September 12, 2008 — Submitted to Trends in Logic VI The Logic BV and Quantum Causality Rick Blute University of Ottawa http://aix1.uottawa.ca/ ∼ rblute/ Prakash Panangaden McGill University http://www.cs.mcgill.ca/ ∼ prakash/ Lutz Straßburger INRIA Saclay– ˆ Ile-de-France http://www.lix.polytechnique.fr/ ∼ lutz/ Abstract We describe how a logic with commutative and non- commutative connectives can be used for capturing the essence of discrete quantum causal propagation. 1 Causal graphs and locative slices In this note we describe how the kinematics of quantum causal evolution can be captured by the logic BV [2]. The setting is discrete quantum mechanics. We imagine a fi- nite “web” of spacetime points. They are viewed as vertices in a directed acyclic graph (DAG); the edges of the DAG represent causal links mediated by the propagation of mat- ter [1]. The fact that the graph is acyclic captures a basic causality requirement: there are no closed causal trajecto- ries. The DAG represents a discrete approximation to the spacetime on which a quantum system evolves. The graph is technically a dangling graph; there is a set of half edges— in addition to the ordinary edges—divided into two disjoint subsets: the incoming edges and the outgoing edges. An incoming edge has no initial point but has a terminal point, and dually for outgoing edges. A pre-causal graph G consists of a quadruple (V G , B G , I G , O G ), where V G is a set of vertices, B G is a set of directed (binary) edges, I G is a set of incoming edges, and O G is a set of outgoing edges, such that V G , B G , I G , and O G are pairwise disjoint (and finite), and two functions source G : B G ∪O G →V G and target G : B G ∪I G →V G , called source and target, respectively. The elements of the set E G = B G ∪I G ∪O G are called edges. On this set we define the precedence relation ≺ G ⊆E G ×E G as e 1 ≺ e 2 iff target(e 1 )= source(e 2 ) The ancestor relation < G is the transitive closure of ≺ G . If < G is irreflexive we say that G is a causal graph.A slice S in a causal graph G is an anti-chain in ≤ G . A slice does not need to be maximal. To each edge in a causal graph we can associate a Hilbert space H and a density matrix ρ associated with the subsys- tem on that edge. At each vertex we imagine that we have an interaction, which may be any one of the following: sub- systems come together, a subsystem breaks into pieces, a subsystem is subject to a unitary transformation, a subsys- tem is subject to a measurement, or a subsystem is partly discarded. When subsystems come together we form the tensor product of their state spaces. If they have no inter- action we form the tensor product of their density matrices, otherwise we have a unitary operator acting on the com- bined density matrices. When a system breaks apart we can have a single density matrix for all the pieces; if, however, we wish to separate the density matrices of the individual components we compute partial traces, which has the ef- fect of removing information about nonlocal correlations. In fact, the presence of non-local correlations is what dis- tinguishes this from Petri nets. Density matrices can be associated with any slice. If we keep all the data associated with maximal slices then we cannot guarantee that information does not propagate be- tween acausal paths. One solution to guaranteeing causal propagation is to only propagate along the individual edges. In this scheme we would only allow the operators (com- pletely positive maps) at the vertices to act on density ma- trices associated with single edges. This would indeed guar- antee causal propagation but would kill all nonlocal corre- lations. The solution to the problem of ensuring causal evo- lution while preserving important non-local correlations is to work with locative slices, which are defined below. Let G be be a causal graph, and let S⊆E and let v ∈V be a vertex such that target −1 (v) ⊆S . Then the set S ′ = S\target −1 (v) ∪source −1 (v) is called the propagation of S through v. Clearly, if S is a slice, then the propagation of S through v is also a slice. We say a slice S ′ is reachable from a slice S if there are an n ≥ 0 and slices S 0 ,..., S n ⊆E and vertices v 1 ,...,v n ∈ V such that for all i ∈{1,...,n} we have that S i is the propagation of S i−1 through v i , and S = S 0 and S ′ = S n . A slice S in a causal graph G is called locative, if it is reachable from a slice I ′ ⊆I . The point is that if S is locative then its density matrix can be computed without ever comuting partial traces: no information is lost.