Quantum computing and AIMiddleLesson 3055 min read

Entanglement without the mystery

Two entangled qubits give perfectly matched results, even far apart. Learn what the Bell state is, what the correlations mean, and why no message travels faster than light.

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In 60 seconds

Entanglement without the mystery

Two entangled qubits give perfectly matched results, even far apart. Learn what the Bell state is, what the correlations mean, and why no message travels faster than light.

1/5
In simple words
Imagine two friends, one in Chennai and one in Delhi, each with a special coin. When they toss them, each coin lands heads or tails at random. But when they phone each other later, they find their coins always matched. Neither friend could choose the result, so neither could send a message this way. Entangled qubits behave a bit like this, and the real version is even stranger than any trick with pre-arranged coins.
With two qubits there are four basic outcomes: 00, 01, 10 and 11. A two-qubit state needs four amplitudes, one for each. We write it as a|00⟩ + b|01⟩ + c|10⟩ + d|11⟩, with |a|² + |b|² + |c|² + |d|² = 1.
Some two-qubit states can be split into "state of qubit A" times "state of qubit B". Those are called product states. States that cannot be split this way are entangled.

The Bell state

The most famous entangled state is this Bell state:
A Bell statetext
(|00⟩ + |11⟩) / √2

amplitudes for 00, 01, 10, 11:  1/√2, 0, 0, 1/√2
probabilities:                  0.5,  0,  0,  0.5
Measure both qubits and you get 00 half the time and 11 half the time. Never 01 or 10. Each qubit alone looks like a fair random coin. But the two results always agree.
StateP(00)P(01)P(10)P(11)Results agree
Bell state (|00⟩ + |11⟩)/√20.5000.5Always
Two separate equal superpositions0.250.250.250.25Half the time

Is it just hidden instructions?

A natural idea: maybe the two qubits secretly agreed on an answer when they were made, like two sealed envelopes with the same card. For measurements in one fixed way, that idea would work. But qubits can be measured in different ways (different "bases"). The physicist John Bell showed that if the results were fixed in advance, the correlations across different measurement choices would have to stay below a certain limit. Quantum theory predicts they can go above that limit, and careful experiments agree with quantum theory.

Why this cannot send a message

  1. 1

    Share

    Alice and Bob each take one qubit of a Bell pair and travel far apart.
  2. 2

    Alice measures

    She gets 0 or 1 at random. She cannot choose which.
  3. 3

    Bob measures

    He gets 0 or 1 at random, 50/50, whatever Alice did.
  4. 4

    Compare

    Only when they compare notes, over a normal phone or internet link, do they see the match.
Bob's own results are always 50/50, whether Alice measured or not, and whatever way she measured. So nothing Alice does changes what Bob sees on his own. The correlation is real, but it only shows up when both lists are brought together, and that needs an ordinary signal that travels no faster than light. This is known as the no-communication theorem.
Do this
Entanglement is useful. It is a resource for quantum algorithms, for quantum key distribution (a way to share secret keys where eavesdropping can be detected), and for quantum teleportation, which moves a qubit's state using entanglement plus two ordinary bits of classical communication. Nothing physical or informative travels faster than light in any of these.
Try this. The state 0.5|00⟩ + 0.5|01⟩ + 0.5|10⟩ + 0.5|11⟩ is a product state. Check that it equals (0.707|0⟩ + 0.707|1⟩) for qubit A times the same for qubit B.

Watch and read more

Lab

A simulation that samples measurements of a Bell state and a product state, and measures how often the results agree and what Bob sees alone.

~25 min

The problem

Run this in the browser, or with Python 3 and numpy. Each result is a 2-character string: the first character is Alice's bit, the second is Bob's.

  1. Finish agreement(results): the fraction of results where Alice's and Bob's bits are equal.
  2. Finish bob_zero(results): the fraction of results where Bob's bit is 0.
  3. Run it. Compare the Bell state with the product state. Write one sentence on what is different, and one sentence on why Bob's number alone cannot carry a message.

Practice lab

Edit the code and run it. Python runs in your browser — nothing is sent to our server.

Output

No output. Add print(...) to see values.

You are done when

Hard questions

Try to answer before you reveal. If you can answer these, you understood the lesson.

Q1This simulation just samples 00 or 11 from a list. Doesn't that prove entanglement is only hidden pre-set answers?Reveal

No. Our simulation only measures both qubits in one fixed way, and for that single measurement pre-set answers can copy the results. Bell's insight was to compare several different measurement settings on each side. Pre-set answers must obey a limit on the combined correlations; entangled qubits can exceed it. A full simulation of that needs rotating each qubit before measuring, which you could add with the matrices from the next lesson.

Q2Alice measures her half of many Bell pairs before Bob measures his. Can Bob tell, from his results alone, that she did?Reveal

No. Bob's results are 50/50 whether Alice measured first, measured later, measured in another basis, or did nothing. Only by comparing his list with Alice's list, which needs an ordinary slower-than-light channel, can they see the correlation. This is the no-communication theorem in action.

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Questions people ask

Does measuring one qubit instantly change the other?

After Alice measures and gets 0, the right description of Bob's qubit (given her result) is |0⟩. But Bob cannot detect any change on his side, because his results stay 50/50 unless he learns Alice's result. Physicists interpret this in different ways, but all agree it cannot be used to send information.

Is entanglement the same as correlation?

Entanglement gives correlations, but not every correlation is entanglement. Two sealed envelopes with matching cards are correlated classically. Entangled qubits can show correlations, across different measurement choices, that no such pre-set scheme can reproduce.

Can entanglement be used for faster internet?

No. Because no information travels by entanglement alone, it cannot make messages arrive sooner. Its uses are in security, sensing and computation.

Does entanglement last forever?

No. Noise from the environment destroys it, often quickly. Keeping entanglement alive long enough to use is a major engineering challenge.

Lesson test

5 questions. Get 3 right (60%) to pass and complete this lesson.

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