Put two qubits together and the state space jumps to four dimensions. Most of those states can't be split back into "this qubit and that qubit" — and that inseparability is entanglement, the engine of everything left in the course.
↩ before you start · keep these handy
·From Ch. 2: one qubit is α|0⟩ + β|1⟩, written as the column [α, β], with |α|²+|β|² = 1.
·From Ch. 3: a measurement collapses the state to one outcome, with probability = amplitude squared.
·From 0.1: a list of numbers stacked in a column is a vector — that's all a state really is.
🔑 symbol decoder · every new mark, in plain words
⊗the tensor product — the recipe for gluing two separate qubits into one joint state.|00⟩both qubits are 0 at once — shorthand for |0⟩⊗|0⟩. Likewise |01⟩, |10⟩, |11⟩ name the four ways two bits can land.|Φ⁺⟩"phi-plus" — the workhorse Bell state (|00⟩+|11⟩)/√2.separablea joint state that splits back into qubit-A × qubit-B. The qubits are independent.entangleda joint state that does not split, no matter how you try — the whole point of this chapter.1/√2≈ 0.707 — the normalizer that keeps the squared amplitudes adding to 1.
§1
Correlated, or just random?
In the Bell state, each qubit alone looks like a fair coin — but the pair always agrees. Flip to independent and the same marginal coins now land in all four corners. Run the grid and compare.
🧤 everyday picture
Split a pair of gloves into two sealed boxes and ship one to Alice, one to Bob across the galaxy. The moment Alice opens hers and sees left, she knows Bob's is right — every single time. The certainty lived in the pairing, not in either box. Entanglement is exactly this kind of locked-in agreement — except the boxes weren't even "left" or "right" until one was opened.
▸ runmeasure(q0,q1)
{{ c.lbl }}
{{ c.count }} · {{ c.pct }}%
{{ bellFormula }}
{{ bellTotal }} paired shots
recapA Bell pair's halves are each a fair coin alone, yet the pair always agrees — the correlation is the whole story.
§2 · the mathematics
Combining states: the tensor product
Two qubits live in a 4-dimensional space with basis |00⟩, |01⟩, |10⟩, |11⟩. You build a combined state from two singles with the tensor product ⊗ — multiply every entry of one vector by the whole other vector:
ab
⊗
cd
=
acadbcbd
Any state you can write this way is separable — the two qubits are independent. The catch: not every 4-vector factors.
4.= (|00⟩ + |10⟩)/√2 — built from parts, so it's separable.
recapThe tensor product ⊗ glues two qubits into one 4-slot state; states built this way are separable.
§3 · the mathematics
A state that won't factor
The Bell state has amplitude 1/√2 on |00⟩ and |11⟩, zero elsewhere:
|Φ⁺⟩ = (|00⟩ + |11⟩)/√2 =
1/√2001/√2
Try to write it as (a,b) ⊗ (c,d): you'd need ac = bd = 1/√2 but ad = bc = 0 — impossible. No description of qubit 0 alone and qubit 1 alone can reproduce it. That's entanglement: the information lives in the correlations, not the parts. Measuring one instantly fixes the other, which is exactly what Chapter 7 puts to work.
2.match slots: ac = 1/√2, bd = 1/√2, ad = 0, bc = 0
3.ad = 0 ⇒ a = 0 or d = 0 — but a = 0 kills ac, and d = 0 kills bd
4.contradiction — no (a,b),(c,d) exist. The state is entangled.
recap|Φ⁺⟩ cannot be split into A×B — its information sits entirely in the correlation.
§4
Each qubit alone is pure noise
Here is the strangest part. Look at just one half of a Bell pair and you see a perfect coin — 50/50, no pattern whatsoever. Nothing about qubit 0 on its own betrays that it is entangled. The structure is entirely in the agreement between the two; it does not live in either one.
Measure Alice's qubit across many pairs. Her tally stays balanced while the pair agreement stays pinned at 100%.
▸ runmeasure(q0) only
Alice's qubit, measured alone
landed 0{{ maP0 }}%
landed 1{{ maP1 }}%
{{ maTotal }} pairs measured
pairs that agreed
{{ agreePct }}%
Bob always matched Alice
Mathematically, qubit 0 on its own is a maximally mixed state — the entanglement has hidden every trace of itself inside the joint description. You cannot find it locally; you can only find it by comparing.
recapEach half of a Bell pair is maximally mixed alone — you can only find the entanglement by comparing the two.
§5
The four Bell states
There are exactly four maximally-entangled two-qubit states, and they split into two correlation classes: always agree and always disagree. Within each class a hidden minus sign — a relative phase — tells the pair apart. Tap one to see its allowed outcomes.
{{ g.lbl }}
{{ bellRuleTag }}
{{ bellRuleText }}
recapThere are exactly four Bell states — two that always agree, two that always disagree — set apart by a hidden phase.
⚠ common misconception
“Entanglement sends signals faster than light.” It doesn't. As §4 shows, Alice's outcomes are pure noise no matter what Bob does — she can't steer them, so she can't encode a message. The correlation only appears once the two lists are compared over an ordinary (light-speed) channel. That's exactly why teleportation, next chapter, still needs a phone call.
✓ you can now
✓glue two qubits into a 4-slot joint state with the tensor product ⊗
✓test whether a two-qubit state is separable or entangled by trying to factor it
✓explain why each half of a Bell pair looks like noise, and why entanglement sends no signal