Entanglement & Measurement

Entanglement is perhaps the strangest feature of quantum mechanics, what Einstein called "spooky action at a distance." Two particles can be correlated in ways that have no classical explanation, and these correlations persist regardless of how far apart the particles travel.

Composite Systems

When we have two quantum systems, the combined state space is the tensor product of the individual spaces.

Tensor Products

If system A has states $|a_1\rangle, |a_2\rangle, \ldots$ and system B has states $|b_1\rangle, |b_2\rangle, \ldots$, the combined system has states:

$$|a_i\rangle \otimes |b_j\rangle = |a_i, b_j\rangle = |a_i b_j\rangle$$

A general state is a superposition:

$$|\Psi\rangle = \sum_{i,j} c_{ij} |a_i\rangle \otimes |b_j\rangle$$

Product States vs. Entangled States

A product state (or separable state) can be factored:

$$|\Psi\rangle = |\phi\rangle_A \otimes |\chi\rangle_B$$

An entangled state cannot be written as a product. The subsystems do not have independent states.

The Bell States

The maximally entangled states of two qubits. They form a complete basis for the two-qubit space.

Perfect Correlations

Consider $|\Psi^-\rangle$: if you measure particle A and find spin up, particle B is instantly spin down, no matter how far apart they are. The outcomes are perfectly anti-correlated.

But here's the twist: before measurement, neither particle has a definite spin. The correlation exists without predetermined values.

The EPR Paradox

In 1935, Einstein, Podolsky, and Rosen argued that quantum mechanics must be incomplete.

The Argument

EPR defined a criterion for physical reality: if we can predict a property with certainty without disturbing the system, that property is "real."

For an entangled pair:

1. Measure spin-z on particle A → we know B's spin-z with certainty
2. Alternatively, measure spin-x on A → we know B's spin-x with certainty
3. These measurements on A don't disturb B (locality assumption)
4. Therefore, B "really" has both spin-z and spin-x values
5. But quantum mechanics says B can't have definite values for both simultaneously!

EPR concluded: quantum mechanics is incomplete. There must be "hidden variables" determining the outcomes.

Bell's Theorem

In 1964, John Bell showed that the debate could be settled experimentally. Local hidden variable theories make predictions that differ from quantum mechanics.

The Setup

Two particles fly apart. Each reaches a detector that can be set to measure spin along one of several directions. The measurement outcomes are +1 or -1.

The Bell Inequality (CHSH form)

Define the correlation function $E(a,b)$ as the average product of outcomes when A uses setting $a$ and B uses setting $b$.

For any local hidden variable theory:

Quantum Mechanics Violates the Inequality

For the singlet state $|\Psi^-\rangle$ with optimal angle choices:

$$|E(a,b) - E(a,b') + E(a',b) + E(a',b')| = 2\sqrt{2} \approx 2.83$$

Quantum mechanics predicts a value exceeding 2. Experiments consistently confirm the quantum prediction.

The Measurement Problem

Quantum mechanics describes two types of evolution:

1. Unitary evolution: The Schrödinger equation evolves states smoothly and deterministically
2. Measurement: The state "collapses" to an eigenstate, probabilistically

But what exactly constitutes a "measurement"? When does collapse happen? This is the measurement problem.

Schrödinger's Cat

Schrödinger's famous thought experiment highlights the problem. A cat in a box is entangled with a radioactive atom:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}(|\text{undecayed}\rangle|\text{alive}\rangle + |\text{decayed}\rangle|\text{dead}\rangle)$$

Before we look, is the cat in a superposition of alive and dead? When does the superposition resolve?

Different interpretations give different answers to this question (see next lesson).

Decoherence

Decoherence explains why we don't see quantum superpositions in everyday life, though it doesn't fully solve the measurement problem.

The Mechanism

Real systems interact with their environment: air molecules, photons, stray electromagnetic fields. These interactions cause entanglement with the environment:

$$|\psi\rangle|E_0\rangle \to \alpha|a\rangle|E_a\rangle + \beta|b\rangle|E_b\rangle$$

When the environment states $|E_a\rangle$ and $|E_b\rangle$ become orthogonal, interference between $|a\rangle$ and $|b\rangle$ vanishes, for any practical observation of the system alone.

The Density Matrix

Tracing over the environment, the system's reduced density matrix evolves from pure to mixed:

$$\rho = |\alpha|^2|a\rangle\langle a| + |\beta|^2|b\rangle\langle b|$$

This looks like a classical probability distribution. The off-diagonal "coherence" terms (the entries that encode quantum interference between $|a\rangle$ and $|b\rangle$) have decayed away, leaving only classical probabilities on the diagonal.

Decoherence Timescales

Decoherence is extremely fast for macroscopic objects:

System	Decoherence time
Large molecule in air	~10⁻³¹ seconds
Dust grain	~10⁻³⁶ seconds
Cat	~10⁻⁴² seconds
Superconducting qubit (engineered isolation)	~10⁻⁶ seconds

This explains why quantum effects are hard to observe macroscopically. Any superposition decoheres almost instantly.

Quantum Information

Entanglement isn't just a philosophical curiosity. It's a resource for information processing.

Quantum Teleportation

Using a shared entangled pair plus classical communication, one can transmit a quantum state without physically sending the particle. The original state is destroyed in the process.

Quantum Cryptography

Entanglement enables fundamentally secure key distribution. Any eavesdropping disturbs the entangled state, revealing the intruder.

Quantum Computing

Entanglement provides computational power beyond classical computers. An $n$-qubit register can exist in a superposition of $2^n$ states, with entanglement correlating them in useful ways.