Quantum Entanglement

What Is Entanglement?

Quantum entanglement is a correlation between qubits that has no classical analogue. When two qubits are entangled, the state of the composite system cannot be written as a tensor product of individual qubit states:

Entanglement is a fundamental resource for quantum computing, quantum cryptography, and quantum teleportation.

Bell States

The four Bell states form a maximally entangled basis for two qubits:

These states are maximally entangled — measuring one qubit immediately determines the state of the other, regardless of spatial separation.

The EPR Paradox

In 1935, Einstein, Podolsky, and Rosen argued that quantum mechanics must be incomplete because it allows "spooky action at a distance." Their argument:

1. If we can predict with certainty the value of a physical quantity without disturbing the system, then there exists an element of reality corresponding to that quantity.
2. For the Bell state , measuring qubit A in the Z-basis tells us qubit B's state with certainty.
3. But we could equally well measure in the X-basis.
4. Therefore, both and must be elements of reality — but they cannot simultaneously have definite values.

EPR concluded that hidden variables must exist. However, Bell's theorem (1964) showed that no local hidden variable theory can reproduce all predictions of quantum mechanics.

Bell's Inequality

Bell derived an inequality that any local hidden variable theory must satisfy:

where is the CHSH correlation function:

Quantum mechanics predicts violations up to , known as the Tsirelson bound. Experimental violations have confirmed Bell inequality violations, ruling out local hidden variable theories.

No-Cloning Theorem

The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. Formally, there is no unitary operator such that:

Proof sketch: Assume such a exists. Then for two states and :

This implies , meaning can only clone orthogonal states. Therefore, no universal cloning machine exists.

The no-cloning theorem is the foundation of quantum key distribution — an eavesdropper cannot copy quantum states without detection.

Entanglement Measures

The most common measure of entanglement for pure bipartite states is the von Neumann entropy of the reduced density matrix:

For a Bell state, , giving ebit (entanglement bit).

For mixed states, entanglement is more subtle. The entanglement of formation and concurrence are commonly used measures.

Python: Creating Entanglement

import numpy as np

def create_bell_state():
    # Create the |Phi+> Bell state.
    KET_0 = np.array([1, 0], dtype=complex)
    KET_1 = np.array([0, 1], dtype=complex)

    # |00>
    psi = np.kron(KET_0, KET_0)

    # Apply Hadamard to first qubit
    H = (1/np.sqrt(2)) * np.array([[1,1],[1,-1]], dtype=complex)
    psi = np.kron(H, np.eye(2)) @ psi

    # Apply CNOT
    CNOT = np.array([[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]], dtype=complex)
    psi = CNOT @ psi

    return psi

bell = create_bell_state()
print(f"Bell state: {bell}")
print(f"|00> coeff: {bell[0]:.4f}")
print(f"|11> coeff: {bell[3]:.4f}")

The output confirms equal amplitudes for and , verifying successful entanglement creation.

Quantum Teleportation Protocol

The teleportation protocol transfers an unknown state using one ebit and two classical bits:

1. Alice and Bob share
2. Alice applies CNOT to her qubits, then Hadamard to the unknown state
3. Alice measures both qubits, sending 2 classical bits to Bob
4. Bob applies to recover

The protocol uses the identity:

Entanglement as a Resource

In the resource-theoretic framework:

• Free operations: LOCC (Local Operations and Classical Communication)
• Free states: separable states
• Resource: entanglement (measured in ebits)

Applications enabled by entanglement:

• Quantum teleportation (1 ebit + 2 cbits)
• Superdense coding (1 ebit + 1 qubit = 2 cbits)
• Quantum key distribution
• Quantum dense coding
• Entanglement-assisted communication