Table of Contents
- Introduction
- What is Quantum Entanglement?
- Historical Context and EPR Paradox
- Entanglement vs Classical Correlations
- Mathematical Representation of Entanglement
- Bell States and Maximally Entangled States
- Schmidt Decomposition
- Criteria for Entanglement
- Local Operations and Classical Communication (LOCC)
- Entanglement as a Resource Theory
- Entanglement Measures
- Entanglement Entropy
- Concurrence and Negativity
- Monogamy of Entanglement
- Entanglement Distillation
- Entanglement Swapping
- Entanglement in Quantum Teleportation
- Entanglement in Superdense Coding
- Entanglement in Quantum Cryptography
- Entanglement in Quantum Algorithms
- Entanglement and Quantum Error Correction
- Entanglement in Many-Body Physics
- Experimental Realization of Entangled States
- Limitations and Decoherence
- Conclusion
1. Introduction
Entanglement is one of the most fundamental and non-classical features of quantum mechanics. More than just a strange phenomenon, it is now recognized as a key resource for quantum computing, quantum communication, and quantum information processing.
2. What is Quantum Entanglement?
Entanglement occurs when the quantum state of two or more particles cannot be described independently of each other, even when separated by large distances. Measurement of one instantly affects the state of the other — a phenomenon that Einstein famously called “spooky action at a distance.”
3. Historical Context and EPR Paradox
In 1935, Einstein, Podolsky, and Rosen (EPR) challenged the completeness of quantum mechanics, suggesting that entanglement implies hidden variables. Bell’s theorem later disproved local hidden variable theories through experimental violations of Bell inequalities.
4. Entanglement vs Classical Correlations
Classically correlated systems follow local realism and obey:
\[
P(a, b) = \sum_\lambda P(a|\lambda)P(b|\lambda)P(\lambda)
\]
Entangled systems violate this factorization.
5. Mathematical Representation of Entanglement
A bipartite pure state \( |\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B \) is entangled if it cannot be written as:
\[
|\psi\rangle \neq |\psi_A\rangle \otimes |\psi_B\rangle
\]
6. Bell States and Maximally Entangled States
Four maximally entangled two-qubit Bell states:
\[
|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle), \quad
|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)
\]
These are used extensively in quantum communication.
7. Schmidt Decomposition
Any pure bipartite state can be written as:
\[
|\psi\rangle = \sum_{i} \lambda_i |u_i\rangle_A \otimes |v_i\rangle_B
\]
If more than one \( \lambda_i \) is non-zero, the state is entangled.
8. Criteria for Entanglement
- Peres-Horodecki criterion: Check positivity under partial transpose
- Entropy tests: Non-zero entropy of reduced state indicates entanglement
9. Local Operations and Classical Communication (LOCC)
Entanglement cannot be increased via LOCC — operations that only involve local gates and classical communication. This constraint gives rise to entanglement monotones.
10. Entanglement as a Resource Theory
Just like energy or information, entanglement can be treated as a resource:
- It cannot be created freely under LOCC
- Can be consumed to perform tasks like teleportation or dense coding
11. Entanglement Measures
Quantify how “entangled” a state is:
- Entanglement Entropy
- Concurrence
- Negativity
- Logarithmic Negativity
- Entanglement of Formation
12. Entanglement Entropy
For a pure state \( |\psi\rangle \), the entropy of subsystem \( A \) is:
\[
S(\rho_A) = -\text{Tr}(\rho_A \log \rho_A)
\]
A non-zero value implies entanglement.
13. Concurrence and Negativity
- Concurrence (for two qubits):
\[
C(\rho) = \max(0, \lambda_1 – \lambda_2 – \lambda_3 – \lambda_4)
\]
- Negativity:
\[
N(\rho) = \frac{|\rho^{T_B}|_1 – 1}{2}
\]
14. Monogamy of Entanglement
If two qubits are maximally entangled, they cannot be entangled with a third. This property ensures security in quantum cryptography.
15. Entanglement Distillation
Process of extracting high-quality entangled pairs from noisy entangled states using LOCC and error correction.
16. Entanglement Swapping
Creating entanglement between particles that never interacted by using intermediate entanglement and Bell measurement.
17. Entanglement in Quantum Teleportation
Teleportation uses entanglement and classical communication to transmit quantum states:
\[
|\psi\rangle \rightarrow |\psi\rangle_{\text{remote}}
\]
18. Entanglement in Superdense Coding
One entangled qubit allows transmission of two classical bits of information.
\[
\text{1 entangled qubit + 1 qubit} \Rightarrow 2 \text{ bits}
\]
19. Entanglement in Quantum Cryptography
Used in:
- Quantum Key Distribution (QKD)
- Device-independent protocols
- Ekert protocol (E91)
20. Entanglement in Quantum Algorithms
While not always explicitly required, entanglement is often a hidden enabler in algorithms like:
- Shor’s factoring
- Grover’s search
- Quantum simulation
21. Entanglement and Quantum Error Correction
Quantum error-correcting codes rely on multi-partite entanglement to encode and protect logical information.
22. Entanglement in Many-Body Physics
- Key to understanding phase transitions
- Area-law scaling of entanglement entropy
- Basis of tensor network methods (e.g., MPS, PEPS)
23. Experimental Realization of Entangled States
Technologies:
- Trapped ions
- Superconducting circuits
- Photons via spontaneous parametric down-conversion
- NV centers in diamond
24. Limitations and Decoherence
Entangled states are fragile:
- Susceptible to decoherence
- Require error correction or robust design
- Entanglement sudden death: complete loss due to noise
25. Conclusion
Quantum entanglement is not just a theoretical curiosity — it’s a powerful resource. From teleportation and quantum cryptography to algorithms and error correction, entanglement is central to quantum information science. Mastering its manipulation and preservation is key to unlocking the full potential of quantum technologies.
