Level 5 · Advanced undergraduate
Objective. Calculate measurement probabilities from normalized amplitudes.
Before this lesson. L3-01, L3-02
Hands-on lab. Qubits and the Bloch sphere
Learn¶
A qubit is a quantum system with two basis states. Write these states as |0> and |1>. A pure state is a normalized complex vector. Its amplitudes describe how the state relates to a selected measurement basis.
The Born rule converts amplitudes to probabilities. Square the magnitude of each amplitude. Add the probabilities to check normalization. A negative amplitude is permitted. A negative probability is not permitted. The measurement basis must always be stated.
A measurement gives one outcome per trial. It does not expose all amplitudes from one system. Estimate probabilities with many equally prepared systems. Use other measurement bases to learn relative phase. Measurement statistics alone do not establish a causal explanation.
Practice¶
Solution
The squared magnitude of 2i/sqrt(5) is 4/5. Thus P(1) is 0.8. The expected count is 250 times 0.8, or 200. The imaginary phase does not change this basis probability.
Check your understanding¶
What must sum to one for a normalized pure state?
A. The amplitudes
B. The squared amplitude magnitudes
C. The measurement counts
Show the answer
B. The squared amplitude magnitudes
Probabilities are squared magnitudes. Their sum is one. Complex amplitudes themselves can cancel and need not sum to one.
Sources¶
This lesson draws on Basics of quantum information, n.d.Preskill, n.d..
Next: L5-02 · Gates and circuit order
- Basics of quantum information. (n.d.). https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information
- Preskill, J. (n.d.). Course Information for Physics 219/Computer Science 219 Quantum Computation (Formerly Physics 229). California Institute of Technology course page with lecture notes. https://www.preskill.caltech.edu/ph219/