Born Rule via Spectral Measures

The Born rule pairs a state with the spectral projections of an observable to produce an ordinary probability measure. The spectral theorem supplies the events; the probability postulate supplies their physical interpretation. No density, eigenbasis, or finite-moment assumption is required to define the probabilities.

Helpful background. Projection-Valued Measures supplies the event measure. The Born Rule introduces the probability postulate; here we establish how it defines a probability measure and which moments exist.

Vector states give normalized measures

For a self-adjoint $A$ and normalized $\psi\in\mathcal H$, define

$$\mu_\psi^A(\Delta) =\langle\psi,E_A(\Delta)\psi\rangle =\|E_A(\Delta)\psi\|^2.$$

Projection positivity gives nonnegative values, and $E_A(\mathbb R)=I$ gives normalization. For disjoint sets $\Delta_n$, strong additivity and continuity of the inner product give countable additivity.

The last equality also gives an orthogonal-decomposition interpretation: probability is the squared norm of the component selected by the event. This remains true for a continuum of possible values without assigning a nonzero probability to each exact point.

Every normalized Hilbert-space vector is allowed in this formula. In particular, $\psi$ need not belong to $D(A)$, because every spectral projection is bounded.

Density operators and the trace formula

For a density operator $\rho\geq0$ with $\operatorname{Tr}\rho=1$, the probability is

$$\mu_\rho^A(\Delta)=\operatorname{Tr}(\rho E_A(\Delta)).$$

The product is trace-class because a trace-class operator times a bounded operator is trace-class. Positivity can be seen by writing the trace as $\operatorname{Tr}(\rho^{1/2}E_A(\Delta)\rho^{1/2})$.

To verify countable additivity, use the positive spectral decomposition

$$\rho=\sum_k r_k|\psi_k\rangle\langle\psi_k|, \qquad r_k\geq0,\quad \sum_k r_k=1.$$

Then

$$\mu_\rho^A(\Delta)=\sum_k r_k\mu_{\psi_k}^A(\Delta).$$

For disjoint events, Tonelli’s theorem permits exchanging the two nonnegative sums. Each vector measure is countably additive, so their mixture is as well. This argument explains why normal density-operator states produce countably additive probabilities even in infinite dimensions.

The decomposition is not unique as a convex ensemble, but the trace formula depends only on $\rho$. Distinct preparations with the same density operator have the same statistics for this observable and every other system observable.

Discrete and continuous formulas are representations

If $A$ has spectral atoms $\lambda_j$ with projectors $P_j$, then

$$\mathbb P_\rho(A=\lambda_j)=\operatorname{Tr}(\rho P_j).$$

For a degenerate eigenspace this sums over the entire projection; picking one eigenvector would omit allowed outcomes in that eigenspace.

For position on $L^2(\mathbb R)$, the vector-state measure is $d\mu_\psi^Q(x)=|\psi(x)|^2\,dx$. A density-operator decomposition gives

$$w_\rho(x)=\sum_k r_k|\psi_k(x)|^2$$

almost everywhere, with $\int w_\rho(x)\,dx=1$. This avoids assuming that an arbitrary trace-class integral kernel has a pointwise-defined diagonal.

Not every self-adjoint observable has a probability density with respect to Lebesgue measure. Atomic and singular continuous contributions may occur. The measurable-set formula is the general object from which a density may be derived when absolute continuity is proved.

Moments require additional integrability

For bounded Borel $f$,

$$\int f(a)\,d\mu_\rho^A(a)=\operatorname{Tr}(\rho f(A)).$$

One proves this first for simple functions and then uses bounded approximation. For unbounded $f$, the integral requires an integrability condition and the trace expression requires a corresponding operator or form interpretation.

For a vector state, a finite absolute mean satisfies

$$\int|a|\,d\mu_\psi^A(a)<\infty \quad\Longleftrightarrow\quad \psi\in D(|A|^{1/2}).$$

The second moment is finite exactly on $D(A)$ and equals $\|A\psi\|^2$. A fourth moment is needed to apply $A^2$ as a Hilbert-space operator.

For a nonnegative $A$ and a mixed state, a safe extended expectation is

$$\operatorname{Tr}(\rho A) :=\sup_n\operatorname{Tr}\bigl(\rho\,\min(A,nI)\bigr),$$

where the right side is defined by bounded spectral calculus and can be infinite. This notation need not mean that the unbounded product $\rho A$ has already been proved trace-class. Finite mean energy is an extra property of a state, not a precondition for its normalized energy probability measure.

State support and observable spectrum

The spectrum describes outcomes available to some states. A particular state’s probability measure can be supported on only a proper part of it. For example, position has spectrum $\mathbb R$, but a wavefunction supported in $(0,1)$ gives zero probability outside that interval.

Conversely, the probability of an exact value can vanish even when the value is spectral and arbitrarily small surrounding intervals have positive probability. Probability zero for a singleton is compatible with a continuous measurement variable.

Neither the vector nor mixed-state Born formula specifies a conditional state update. A PVM gives statistics; a measurement instrument supplies the additional transformation rule.

Exercises

For $Ae_n=n e_n$, choose normalized weights $r_n=Cn^{-2}$. Does the state $\rho=\sum r_n|e_n\rangle\langle e_n|$ define a probability measure? Is its mean finite?

Solution

The weights sum to one after normalization, so $\rho$ is a density operator and the outcome probability of $n$ is $r_n$. The mean is $C\sum n^{-1}$, which diverges. All event probabilities remain well-defined despite the infinite first moment.

Show that $\operatorname{Tr}(\rho E)=\operatorname{Tr}(\rho F)$ for all density operators $\rho$ forces equality of two proposed bounded effects $E$ and $F$.

Solution

Testing all rank-one states gives $\langle\psi,(E-F)\psi\rangle=0$ for every vector after rescaling. Polarization recovers every matrix element of $E-F$, hence $E=F$. A single state’s distribution is insufficient; the assertion needs all states.

References

• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016, doi:10.1007/978-3-319-43389-9 — states, effects, probabilities, and instruments.
• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — density operators and spectral probability formulas.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — scalar spectral measures and moment domains.