Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes

Adopt as physical assumptions of the framework the following conditional-update rule for declared ideal measurements. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

(i) Declared measurement. A declared ideal sharp measurement of a system with state space $H$ consists of a finite or countably infinite family $(P_a{)}_{a\in A}$ of bounded orthogonal projections on $H$ (The Hilbert orthogonal projection onto a closed subspace, The orthogonal projection $P_{W}v$ is the $W$-component in $V=W\oplus W^{\perp }$) which are mutually orthogonal, $P_a{P}_b=0$ for $a\mathrm{\ne }b$, and whose strong sum is the identity: the finite or norm-convergent countable sum ${\sum }_a{P}_{a}x=x$ for every $x\in H$ (Projection valued measure clause 4). The family is part of the apparatus description; the model does not claim that every laboratory device is a declared measurement of this form.

(ii) Outcome probabilities. For a preparation $\rho$ the probability of the outcome $a$ is $p_a:=\mathrm{tr}(\rho P_a)\ge 0$, with ${\sum }_a{p}_a=1$; equivalently $p_a=\mathrm{\parallel }{P}_a\psi {\mathrm{\parallel }}^2$ for a pure preparation $[\psi ]$, as in the probability postulate P2. This is the Born law of the declared measurement.

(iii) Conditional update. If the outcome $a$ has been recorded and $p_a>0$, the preparation posterior to the measurement is ${\rho }_a:=\frac{P_a\rho P_a}{p_a},$ the Lueders update. It is a density operator (Density operators and scalar spectral probability clause (a), Rank-one operator calculus): $P_a\rho P_a$ is positive, trace class with trace $p_a$ by the rank-one series expansion of $\rho$ and the trace formulas, and ${\rho }_a$ is self-adjoint, positive, trace class with trace $1$. The update preserves coherence inside the range of a degenerate $P_a$: a superposition of orthogonal vectors within that range is updated to the same relative superposition, and repetition of the same declared measurement on ${\rho }_a$ returns $a$ with probability $1$ (repeatability of the declared projective measurement).

(iv) Discarded outcomes. If the outcomes are recorded but not selected, the posterior ensemble is ${\rho }^{\mathrm{\prime }}={\sum }_a{P}_a\rho P_a,$ which converges in trace norm and is a density operator: for an eigen-expansion $\rho ={\sum }_n{w}_n{T}_{u_n,u_n}$ as in clause (a) of Density operators and scalar spectral probability under the adopted Axiom of Countable Choice, the series ${\sum }_a{P}_a\rho P_a={\sum }_a{\sum }_n{w}_n{T}_{P_a{u}_n,P_a{u}_n}$ is nuclear with trace ${\sum }_a{\sum }_n{w}_n\mathrm{\parallel }{P}_a{u}_n{\mathrm{\parallel }}^2={\sum }_n{w}_n=1$ by the nuclear-series characterization of the trace norm and the absolute convergence of the trace (Nuclear series characterizes trace norm, Trace is absolutely convergent and basis independent); its normalized form ${\rho }^{\mathrm{\prime }}$ has the operator dependence recorded by clause (iii) for each branch.

(v) Limits of the rule. The rule is an assumption of the model for the declared ideal measurement; it is not derived from the Born law of P2, and it is not asserted for every apparatus. In particular: a measurement of an exact eigenvalue of an operator with continuous spectrum is not represented by normalizing a delta distribution; a photon-absorption detector need not leave an outgoing one-photon preparation of the same form; and a unitary system-apparatus model that produces conditional correlations does not by itself select a unique recorded outcome or resolve which branch occurred. Apparatuses outside the declared class require their own models with their own stated assumptions. The example in Finite effects and instruments: normalization and non-determination of the update that equal finite outcome effects can have different updates is proved there only for finite-dimensional $H$; that example is not a representation result for instruments on infinite-dimensional $H$.

(vi) Status. Clauses (i)-(v) are adopted assumptions. They record how the model updates preparations in the declared ideal case and what it deliberately does not claim about arbitrary apparatuses and about the uniqueness of recorded outcomes.

• No preferred-basis claim. The declared family $(P_a)$ belongs to the apparatus model; the postulate gives no rule that selects a preferred family from a system alone.
• Degeneracy. Clause (iii) is the density-operator form of the Lueders rule with the projections $P_a$ themselves, not with one-dimensional sub-projections chosen inside a degenerate eigenspace; the difference is measurable and is treated with the update rule where degenerate measurements occur.
• Discarding versus conditioning. Clause (iv) describes the discarded-outcome ensemble; it is a different object from the selected-branch state of clause (iii), and it destroys the coherence between different branches.