Entanglement & Bell States
The correlation with no classical explanation
By the end of this topic you'll be able to
When the state of a composite system AB cannot be written as the product of a state of A and a state of B, Binney and Skinner say that A and B are entangled. The Bell state |Φ⁺⟩ = (|00⟩+|11⟩)/√2 is the canonical instance: no choice of α, β, γ, δ makes it equal to (α|0⟩+β|1⟩)⊗(γ|0⟩+δ|1⟩).
As they put it, the use of the word 'entanglement' is apt because subsystems are as prone to becoming entangled as the line of a kite — almost any coupling between two systems, evolved for any length of time, will entangle them starting from an unentangled initial condition.
In 1935 Einstein, Podolsky and Rosen proposed a thought experiment with exactly this kind of state, arguing that quantum mechanics must be an incomplete theory: to fully specify a system, they claimed, you would need to know the values of hidden variables that quantum mechanics simply omits.
Binney and Skinner work through the standard version of the experiment: a spinless nucleus decays into an electron and a positron moving in opposite directions, so their spins are equal and opposite by conservation of angular momentum. Alice measures the electron's spin along a direction of her choosing, a, and Bob measures the positron's spin along a direction of his choosing, b.
Alice
Bob
At 0° (same axis), Alice and Bob always get opposite results — measure enough pairs and it's 100% every time, no exceptions. Away from 0°, the anti-correlation weakens: the probability their results differ is , so at 0° that's 0%.
If Alice measures +½ along a, conservation forces her to conclude the positron is −½ along that same direction — and indeed, immediately after her measurement, the amplitude for Bob to find +½ along a has dropped to zero. What EPR found troubling is that this holds for whichever direction a Alice happens to choose, as if the positron somehow knows what she measured.
In 1964 J. S. Bell showed that this worry can be made into an experimental question. Any theory in which a hidden variable fixes each particle's measurement outcome in advance must satisfy a specific inequality relating the correlations Alice and Bob observe for different choices of measurement direction.
Quantum mechanics predicts correlations that violate this Bell inequality, and experiments — beginning with Freedman and Clauser in 1972, and repeated with ever fewer loopholes since — have confirmed the quantum-mechanical prediction. Hidden-variable theories of this kind are therefore ruled out by experiment, not merely by argument.
None of this permits sending a message faster than light: the outcome Alice observes is random and entirely outside her control, so there is no message encoded in it for Bob to decode — only a correlation, revealed after the fact by comparing notes over an ordinary classical channel.
The four Bell states {|Φ⁺⟩,|Φ⁻⟩,|Ψ⁺⟩,|Ψ⁻⟩} form a basis for the 2-qubit space every bit as valid as the computational basis {|00⟩,|01⟩,|10⟩,|11⟩}, related to it by a Hadamard and a CNOT. This 'Bell basis' is the engine behind quantum teleportation and superdense coding, both coming up soon.
Try It Yourself
Alice and Bob share the Bell state |Φ⁺⟩ = (|00⟩+|11⟩)/√2 and both measure in the computational basis. If Alice measures 0, what does that tell you about Bob's outcome, and with what probability?
- 1Before either measurement, the joint state is (1/√2)|00⟩ + (1/√2)|11⟩ — only two of the four possible 2-bit outcomes (00 and 11) have nonzero amplitude at all.
- 2By the measurement postulate, Alice's measurement collapses the joint state onto whichever term is consistent with her outcome. If she measures 0, the only term in the superposition with Alice's qubit equal to 0 is |00⟩ — so the joint state collapses entirely to |00⟩.
- 3That means Bob's qubit is now definitely |0⟩: if he measures next, he gets 0 with probability 1, not 1/2.
- 4Before collapse, Bob's own qubit looked maximally random on its own (P(0)=P(1)=1/2, exactly the mixed state ρ_B = I/2) — Alice's measurement didn't change Bob's qubit physically, but it did instantly update what outcome is now certain for him.
If Alice measures 0, Bob is guaranteed to measure 0 too (probability 1) — the two outcomes are perfectly correlated, even though each one alone, before comparing notes, looked like a fair coin flip.
Reference
| Φ⁺ | Canonical Bell state | |
| Φ⁻ | Bell state, opposite relative phase | |
| Ψ⁺ | Bell state, opposite-outcome correlation | |
| Ψ⁻ | The 'singlet' state, rotation-invariant |
Quick Check
Can entanglement be used to send a message faster than light?
What does Bell's inequality (and its experimental violation) rule out?