1.11 The selector as a projective measurement
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The pair \(\{P,I-P\}\) is a projective measurement because its two members are self-adjoint, idempotent, orthogonal, and sum to the identity. For the conditional update we choose the ideal Lüders instrument, whose unnormalized branches send a density matrix \(\rho\) to \(P\rho P\) and \((I-P)\rho(I-P)\). The projector effects determine the outcome probabilities; this chosen instrument determines the state update. The probability rule and this conditional update can be stated independently of the arithmetic example.
Theorem QA16: Selection probability and conditional state
Let \(P\) be an orthogonal projector on \(\mathbb C^N\) and let \(v\) be a normalized state. For the ideal Lüders instrument just specified, the two probabilities are
When \(p>0\), the normalized selected state is \(Pv/\sqrt p\). When \(1-p>0\), the normalized complementary state is \((I-P)v/\sqrt{1-p}\). The probabilities belong to \([0,1]\) and sum to one.
Proof
Since \(P^\dagger P=P\), the quadratic form equals \(\|Pv\|^2\) and is nonnegative. QA07 applied to a unit vector gives the probability sum and the upper bound. Division by the positive square root produces a state of squared norm one in each nonzero branch. \(\square\)
For the explicit normalized test signal \(v=x/\sqrt{150}\), QA11 gives
The selected conditional state is \(c/\sqrt{48}=\psi\), and the complementary conditional state is \(b/\sqrt{102}\). An exact classical filter applied to the full vector returns \(c\). A projective measurement on the normalized input instead returns a branch with these probabilities. These are compatible uses of the same operator with different observation rules.
The vector \(\chi\) makes noninjectivity concrete: \(P\chi=0\) although \(\|\chi\|=1\). A reversible Fourier change of basis cannot be substituted for this loss of information. Conversely, retaining both complementary packets reconstructs every vector, as QA07 and the publication module explicitly state.
The inherited formal source certifies the concrete projectors, states, and probability identities. It does not provide a general quantum measurement library or a generic theorem implementing the conditional state-update rule. QA16 is a general prose theorem, with the usual finite-dimensional measurement interpretation.