Functions of Observables

A real function of a sharp observable describes a deterministic relabeling of its outcomes. Squaring momentum produces kinetic energy, thresholding position produces a yes/no region event, and binning energy produces a coarse readout. The spectral measure specifies the new probabilities and shows precisely which distinctions the relabeling loses.

Helpful background. Projection-Valued Measures defines sharp events. Functional Calculus constructs functions of operators and determines their domains; here we explore what those functions mean for measurement outcomes.

Relabeling pushes the spectral measure forward

Let $A$ be self-adjoint and $f:\mathbb R\to\mathbb R$ Borel measurable and finite-valued. The observable $B=f(A)$ is self-adjoint on its natural spectral-integrability domain. Its PVM is

$$E_B(\Gamma)=E_A(f^{-1}(\Gamma)).$$

This follows because measuring whether $f(a)$ belongs to $\Gamma$ is the same event as asking whether $a$ belongs to its preimage. Mathematically, indicator functions compose: $\mathbf1_\Gamma(f(A))=(\mathbf1_\Gamma\circ f)(A)$.

For any state $\rho$, the outcome measures obey

$$\mu_\rho^B(\Gamma) =\mu_\rho^A(f^{-1}(\Gamma)).$$

Thus $\mu_\rho^B=f_*\mu_\rho^A$, the ordinary probability pushforward. The theorem applies to atoms and singular measures as well as to densities; a Jacobian formula is a special case that requires additional regularity.

A complex-valued function produces a normal operator, not generally a self-adjoint observable with real outcomes. To interpret it as a pair of real commuting outputs, use the real and imaginary parts and their joint PVM.

Kinetic energy loses momentum direction

For a one-dimensional free particle,

$$H_0=\frac{P^2}{2m}.$$

The event $H_0\in[E_1,E_2]$, with $0<E_1<E_2$, corresponds to the two momentum intervals

$$[-\sqrt{2mE_2},-\sqrt{2mE_1}] \ \cup\ [\sqrt{2mE_1},\sqrt{2mE_2}].$$

Both directions contribute to the same energy bin. The energy observable therefore cannot distinguish right-moving and left-moving components.

If the momentum density is $w(p)$, the energy density for $E>0$ is

$$v(E)=\frac{m}{\sqrt{2mE}} \left[w(\sqrt{2mE})+w(-\sqrt{2mE})\right].$$

Each inverse branch contributes its own absolute Jacobian $|dp/dE|=m/\sqrt{2mE}$. Omitting a branch discards probability. The density can have an integrable threshold singularity at zero even when $w$ is regular there.

This relation concerns outcome probabilities. Whether a particular apparatus physically measures momentum and then discards its sign, or directly measures energy, is a separate instrumental question.

Thresholds and bins remain sharp

For a Borel set $\Delta$, choose $f=\mathbf1_\Delta$. Then $f(A)=E_A(\Delta)$ is a projection with outcomes zero and one.

For a countable partition $\{\Delta_j\}$ and distinct real labels $b_j$, the binning function gives

$$B=\sum_j b_jE_A(\Delta_j)$$

on its natural domain. If the labels are bounded, so is $B$, even when $A$ is unbounded. The labels specify a numerical output convention; the bins specify which physical distinctions remain.

Deterministic coarse graining preserves projection-valuedness because preimages preserve intersections. Noisy post-processing is different. If a stochastic kernel $k(\Gamma|a)$ describes random readout error, the effect becomes

$$F(\Gamma)=\int k(\Gamma|a)\,dE_A(a),$$

which generally is not idempotent. A noisy observable is therefore usually a POVM rather than a PVM.

Lost information and invertible relabeling

If $f$ is injective on the relevant spectral space and has a measurable inverse there, the original events can be recovered from the relabeled observable. In that case $A$ is a measurable function of $B$.

For a noninjective $f$, distinct spectral channels can merge. This can increase multiplicity even if $A$ has multiplicity one. The map $p\mapsto p^2$ is the simplest example. The distinction is carried by the PVM, not just by the numerical set $\sigma(B)$.

For continuous $f$ on the compact spectrum of a bounded $A$, spectral mapping gives $\sigma(f(A))=f(\sigma(A))$. For general Borel functions, raw pointwise images can include values attained only on spectrally null sets; spectral support and essential range are the appropriate replacements.

Coarse probabilities do not determine coarse state updates

Suppose two orthogonal projections $P_1,P_2$ are merged into $P=P_1+P_2$. A direct ideal measurement of the coarse event has unnormalized Lüders update $P\rho P$. Measuring the fine outcomes and forgetting which occurred instead gives

$$P_1\rho P_1+P_2\rho P_2.$$

The probabilities agree, since both traces equal $\operatorname{Tr}(\rho P)$. The states can differ because the second procedure removes coherence between the two subspaces, whereas the first can retain it. A function of an observable specifies outcome events, not a unique measurement instrument.

Exercises

For $\psi=(|1\rangle+|2\rangle)/\sqrt2$, compare the two updates above when $P_j=|j\rangle\langle j|$.

Solution

The coarse projector is the identity on the two-dimensional subspace, so direct Lüders conditioning leaves the pure state $|\psi\rangle\langle\psi|$. Fine measurement followed by forgetting gives the mixed state $\tfrac12(|1\rangle\langle1|+|2\rangle\langle2|)$. Both assign probability one to the coarse event but differ on measurements sensitive to the relative phase.

Show that the free-particle energy density integrates to the total momentum probability.

Solution

Apply $E=p^2/(2m)$ separately on $p>0$ and $p<0$. Each branch transforms its Jacobian back to $dp$, giving $\int_0^\infty v(E)\,dE=\int_{\mathbb R}w(p)\,dp$. For an absolutely continuous momentum measure there is no atom at $p=0$ to add separately.

References

• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016, doi:10.1007/978-3-319-43389-9 — post-processing, sharpness, and instruments.
• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — functions of observables and spectral representations.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — spectral pushforwards and free-particle representation.