Definition: Five operational requirements

The theory $\mathsf{T}$ consists of:

$(\mathrm{C}{)}^{\star }$: composability.

For disjoint $S,T\subseteq L$, successive readout in either order and joint readout of $S\cup T$ yield the same ordered joint law, with all other names returned. On a positive first branch,

$${{\mathsf{P}}^{\mathsf{S}\cup \mathsf{T}}}_{\psi }(\kappa ,\lambda )={{\mathsf{P}}^{\mathsf{S}}}_{\psi }(\kappa ){{\mathsf{P}}^{\mathsf{T}}}_{{{\mathsf{U}}^{\mathsf{S}}}_{\psi }(\kappa )}(\lambda ).$$

A zero first weight gives zero joint weight. In particular,

$${{\mathsf{P}}^{\mathsf{S}}}_{\psi }={\mathrm{marg}}_S{{\mathsf{P}}^{\mathsf{S}\cup \mathsf{T}}}_{\psi }.$$

Only the classical joint laws, not complete residual instruments, are equated.

$(\mathrm{NS}{)}^{\star }$: no-signaling.

For every bipartition, the terminal classical law of either local program is independent of the other program and of the admissible interleaving. Local programs may allocate, apply gates, read out, use classical control, and return quantum spectators. They exchange neither names nor messages.

$(\mathrm{N}{)}^{\star }$: nullity.

For every $\psi ,S,\kappa$,

$${M^S}_{\kappa }\psi =0⟹{{\mathsf{P}}^{\mathsf{S}}}_{\psi }(\kappa )=0.$$

The experiment reads $S$ and returns its complement. This is a zero-support condition, not a vacuum condition or a Born probability formula.

$(\mathrm{K}{)}^{\star }$: continuity.

At every fixed finite live register and measured subset,

$$[\psi ]⟼{{\mathsf{P}}^{\mathsf{S}}}_{\psi }$$

is continuous from ray space to the finite probability simplex, equivalently in total variation. No continuity of $\mathsf{U}$ is assumed.

$(\mathrm{R}{)}^{\star }$: records.

For every configuration of Definition def:record,

$${\mathrm{Obs}}_{\mathsf{S}}[\psi ,t_{S,u}]={\mathrm{Obs}}_{\mathsf{S}}[\psi ,{\mathrm{defer}}_S(t_{S,u})].$$

The continuation consumes the entire complementary register and does not use the first reported outcome.