The statement
Explanation
Probability of outcome $i$ for a POVM $\{E_i\}$ with $E_i\ge0$, $\sum_iE_i=I$.
For a pure state and projector this is $\langle\psi|P_i|\psi\rangle$.
Conditions
A compact formula is not a claim of unrestricted validity. Before applying this statement, check the assumptions stated in its wording (dimension, finiteness, equilibrium, model regime) and in the explanation above.
Deeper understanding
A density operator encodes outcome probabilities, including statistical mixtures; distinguish what a single measurement reveals from what repeated measurements can estimate.
What this result tells us
For a general POVM, replace orthogonal projectors by positive effects $E_i$ satisfying $\sum_i E_i=I$.
Mathematical reading guide
Read the equation from left to right: identify its premises, the operation being performed, and the resulting conclusion or bound. When a symbol represents an operator, verify its domain and whether Hermiticity, unitarity, positivity or finite dimensionality is assumed. When probabilities or entropies appear, check normalization and the logarithm convention.
Where it matters
Quantum communication, QKD, tomography, readout calibration and state discrimination.
Worked example and interpretation
For $|\psi\rangle=\sqrt{3}/2|0\rangle+1/2|1\rangle$, measurement with $E_0=|0\rangle\langle0|$ gives $p(0)=3/4$.
Choose a qubit or pair of qubit states, construct the density matrix and compute measurement probabilities or distinguishability.
Check your understanding
Questions to work through
- What are the assumptions and the physical system for Born rule?
- Which quantity is invariant, bounded, measurable or transformed in this statement?
- Can you demonstrate the claim on the simplest valid example, and identify a situation where its assumptions fail?
- How does this result connect to quantum states and measurement in an experiment or simulation?
Suggested approach to the answers
First identify the hypotheses in the original statement. Use the displayed equation and the worked example above; distinguish an exact theorem from an approximation, and state the operational meaning of each quantity. For the final question, relate the mathematical result to quantum communication, qkd, tomography, readout calibration and state discrimination.