Classical vs Quantum Correlations

Overview

We have seen correlated outcomes in Bell states, and we have discussed mixtures and density matrices. This page ties those ideas together.

Goal: clearly separate three concepts:

• Classical correlation (explained by shared randomness),
• Quantum correlation from entanglement (not explained by shared randomness),
• and the role of measurement choice in revealing the difference.

We keep this conceptual: no Bell-inequality math yet.

Intuition

Correlation means “the outcomes of two measurements are related.” Correlation alone is not mysterious; classical variables can be correlated easily.

A standard classical story is shared randomness:

• A hidden variable $\lambda$ is sampled (like a shared coin flip).
• Each side’s outcome is a function of that $\lambda$ and the local measurement setting.

This can produce strong correlations, even perfect ones.

Entanglement is different because the joint state can be pure and fixed, yet each subsystem locally looks mixed. The correlations are not coming from “each subsystem carrying its own hidden instruction list” in a way that works for all possible measurement choices.

The important idea here is not faster-than-light influence. It is incompatibility of explanations: there are quantum correlations that cannot be reproduced by any shared-randomness model that assigns outcomes based only on local settings.

Formal Description

Classical correlated mixture (separable mixed state)

Consider the mixed state that says:

• with probability $1/2$, the pair is in $|00\rangle$,
• with probability $1/2$, the pair is in $|11\rangle$.

As a density matrix:

$$\rho_{\text{class}} = \tfrac{1}{2}|00\rangle\langle 00| + \tfrac{1}{2}|11\rangle\langle 11|.$$

This state has perfect correlation in the computational basis: outcomes always match.

But it is not entangled. It is a mixture of product states, which means it can be explained by shared randomness: the hidden variable is “whether we prepared 00 or 11.”

Entangled pure state (Bell state)

Now compare the pure entangled state:

$$|\Phi^+\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle+|11\rangle),$$

with density matrix

$$\rho_{\text{ent}} = |\Phi^+\rangle\langle\Phi^+|.$$

In the computational basis, this also produces perfect matching outcomes.

So if you only ever measure in the computational basis, you cannot tell whether the correlations came from $\rho_{\text{class}}$ or $\rho_{\text{ent}}$.

The difference appears when you change what you measure.

What changes when you change basis

Measurement choice matters because classical mixtures and entangled states respond differently to basis changes.

• In a classical correlated mixture, the correlation story is tied to a specific preparation of definite values.
• In an entangled pure state, the correlation is built into amplitudes and phases, and it can persist (in the appropriate way) across multiple measurement bases.

A full quantitative separation is made using Bell inequalities, which we will study later. For now, the core message is: entanglement is not defined by correlation in one basis; it is defined by the impossibility of factorizing the joint description into independent local descriptions.

Worked Example

Compare $\rho_{\text{class}}$ and $\rho_{\text{ent}}$ using a simple measurement idea.

Let both parties measure either:

• in the computational basis $\{|0\rangle,|1\rangle\}$, or
• in the $\{|+\rangle,|-\rangle\}$ basis, where

$$|+\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\quad |-\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle).$$

For the entangled state $|\Phi^+\rangle$, one can show (by rewriting in the $\{|+\rangle,|-\rangle\}$ basis) that:

$$|\Phi^+\rangle = \tfrac{1}{\sqrt{2}}(|++\rangle + |--\rangle).$$

So if both measure in the $\{|+\rangle,|-\rangle\}$ basis, outcomes still match perfectly.

For the classical mixture $\rho_{\text{class}}$, the preparation is either $|00\rangle$ or $|11\rangle$. In either case, measuring each qubit in the $\{|+\rangle,|-\rangle\}$ basis produces independent random outcomes with $P(+)=P(-)=1/2$ for each qubit.

Averaging over the mixture, the outcomes are not perfectly matched in that basis.

This illustrates the key conceptual separation:

• same perfect correlation in one basis,
• different behavior under a basis change.

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What’s Next

We now have the full Foundations vocabulary: pure states, mixed states, density matrices, measurement rules, composition, and entanglement. Next is the final page: a concise summary of what you’ve learned and how to transition to gates and circuits without changing the underlying rules.