Probability and Measurement Conventions

The default probability notation is $P(a)$ for the probability of outcome $a$ when the measurement context is clear. Conditional probabilities may be written $P(a\mid b)$ or $P(a\mid\psi)$ when the condition is explicit.

Probabilities are real, nonnegative, and normalized:

$$P(a)\ge0, \qquad \sum_a P(a)=1$$

for a discrete exhaustive outcome set. For continuous outcomes, probabilities are assigned to regions by integrating a probability density.

Projective Measurements

For a projective measurement with projectors $P_a$, the convention is

$$P_aP_b=\delta_{ab}P_a, \qquad \sum_a P_a=I.$$

For a normalized pure state,

$$P(a)=\langle\psi\vert P_a\vert\psi\rangle.$$

For a density operator,

$$P(a)=\operatorname{Tr}(\rho P_a).$$

When $P$ could mean both probability and projector, use prose or typography to disambiguate: $P(a)$ is a probability, while $P_a$ is a projector.

General Measurements

For a POVM, effects are written $E_a$ and satisfy

$$E_a\ge0, \qquad \sum_a E_a=I.$$

The probability rule is

$$P(a)=\operatorname{Tr}(\rho E_a).$$

Effects determine outcome probabilities. State updates require additional measurement-operation data, not only the POVM effects.

State Update

For an ideal projective measurement with outcome $a$, the default density-operator update is Lüders’ rule:

$$\rho\mapsto \frac{P_a\rho P_a} {\operatorname{Tr}(\rho P_a)}$$

when $P(a)>0$. Pure-state versions are the corresponding normalized projections. More general measurements use Kraus operators or instruments, which must be specified before writing an update rule.

Continuous Outcomes

For a position-space wavefunction, the convention is

$$P(X\in R)=\int_R \lvert\psi(x)\rvert^2\,dx$$

in one dimension, with the appropriate measure in other coordinates. The density $\lvert\psi(x)\rvert^2$ is not itself the probability of the exact point $x$.

Common Mistakes

• Applying the Born rule without saying what measurement is being performed.
• Confusing probability amplitudes with probabilities.
• Treating a probability density as a probability.
• Using a projective update rule for a nonprojective measurement.
• Assuming a POVM effect alone determines the post-measurement state.
• Reading an interpretation of quantum probability into notation that only states operational probabilities.

References

• M. Born, “Zur Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik 37, 863–867, 1926.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. Let $P_0=\lvert0\rangle\langle0\rvert$ and $\lvert\psi\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$. Compute $P(0)$.

Solution

Using the projective Born rule,

$$P(0)=\langle\psi\vert P_0\vert\psi\rangle=\frac12.$$

2. Why is $P(a)=\operatorname{Tr}(\rho E_a)$ not enough to determine the post-measurement state?

Solution

The effect $E_a$ determines the probability of outcome $a$, but different physical measurement procedures can have the same effect and different state updates. The update requires an instrument or a set of Kraus operators, not just the POVM effect.