Manuscript X-K-ST@1 · X-K-ST.tex · section · 93c39c0f

Typed concatenation

The multi-instrument composition law › Typed concatenation · lines 826–862

X-K-ST@1#eq:typedcomposition

For a nonadaptive list of complete stage channels, with the full history and memory included in the successive spaces,

Γ_{j} : A_{j - 1} ⟶ A_{j}, j = 1, \dots, m,

the complete channel is

C = Γ_{m} \circ \dots \circ Γ_{1} .

For compatible cut indices $i \leq j \leq k$ , define

T (k, i) = Γ_{k} \dots Γ_{i + 1} .

Then

T (k, j) T (j, i) = T (k, i) .

This is an operational channel identity between specified cuts.

A readout of $C_{1}$ , a new supply $E_{2}$ , and a readout of $C_{2}$ can occur in a product such as

M_{C_{2}} E_{2} V M_{C_{1}} .

By contrast, $M_{C} M_{C}$ is not well-typed on the destructive interfaces. The first map has no quantum $C$ -output En09.

Displayed formulas in this passage