Quantum networks theory

Abstract

The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates in a robust manner. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the neighbourhood and partitioning between systems become both quantum, dynamical, and logical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.

Figures (11)

• Figure 1: Necessity of the name algebra. Left: naming vertices is necessary in order to track alignment across quantum superpositions. Right/grey: A quantum evolution may split $u$ into $u.l$ and $u.r$. As the inverse evolution merges them back we need $( u.l\lor u.r) =u$. Right/blue: The inverse quantum evolution may also merge vertices $u$ and $v$ into $( u\lor v)$. As the forward evolution splits them back we need $( u\lor v) .l=u$ and $( u\lor v) .r=v$.
• Figure 2: Tensor decomposition operation on a superposition of geometries. Top: Connectivity is encoded within the names of the nodes, e.g. $\raisebox{-0.5pt}{\castlinghyphen} x$ being connected to node $x \vee \raisebox{-0.5pt}{\castlinghyphen} y$ in the first branch of the superposition. In this first branch node $y$ lies within radius $1$ of node $\raisebox{-0.5pt}{\castlinghyphen} x$, in the other they are disconnected. Bottom: As this ${\hbox{\cr$\rightharpoonup$\cr$\chi$\cr}} {\hbox{\cr$\rightharpoonup$\cr$\chi$\cr}} {\hbox{\cr[.9]{$\rightharpoonup$}\cr$\chi$\cr}} {\hbox{\cr[.8]{$\rightharpoonup$}\cr$\chi$\cr}} ^1:= {\hbox{\cr$\rightharpoonup$\cr$\zeta$\cr}} {\hbox{\cr$\rightharpoonup$\cr$\zeta$\cr}} {\hbox{\cr[.9]{$\rightharpoonup$}\cr$\zeta$\cr}} {\hbox{\cr[.8]{$\rightharpoonup$}\cr$\zeta$\cr}} _{\raisebox{-0.5pt}{\castlinghyphen} x}^{2}$ selects the oriented radius two neighbours of $\raisebox{-0.5pt}{\castlinghyphen} x$, node $y$ finds itself in a superposition of falling left or right of the tensor.
• Figure 3: Graphs. Left: A system with state 'white' and name $u=(( 3.l\ \lor 8.rl) \lor \raisebox{-0.5pt}{\castlinghyphen} 2)$. Right: A system with state 'black' and name $v=( 2\lor 4)$. Middle/grey: Here we decided to interprete $u.r=\raisebox{-0.5pt}{\castlinghyphen} 2$ and $v.l=2$ as the presence of an unoriented edge $\{u,v\}$. Middle/blue: We could have chosen to interprete it as an oriented edge $( u,v)$ instead. Middle: In both cases, geometry is derived from relative information that is already present within systems, and which is invariant under renamings.
• Figure 4: Comprehension of restrictions $\zeta \ \sqsubseteq \ \chi$ demands condition Eq. \ref{['eq:comprehension']}, which states that for any $G$, $H$, equality outside the small restriction $\zeta$ (i.e. whether $G_{\overline{\zeta }} =H_{\overline{\zeta }}$) may be decomposed as both equality outside $\zeta$ but inside $\chi$ (i.e. whether $G_{\chi \overline{\zeta }} =H_{\chi \overline{\zeta }}$), and equality outside $\chi$ (i.e. whether $G_{\overline{\chi }} =H_{\overline{\chi }}$). Condition Eq. \ref{['eq:comprehension']} may fail if a difference lying within $\zeta$ influences the way $\chi$ partitions the outside of $\zeta$. The condition holds in most relevant cases as shown by Prop. \ref{['prop:npcomprehension']}. It is needed to establish Lem. \ref{['lem:tracetrace']}.
• Figure 5: Generalized partial trace. Across figures $v:=y\lor \raisebox{-0.5pt}{\castlinghyphen} z$, restriction $\zeta _{v}$ retains vertex $v$. Top: the ket and bra do not coincide on the complement of the neighbourhood, this goes to zero. Middle: the ket and bra coincide beyond first neighours, this goes to the restriction of the ket and bra on first neighbours. Below: with oriented edges, the neighbours are those which can signal to $v$. Overall: the question of what to do with edges of the frontier zone does not arise, as edges are derived information.

Theorems & Definitions (66)

• Definition 1: Name algebra
• Definition 2: Graphs
• Definition 3: Induced edges
• Definition 4: Renaming and renaming-invariance
• Definition 5: Name-preservation
• Definition 6: Restrictions, partial trace, comprehension
• Example 1: Namewise restriction
• Example 2: Statewise restriction
• Example 3: A pointwise restriction
• Example 4: Marked restriction