presence of nonlocal correlations in entangled systems, violating classical realism.

---

8. Separability and Mixed State Entanglement

For mixed states \( \rho_{AB} \), entanglement is defined via separability:

• A state is separable if:

\[
\rho_{AB} = \sum_i p_i \, \rho_A^{(i)} \otimes \rho_B^{(i)}
\]

• Otherwise, it is entangled.

Distinguishing mixed entangled states from separable ones is computationally difficult and central to quantum information theory.

---

9. Entanglement Measures

To quantify entanglement, several measures exist:

• Entropy of Entanglement (for pure states):

\[
S(\rho_A) = -\text{Tr}(\rho_A \log \rho_A)
\]

• Concurrence and Negativity (for mixed states)
• Entanglement of Formation
• Logarithmic Negativity

These tools help analyze resources in quantum protocols.

---

10. Entanglement Swapping and Monogamy

• Entanglement Swapping: Entanglement can be created between particles that have never interacted directly, by performing joint measurements.
• Monogamy of Entanglement: A quantum system maximally entangled with one system cannot be entangled with another—unlike classical correlations.

---

11. Decoherence and Loss of Entanglement

Entanglement is fragile. Environmental interactions cause decoherence, reducing entanglement and producing mixed states. Protecting entanglement is crucial in quantum computing and communication.

---

12. Experimental Realizations

• Photon pairs via spontaneous parametric down-conversion (SPDC)
• Trapped ions and superconducting qubits
• Nitrogen-vacancy centers in diamonds
• Atom-photon entanglement

These setups confirm entanglement over kilometers and violate Bell inequalities under strict conditions.

---

13. Applications in Quantum Technologies

• Quantum teleportation: transferring states using entanglement and classical communication.
• Superdense coding: sending two classical bits using one entangled qubit.
• Quantum key distribution (QKD): e.g., Ekert protocol.
• Quantum error correction and entanglement-assisted communication.

---

14. Interpretational Implications

Entanglement challenges classical notions of locality and separability. Interpretations like:

• Many-worlds
• Relational quantum mechanics
• Objective collapse theories

seek to explain entanglement’s deep implications.

---

15. Conclusion

Quantum entanglement is one of the most profound and non-classical features of quantum mechanics. It underpins both foundational debates and practical technologies, reshaping our understanding of information, causality, and reality. A firm grasp of its basics is essential for any serious student of quantum science.

---