Multi-Qubit Measurement Probabilities

1. The Born Rule for Multiple Qubits

The rule for calculating measurement probabilities in multi-qubit systems is a natural extension of the single-qubit case. A system of $n$ qubits is described by a state vector with $2^n$ complex amplitudes, one for each possible classical outcome.
$|q⟩={\alpha }_0|0\dots 00⟩+{\alpha }_1|0\dots 01⟩+\cdots +{\alpha }_{2^n-1}|1\dots 11⟩={\sum }_{j=0}^{2^n-1}{\alpha }_j|j⟩$
Here, $|j⟩$ is the basis state corresponding to the binary representation of the integer $j$. For example, in a 3-qubit system, $|5⟩$ is shorthand for $|101⟩$.

The Born rule states that the probability of measuring the system and finding it in the specific basis state $|j⟩$ is the squared magnitude of its corresponding amplitude, ${\alpha }_j$.
$P(j)=|{\alpha }_j{|}^2$

Just as with a single qubit, the sum of all these probabilities must equal 1.
${\sum }_{j=0}^{2^n-1}|{\alpha }_j{|}^2=1$

The following resource provides a detailed walkthrough of this principle, using a specific 3-qubit superposition state called the W-state as an example.

2. Partial Measurements: Measuring a Single Qubit

A more common and interesting scenario is when we measure only a subset of the qubits in a circuit. For example, in a 3-qubit system, what is the probability of measuring the first qubit and getting a 1, regardless of what the other two are?

The rule is straightforward: you sum the probabilities of all the full-state outcomes that match your partial measurement criterion.

Let's continue with the W-state from the previous section:
$|\text{w}⟩=\frac{1}{\sqrt{3}}|001⟩+\frac{i}{\sqrt{3}}|010⟩-\frac{1}{\sqrt{3}}|100⟩$
(Remember that in Qiskit's little-endian convention, this would be written with qubit 0 on the right, so the states are $|q_2{q}_1{q}_0⟩$).

Let's find the probability of measuring the last qubit, $q_0$, and getting 0.

1. Identify all basis states in the superposition where $q_0$ is 0: These are $|010⟩$ and $|100⟩$.
2. Find the probability of measuring each of those states:
   • $P(010)=|\frac{i}{\sqrt{3}}{|}^2=\frac{1}{3}$
   • $P(100)=|\frac{-1}{\sqrt{3}}{|}^2=\frac{1}{3}$
3. Sum these probabilities:
   • $P(q_0=0)=P(010)+P(100)=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$

This process is explained very clearly in the following short video.

State Collapse after Partial Measurement

Just as with a full measurement, a partial measurement also collapses the quantum state. However, it only collapses it partially. The superposition is eliminated with respect to the qubit that was measured, and the overall state must be renormalized so the new probabilities sum to 1.

The next section of the reading you started provides the formal procedure for this.

3. A Practical Example: From Math to Code

It's one thing to do these calculations on paper, but it's another to see them match a real simulation. Sometimes, a mismatch can reveal subtle but crucial details about the quantum computing framework you're using.

Let's analyze a common scenario where a developer's calculations don't match their Qiskit simulation results.

This example is a valuable lesson in bridging theory and practice. The mathematical rules for probability are universal, but their application in a specific framework like Qiskit requires careful attention to conventions like gate definitions and qubit ordering.

Conclusion

In this lesson, we generalized the concept of measurement from single to multi-qubit systems. You now have the tools to predict the outcome of any measurement on any quantum circuit, given its state vector.

Key Takeaways:

• Full State Measurement: The probability of measuring a multi-qubit system in a specific basis state $|j⟩$ is given by the Born rule: $P(j)=|{\alpha }_j{|}^2$, where ${\alpha }_j$ is the amplitude of that state.
• Partial State Measurement: To find the probability of a specific outcome for a subset of qubits (e.g., $q_0=1$), you sum the probabilities of all the full basis states that are consistent with that outcome.
• Post-Measurement Collapse: After a partial measurement, the state collapses. Components of the superposition that are inconsistent with the measurement outcome are eliminated, and the new state vector is renormalized.
• Theory vs. Practice: When implementing circuits, it is crucial to pay attention to the specific conventions of your framework (like Qiskit's little-endian ordering) to ensure your theoretical calculations match the simulated results.

Preview of the Next Lesson:
We have now covered all the essential theoretical pillars of the quantum circuit model: representing states with vectors, manipulating them with unitary gates, and predicting measurement outcomes. You're ready to start building. In the next lesson, we will begin Module 3 by focusing on the practical side of things: setting up your own Qiskit development environment. This will be our first step towards writing and running your own quantum programs from scratch.