Consistent circuits for indefinite causal order

TL;DR

The paper develops a general method to construct and certify quantum processes with indefinite causal order by extending the quantum circuit formalism to routed graphs equipped with Boolean route constraints. It proves a central theorem: any routed graph that is bi-univocal and whose branch graph has only weak loops yields a valid skeletal superunitary, thereby guaranteeing a consistent process even in the presence of feedback loops. The authors demonstrate the framework by reconstructing several key exotic processes (the quantum switch, the quantum 3-switch, the Grenoble process, and the Lugano process) and argue that all unitarily extendible processes can be obtained via fleshing out a valid routed graph. They also connect the supermap and process-matrix viewpoints, discuss implications for broader classes of higher-order quantum processes, and outline future directions toward a top-down program for decomposing known processes into routed graphs. The work reinstates compositionality for a broad class of indefinite-causal-order processes and suggests a unifying, graph-based route to exploring logical consistency in higher-order quantum dynamics.

Abstract

Over the past decade, a number of quantum processes have been proposed which are logically consistent, yet feature a cyclic causal structure. However, there is no general formal method to construct a process with an exotic causal structure in a way that ensures, and makes clear why, it is consistent. Here we provide such a method, given by an extended circuit formalism. This only requires directed graphs endowed with Boolean matrices, which encode basic constraints on operations. Our framework (a) defines a set of elementary rules for checking the validity of any such graph, (b) provides a way of constructing consistent processes as a circuit from valid graphs, and (c) yields an intuitive interpretation of the causal relations within a process and an explanation of why they do not lead to inconsistencies. We display how several standard examples of exotic processes, including ones that violate causal inequalities, are among the class of processes that can be generated in this way; we conjecture that this class in fact includes all unitarily extendible processes.

Figures (30)

• Figure 1: Diagrammatic representation of a monopartite superunitary. $\mathcal{S}$ is a linear map from unitaries of the form $U: \mathcal{H}_{X^\rmfamily\upshape{\upshape in}} \otimes \mathcal{H}_{A^\rmfamily\upshape{\upshape in}} \rightarrow \mathcal{H}_{X^\rmfamily\upshape{\upshape out}} \otimes \mathcal{H}_{A^\rmfamily\upshape{\upshape out}}$ that maps the ingoing space for any pair of ancillary systems $X^\rmfamily\upshape{\upshape in}$ and $X^\rmfamily\upshape{\upshape out}$ to unitaries of the form $(\mathcal{I} \otimes \mathcal{S})(U): \mathcal{H}_{X^\rmfamily\upshape{\upshape in}} \otimes \mathcal{H}_P \rightarrow \mathcal{H}_{X^\rmfamily\upshape{\upshape out}} \otimes \mathcal{H}_F$. This definition can be extended in an obvious way for supermaps acting on an arbitrary finite number of unitary operators.
• Figure 2: Circuit decomposition of the switch. The left term in the sum projects onto the $\ket{0}$ state of the control and implements Alice's transformation before Bob's. The right term has a similar interpretation. Formally, the wire bent into a 'U' shape can be interpreted as the unnormalised Bell ket $\ket{00}+\ket{11}$, and the upside-down 'U' as the corresponding bra.
• Figure 3: The action of the switch on a pair of unitary operators. The order of implementation of unitaries on the target system and ancillas is coherently controlled.
• Figure 4: Routed circuit decomposition of the switch, using index matching. $W$ is the routed unitary defined in (\ref{['sup channels def']}). The wires bent into 'cup' or 'cap' shapes represent the (unnormalised) perfectly correlated entangled states coecke_kissinger_2017. Overall, the '$i=0$' sectors correspond to the branch where Alice's intervention is implemented before Bob's, while '$i=1$' corresponds to the branch where Bob's intervention is implemented before Alice's. Thus the cycle is constructed from two acyclic components corresponding to definite orders of implementation.
• Figure 5: Skeletal supermap for the switch. The nodes suffer sectorial constraints represented by the index-matching of the input and output wires, making this a routed supermap. The routed circuit for the switch in Figure \ref{['fig:switch routed circuit']} is obtained by inserting unitary transformations into the nodes $P$ and $F$, and the monopartite superunitary (\ref{['comb']}) into $A$ and $B$.

Theorems & Definitions (39)

• Definition 3.1: Indexed and routed graphs
• Definition 3.2: Branched routes
• Definition 3.3: Skeletal supermap associated to a routed graph
• Definition 3.4: Augmenting
• Definition 3.5: Consistent assignment
• Definition 3.6
• Definition 3.7: Branch graph
• Theorem 3.1
• Corollary 3.1
• Conjecture 1