Completeness of Graphical Languages for Mixed States Quantum Mechanics

Complete ZX-calculi for the stabiliser fragment in odd prime dimensions

“…In this last section we use the work of [Car+19] to extend our completeness result to the mixed-state case. This allows us to provide a clear connection to the matricial formalism introduced at the beginning of this paper in link with the representation of the metaplectic group.…”

“…We now extend our completeness result form Stab p to CPM(Stab p ), the category of completely positive maps corresponding to mixed state stabiliser quantum mechanics, see [Sel07; CK17] for a formal definition. We will rely on the discard construction of [Car+19] to define a graphical language (ZX Stab p ) . It consists in adding to the equational theory one generator, the discard : 1 → 0 and equations stating that this generator erases all isometries.…”

The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

“…In [5], a construction is given to extend any †-compact PROP for pure quantum mechanics to another †-compact PROP for quantum mechanics with environment. This new formalism can also be understood as the previous one, but where on top of it, one can discard the qubits.…”

“…As you can see, this rule is not easy to apply. Thankfully, the last part of [5] is devoted to showing that the big meta rule can sometimes be replaced by a few small ones. The idea is that, in some cases (in particular in the Clifford fragment), all the isometries can be generated from a finite set of generators.…”

“…In Section 7, we enrich the whole formalism using the discard construction [5], so as to be able to represent completely positive maps, as well as the operator of partial trace. Again, one can consider the Clifford fragment of this formalism.…”