Quantum foundations · Independent lesson
An amplitude is not a probability with extra decoration
An independent QubitWire lesson on this topic, based on the source reading below. It is not a transcript or a summary of the recording.
Quantum information uses amplitudes to describe a state relative to a chosen basis. Measurement probabilities come from the squared magnitudes of those amplitudes. Relative phases also matter because later operations can bring contributions together, allowing them to reinforce or cancel before a measurement is made.
This distinction explains why listing only the probabilities of a measurement in one basis does not describe everything relevant about a quantum state. Two states can yield the same distribution in that basis and different results after another operation or in another basis. A mathematical description is useful because it keeps the information needed to predict those later experiments.
For multiple qubits, the state refers to joint possibilities. For a pure multi-qubit state, failure to factor into separate single-qubit state vectors signals entanglement. Mixed states need a more general test that also distinguishes classical correlations. That does not allow a user to read every amplitude in one measurement. Extracting information still requires a specified measurement and, generally, repeated preparations.
The summer-school lecture’s topic is a good place to move carefully between notation and experiment. For every vector, ask what preparation it represents. For every probability, ask which measurement produces it. That habit makes the mathematics less detached from the results a program returns.
Questions to work through
Why can equal probabilities hide a difference?
The states may have different relative phases. A later operation can convert that phase difference into a difference in measured probabilities.
Does measurement reveal the complete state?
A single measurement returns an outcome for the selected measurement. Reconstructing a state generally requires many preparations and multiple measurement settings.
How can a global phase differ from a relative phase?
Multiplying the entire state by one phase does not change observable predictions. Changing phases between components can change interference and later outcomes.
Source reading
- youtube.com ↗
https://www.youtube.com/watch?v=-reX5fajpvg
- quantum.cloud.ibm.com ↗
https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information