Finite-dimensional conventions

All matter, code, and record spaces used in channel statements are finite-dimensional. Write $L(\mathcal{H})$ for the operator space and $D(\mathcal{H})$ for its density operators. A channel is a completely positive, trace-preserving linear map. Its adjoint satisfies

Channel adjoints are unital and completely positive Watrous2018.

A classical register has a distinguished basis. An admissible joint state is block diagonal in that register, not merely diagonal after taking its marginal. Declared classical operations preserve this joint block-diagonal state space. Their individual Kraus operators need not be diagonal; classical permutations are allowed.

For a unital ${}^{\ast }$-subalgebra $\mathcal{A}$, define

Equality of these functionals means equality on the named observable interface.

Our distance conventions are

The trace norm itself has no factor $1/2$. Outcome sets and complete history alphabets are finite unless explicitly stated otherwise.