Approaches to Quantum Gravity

132 F. Markopoulouvertex x ∈ V (Ɣ) a finite-dimensional Hilbert space H(x) and/or a matrix algebraA(H(x)) (or A(x) for short) of operators acting on H(x). It is best to regard thealgebras as the primary objects, but we will not make this distinction here.If two vertices, x and z, are unrelated, their joint state space isH (x ∪ z) = H(x) ⊗ H(y). (9.1)If vertices x and y are related, let us for simplicity say by a single edge e,weshallthink of e as a change of the quantum systems of the source of e into a new set ofquantum systems (the range of e). It is then natural to assign to each e ∈ E(Ɣ) acompletely positive map e : e : A(s(e)) −→ A(r(e)), (9.2)where A(x) is the full matrix algebra on H(x). Completely positive maps are commonlyused to describe evolution of open quantum systems and generally arise asfollows (see, for example, [27]).Let H S be the state space of a quantum system in contact with an environmentH E (here H S is the subgraph space and H E the space of the rest of the graph).The standard characterization of evolution in open quantum systems starts withan initial state in the system space that, together with the state of the environment,undergoes a unitary evolution determined by a Hamiltonian on the compositeHilbert space H = H S ⊗ H E , and this is followed by tracing out the environmentto obtain the final state of the system.The associated evolution map : A(H S ) → A(H S ) between the correspondingmatrix algebras of operators on the respective Hilbert spaces is necessarilycompletely positive (see below) and trace preserving. More generally, the map canhave different domain and range Hilbert spaces. Hence the operational definitionof quantum evolution from a Hilbert space H 1 to H 2 is as follows.Definition 2 Completely positive (CP) operators. A completely positive operator is a linear map : A(H 1 ) −→ A(H 2 ) such that the mapsare positive for all k ≥ 1.id k ⊗ : M k ⊗ A(H 1 ) → M k ⊗ A(H 2 ) (9.3)Here we have written M k for the algebra A(C k ).Consider vertices x, y, z and w in Ɣ. There are several possible connecting paths,such as