Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes

Positive operators and effects on the assigned Hilbert space. For a bounded operator $F:H\to H$, write $F\ge 0$ when $F=F^{\ast }$ and $⟨Fx,x⟩\ge 0$ for every $x\in H$. Write $0\le F\le I$ when $F$ and $I-F$ are positive; equivalently $0\le ⟨Fx,x⟩\le \mathrm{\parallel }x{\mathrm{\parallel }}^2$ for every $x\in H$. Thus an effect on $H$ is a bounded operator satisfying these conditions. These definitions apply whether or not $H$ is finite-dimensional; in finite dimensions they agree with Non-negative and positive operators.

Mathematical normalization of finite effects (conditional on P1). Let $H$ be any nonzero Hilbert space, let $\rho$ be a density operator on $H$, and let $(F_a{)}_{a=1}^p$ be effects with ${\sum }_a{F}_a=I$. By Density operators and scalar spectral probability clause (a), write $\rho ={\sum }_n{w}_n{T}_{u_n,u_n}$ in trace norm with $w_n\ge 0$, ${\sum }_n{w}_n=1$, and $(u_n)$ orthonormal. The rank-one trace formula gives $\mathrm{tr}(T_{u_n,u_n}{F}_a)=⟨F_a{u}_n,u_n⟩$ (Rank-one operator calculus), and the trace-class ideal bound makes $T\mapsto \mathrm{tr}(T{F}_a)$ continuous in trace norm (Trace class is a two sided Banach operator ideal). Hence $\mathrm{tr}(\rho F_a)={\sum }_n{w}_n⟨F_a{u}_n,u_n⟩\ge 0$; summing over the finite index $a$ gives ${\sum }_a\mathrm{tr}(\rho F_a)={\sum }_n{w}_n⟨I{u}_n,u_n⟩=1$. This is a mathematical normalization calculation for the probabilities in clause (iv), not an additional physical measurement assumption.

Adopt as physical assumptions of the framework the following statements about sharp and unsharp real-valued quantities. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). The latter is the explicit hypothesis used to obtain the spectral PVM of a general self-adjoint operator from Spectral theorem for unbounded self-adjoint operators (PVM form) in clause (ii); consumers invoking this construction inherit it.

(i) Sharp observables. An ideal sharp real-valued quantity (observable) of a system with state space $H$ is a densely defined self-adjoint operator $A:D(A)\subset H\to H$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators), meaning $D(A)=D(A^{\ast })$ and $A\psi =A^{\ast }\psi$ on this common domain, together with a declared unit in which its real spectral coordinate is read; the unit is part of the assignment, and no numerical value of the coordinate is asserted by the assignment. Symmetry of $A$ on a small domain is not sufficient for self-adjointness, and the domain is a declared part of the observable.

(ii) Spectral probability. Let $E_A$ be the spectral projection-valued measure of the self-adjoint operator $A$ on the Borel $\sigma$-algebra of $\mathbb{R}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, The Borel sigma-algebra of a topological space). For a preparation $\rho$ and a Borel set $B\subseteq \mathbb{R}$ the probability of the outcome in $B$ is ${\mu }_{\rho }^A(B):=\mathrm{tr}(\rho E_A(B)),\text{and for a pure preparation }[\psi ]\text{ it is }{\mu }_{\psi }^A(B)=\mathrm{\parallel }{E}_A(B)\psi {\mathrm{\parallel }}^2\mathrm{.}$ By clause (b) of Density operators and scalar spectral probability the assignment $B\mapsto {\mu }_{\rho }^A(B)$ is a probability measure on the Borel sets (Probability measures and probability spaces, Scalar and complex measures from a pvm). It is the general form of the Born rule adopted here.

(iii) Nature of the outcome law. The probability law is a measure, not necessarily a density with respect to Lebesgue measure and not necessarily a finite or countable sum of point masses: a self-adjoint operator with continuous spectrum ordinarily has no normalizable eigenvector at a point of its continuous spectrum, so an "exact value" of a continuum quantity is not represented by a normalized state. An ideal point value is an idealization; a real detector reports a finite-resolution outcome, that is, the probability of a declared Borel set $B$ such as a bin, and the recorded-event probability is that of (ii) for the bin, not the value of a density at a point.

(iv) Unsharp finite outcomes. Outcomes of a finite-resolution apparatus with finitely many channels are described, when the idealization uses bounded effects, by a finite family $(F_a{)}_{a=1}^p$ of effects on $H$ as defined above, with ${\sum }_a{F}_a=I$. The outcome probabilities are $p_a=\mathrm{tr}(\rho F_a)$. They are normalized for every state space assigned by P1, as shown by the preceding general-$H$ calculation. This unsharp extension is an operational description tied to an apparatus model; it is not obtained from the sharp observable by mathematical manipulation alone, and the effects alone do not determine what happens to the state after the outcome (P3).

(v) Moments are a separate matter. Statements about the mean, variance or higher moments of the outcome law require the corresponding integrability of $t\mapsto t^n$ against ${\mu }_{\rho }^A$ and are not part of this postulate; where they are used they are stated with their hypotheses. In particular the existence of $\mathrm{tr}(\rho A)$ for an unbounded $A$ is not asserted.

(vi) Status. Clauses (i)-(v) are adopted assumptions of the model. They assert nothing about which quantities of a concrete apparatus are sharp, about the physical mechanism producing an outcome or about the completeness of the description; a laboratory record is a classical event whose relation to the model is a tested identification with uncertainty, in the sense of P7.

• Adopted, not derived. The representation of sharp quantities by self-adjoint operators and the Born rule of clause (ii) are assumptions. They are motivated by the successes of the framework but are not deduced from experiment.
• No detector theorem. Clause (iii) explicitly distinguishes an ideal point outcome from a finite-resolution record: the model's binomial or binned probabilities are the operational objects, and a vanishing ideal density at a point does not make a finite bin of positive length impossible.
• Sharpness and units. A declared observable carries a unit; sums and comparisons of outcome coordinates are meaningful only for observables with the same unit. Quantities of the model whose operator status is not declared are numbers or parameters, not observables.