Measuring One of Two Qubits

A two-qubit system lives in a four-dimensional Hilbert space spanned by $|00\rangle$ , $|01\rangle$ , $|10\rangle$ , $|11\rangle$ . When you measure only one of the two qubits you read out partial information — and the result you see, together with the way the amplitude was spread across the state, determines both the probability of each outcome and what state remains afterwards.

Marginal probabilities

Consider the product state

|\psi\rangle = |+\rangle \otimes |0\rangle = \frac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr) \otimes |0\rangle = \frac{1}{\sqrt{2}}\bigl(|00\rangle + |10\rangle\bigr).

The full probability vector has four entries, one per basis state:

P(|00\rangle) = \frac{1}{2}, \quad P(|01\rangle) = 0, \quad P(|10\rangle) = \frac{1}{2}, \quad P(|11\rangle) = 0.

To find the probability of getting outcome $0$ when measuring only qubit 0, you sum over all values of qubit 1 that are consistent with that outcome:

P(\text{qubit 0} = 0) = P(|00\rangle) + P(|01\rangle) = \tfrac{1}{2} + 0 = \tfrac{1}{2}.

Likewise, $P(\text{qubit 0} = 1) = P(|10\rangle) + P(|11\rangle) = \tfrac{1}{2}$ . Both outcomes are equally likely, exactly as the Born rule predicts for a single qubit in $|+\rangle$ .

Post-measurement collapse

Measurement does more than produce a number — it also changes the state. If the instrument reports $0$ for qubit 0, the surviving amplitude is the part of $|\psi\rangle$ compatible with that outcome: only $|00\rangle$ had qubit 0 in $|0\rangle$ . After re-normalizing,

|\psi'\rangle = |00\rangle.

If the instrument reports $1$ , the surviving part is $|10\rangle$ , and

|\psi''\rangle = |10\rangle.

In either case qubit 1 ends up in $|0\rangle$ — because the original state was a product state and measuring qubit 0 carries no information about qubit 1. This clean separation breaks down for entangled states, where reading one qubit can instantly determine the other; we will explore that soon.

Try it

Prepare the two-qubit state $|{+}0\rangle$ by applying a Hadamard to qubit 0, then measure only qubit 0. The grader checks the full probability distribution over all four basis states.

Run your code to see the quantum state.

The Probabilities panel should show equal bars at $|00\rangle$ and $|10\rangle$ , with zeros at $|01\rangle$ and $|11\rangle$ — confirming that qubit 1 was never disturbed.