Quantum Entanglement

Summary

Quantum entanglement occurs when two systems interact such that the joint state cannot be written as a product of individual states. Entangled systems exhibit non-classical correlations; Bell’s theorem proves these correlations cannot be explained by any local hidden-variable theory. Entanglement is the key resource in quantum computing and quantum communication.

Overview

When two quantum systems combine, the state of the composite system lives in the tensor product of their individual Hilbert spaces. Not all composite states are separable (product states) — the superposition principle allows entangled states where measuring one subsystem instantly affects the statistics of the other, regardless of distance. This seemingly paradoxical feature (which Einstein called “spooky action at a distance”) is now experimentally confirmed.

Composite Systems

Tensor Product Hilbert Space

For two quantum systems $A$ and $B$ with Hilbert spaces ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$, the combined system has Hilbert space:

$${\mathcal{H}}_{AB}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$$

If system $A$ is in state ${\psi }_A$ and system $B$ in state ${\psi }_B$, the combined state is the product state ${\psi }_A\otimes {\psi }_B$.

Entangled States

Entanglement

A state $\Psi \in {\mathcal{H}}_{AB}$ is entangled if it cannot be written as a product state ${\psi }_A\otimes {\psi }_B$. For example, if ${\psi }_A,{\varphi }_A\in {\mathcal{H}}_A$ and ${\psi }_B,{\varphi }_B\in {\mathcal{H}}_B$ are valid states, then:

$$\Psi =\frac{1}{\sqrt{2}}\left({\psi }_A\otimes {\psi }_B+{\varphi }_A\otimes {\varphi }_B\right)$$

is a valid joint state that is not separable — it is entangled.

For an entangled state, it is impossible to describe either subsystem $A$ or $B$ by a state vector alone. One must use a reduced density matrix ${\rho }_A={\text{tr}}_B(\Psi {\Psi }^{†})$, which is obtained by tracing over the other system. Knowing ${\rho }_A$ and ${\rho }_B$ individually does not reconstruct $\Psi$.

Bell’s Theorem

Bell's Theorem

If nature operates according to any theory of local hidden variables, then the predictions of that theory are constrained in a quantifiable way (Bell inequalities). Quantum mechanics predicts — and experiments confirm — violations of these inequalities. Therefore, quantum correlations cannot be explained by any local hidden-variable theory.

Bell tests (experiments testing Bell inequalities) have been performed many times; results are consistently incompatible with local hidden variables. This means that the non-classical correlations of entanglement are a fundamental feature of nature, not an artifact of incomplete knowledge.

Key implication: Entanglement does NOT allow faster-than-light communication (proven by the no-communication theorem) — but it does allow correlations that cannot be explained classically.

Quantum Decoherence

When a quantum system interacts with its environment, it becomes entangled with that environment — a process called quantum decoherence. The system’s quantum superpositions effectively disappear (from the perspective of measurements on the system alone), explaining why macroscopic objects do not exhibit quantum behavior.

Decoherence

A quantum system interacting with an environment $E$ evolves from:

$${\psi }_{\text{system}}\otimes {\psi }_E\to \sum _n{c}_n{\varphi }_n^{\text{system}}\otimes {\varphi }_n^E$$

After tracing out the environment, the reduced density matrix of the system becomes approximately diagonal in the “pointer basis”, suppressing quantum interference terms.

Applications

Application	Role of Entanglement
Quantum computing	Entangled qubits provide exponential parallelism for certain problems
Quantum key distribution	Entangled pairs enable cryptographic security based on physics
Superdense coding	1 entangled pair + 1 qubit transmits 2 classical bits
Quantum teleportation	Transfer quantum state using classical bits + shared entanglement

Connections

• Wave Function and Hilbert Space: Entanglement is a direct consequence of the tensor-product structure of composite Hilbert spaces and the superposition principle.
• Schrödinger Equation and Time Evolution: The joint state of entangled particles evolves unitarily under the Schrödinger equation until a measurement disturbs the system.
• QFT Overview: Quantum field theory treats entanglement at the level of field modes; vacuum entanglement plays a role in Hawking radiation and the Unruh effect.

See Also

• Wave Function and Hilbert Space — the state formalism underlying entanglement
• Schrödinger Equation and Time Evolution — unitary time evolution of composite quantum systems
• Uncertainty Principle — another non-classical consequence of the commutation structure of quantum observables
• QFT Overview — entanglement in the field-theory context