Born Rule

The Born rule assigns probabilities to the outcomes of a specified quantum measurement. For a state represented by a density operator $\rho$ and a measurement outcome represented by a positive effect $E_k$,

$$p(k)=\operatorname{Tr}(\rho E_k).$$

The state and the measurement are both required. The rule does not determine which measurement is performed, and the probabilities do not by themselves specify the state after an outcome is recorded.

In Symbols

Projective Measurement

A projective measurement is described by mutually orthogonal projectors $\{\Pi_a\}$ satisfying

$$\Pi_a^\dagger=\Pi_a, \qquad \Pi_a\Pi_b=\delta_{ab}\Pi_a, \qquad \sum_a\Pi_a=I.$$

The probability of outcome $a$ in state $\rho$ is

$$p(a)=\operatorname{Tr}(\rho\Pi_a).$$

For a normalized pure state $\rho=\lvert\psi\rangle\langle\psi\rvert$,

$$p(a) =\langle\psi\vert\Pi_a\vert\psi\rangle =\left\lVert\Pi_a\lvert\psi\rangle\right\rVert^2.$$

If the outcome is nondegenerate, $\Pi_a=\lvert a\rangle\langle a\rvert$, and the familiar amplitude form follows:

$$p(a)=\lvert\langle a\vert\psi\rangle\rvert^2.$$

For a degenerate eigenspace, the full projector must be used. Choosing one basis vector inside that subspace would omit probability assigned to the other vectors.

Generalized Measurement

A positive-operator-valued measure, or POVM, is a set of effects $\{E_k\}$ satisfying

$$E_k\ge0, \qquad \sum_k E_k=I.$$

The same trace rule applies:

$$p(k)=\operatorname{Tr}(\rho E_k).$$

Projective measurements are the special case $E_k=\Pi_k$. An effect determines an outcome probability, but not a unique conditional state change. That extra information belongs to a measurement instrument or a set of measurement operators.

Continuous Outcomes

For a normalized position-space wavefunction,

$$\int_{\mathbb R} \lvert\psi(x)\rvert^2\,dx=1,$$

the probability of finding the position in a measurable region $\Delta$ is

$$\Pr(x\in\Delta) =\int_\Delta \lvert\psi(x)\rvert^2\,dx.$$

$\lvert\psi(x)\rvert^2$ is a probability density, not the probability of the exact point $x$. In an ideal continuous distribution, a single point normally has probability zero even when the density there is nonzero.

More generally, a continuous POVM assigns an effect $E(\Delta)$ to each measurable outcome region:

$$\Pr(\Delta)=\operatorname{Tr}\!\left[\rho E(\Delta)\right].$$

Required Conditions and Checks

For a valid density operator and effect,

$$\rho\ge0, \qquad \operatorname{Tr}\rho=1, \qquad 0\le E_k\le I.$$

These conditions ensure

$$0\le p(k)\le1.$$

Completeness of the measurement gives normalization:

$$\sum_k p(k) =\operatorname{Tr}\left( \rho\sum_kE_k \right) =\operatorname{Tr}\rho =1.$$

For continuous outcomes, the corresponding integral over the full outcome space must equal one. Probabilities are dimensionless; probability densities generally carry inverse units of their integration variable.

Canonical Home

See Born Rule for the canonical conceptual treatment. The compact Born Rule Formula Card collects the equations and checks, while Continuous Born Rule develops probability densities and spectral measures.

State changes conditioned on outcomes belong to Projective Measurement and POVMs: First Encounter. The probability rule and the update rule are logically distinct parts of the measurement formalism.

Context

The Born rule connects the mathematical state to empirical frequencies for a declared measurement. In repeated preparations of the same state followed by the same measurement, observed relative frequencies are expected to approach the assigned probabilities under the usual assumptions of independent trials and stable preparation and measurement procedures.

For an observable with spectral decomposition

$$A=\sum_a a\Pi_a,$$

the Born rule assigns $p(a)=\operatorname{Tr}(\rho\Pi_a)$. The expectation value is then

$$\langle A\rangle =\sum_a a\,p(a) =\operatorname{Tr}(\rho A).$$

The expectation value is an average over the outcome distribution. It need not itself be one of the possible outcomes.

The rule is standard quantum formalism. Interpretations differ over what the state and probabilities represent, but those debates do not change the operational probability formula used for ordinary predictions.

Minimal Example

Let

$$\lvert\psi\rangle =\alpha\lvert0\rangle+\beta\lvert1\rangle, \qquad \lvert\alpha\rvert^2+\lvert\beta\rvert^2=1.$$

A measurement in the basis $\{\lvert0\rangle,\lvert1\rangle\}$ has projectors

$$\Pi_0=\lvert0\rangle\langle0\rvert, \qquad \Pi_1=\lvert1\rangle\langle1\rvert.$$

Therefore

$$p(0)=\lvert\alpha\rvert^2, \qquad p(1)=\lvert\beta\rvert^2.$$

The relative phase between $\alpha$ and $\beta$ does not affect this basis measurement, but it can affect probabilities in another basis. The Born rule is basis independent as a trace formula; the numerical distribution depends on the measurement effects.

Common Aliases

• probability postulate;
• Born probability rule;
• Born interpretation, an older phrase that may carry additional interpretive connotations;
• modulus-squared rule, when restricted to rank-one pure-state amplitudes.

Common Confusions

• Probability rule versus update rule: $p(k)=\operatorname{Tr}(\rho E_k)$ does not determine the post-measurement state.
• Amplitude versus probability: amplitudes add before taking an absolute square when alternatives remain coherent; probabilities add for an appropriate classical mixture or exclusive resolved outcomes.
• Density versus probability: $\lvert\psi(x)\rvert^2$ must be integrated over a region and has units of inverse length in one spatial dimension.
• Outcome versus expectation value: $\langle A\rangle$ is generally not the result of one trial.
• Degenerate outcome versus eigenvector: an eigenspace projector, not an arbitrary vector in that space, represents the unresolved degenerate outcome.
• Observable versus measurement: a self-adjoint observable specifies an ideal projective measurement, while a general laboratory measurement may require a POVM and an instrument.
• Classical ignorance versus quantum state: the trace rule applies to both pure and mixed states, but it does not make every density operator a unique classical ensemble.
• Measurement context: a state alone does not define probabilities until the effects or projectors are specified.

Related Entries

• Wavefunction
• Observable
• Density Matrix
• Probability Amplitudes
• Expectation Values
• Projectors
• Measurement in the Formalism

References

• M. Born, “Zur Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik 37, 863–867 (1926), doi:10.1007/BF01397477.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 4 and 12.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 1.
• P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, 2nd rev. ed., Springer, 1996.