Manuscript I@2 · I.tex · corollary · 85631fa7

Corollary: Finite instrument representation

Stability and represented measurements › Rotated readout and POVMs · lines 2141–2182 · proof included or linked

I@2#cor:instrument

Let

Φ_{x} (ρ) = j \sum M_{x j} ρ M_{x j}^{†}, x, j \sum M_{x j}^{†} M_{x j} = I

be a finite-dimensional instrument with fixed input and quantum output spaces. After finite qubit encodings, an Enter program realizes these maps on its declared classical and quantum outputs. With an untouched reference $B$ , its branch map is $Φ_{x} \otimes id_{B}$ .

Proof. Let $J_{in}$ and $J_{out}$ be isometric qubit encodings of the input and output spaces. Choose a qubit register large enough to accommodate the encoded output, a computational register for $(x, j)$ , and zero padding. On the encoded zero-ancilla input subspace prescribe

J_{in} φ \otimes ∣ 0 ⟩ ⟼ x, j \sum J_{out} M_{x j} φ \otimes ∣ x, j ⟩ \otimes ∣ 0 ⟩_{pad} .

Completeness makes this prescription isometric. Enlarge the zero-ancilla input register if necessary so that input and target are subspaces of the same finite qubit space. Complete orthonormal bases to extend the prescription to a unitary.

Implement the unitary, destructively read the index and padding registers, decode $x$ , delete unreported classical indices by $dropBit$ , and return the encoded quantum output. On the encoded input, the resolved path maps are the encoded $M_{x j}$ , with zero padding fixed. Summing the unreported $j$ gives $Φ_{x}$ . Every operation acts trivially on $B$ , proving the reference claim.

Displayed formulas in this passage

unlabeled · line 2144
$Φ_{x} (ρ) = j \sum M_{x j} ρ M_{x j}^{†}, x, j \sum M_{x j}^{†} M_{x j} = I$
unlabeled · line 2163
$J_{in} φ \otimes ∣ 0 ⟩ ⟼ x, j \sum J_{out} M_{x j} φ \otimes ∣ x, j ⟩ \otimes ∣ 0 ⟩_{pad} .$

Corollary: Finite instrument representation

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