Abstract
The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input systemAto output systemB, systemAcannot influence systemB. Conversely, given a unitaryUwith a no-influence relation from inputAto outputB, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition ofUwith no path fromAtoB. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evidentsimultaneously. To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. Acausally faithfulextended circuit decomposition, representing a unitaryU, is then one for which there is a path from an inputAto an outputBif and only if there actually is influence fromAtoBinU. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.
Highlights
Understanding causal structure in quantum theory is important to the foundations of quantum theory, as well as to applications in the field of quantum information science
Each of the examples discussed so far has the feature that if the assumed causal constraints are the only ones that the respective unitary transformation satisfies, the causal structure of the unitary transformation is made explicit in a circuit diagram, such that in the diagram, there is a path from a given input to a given output if and only if that input has a causal influence on that output
Given an extended circuit diagram representing a unitary U : HA1 ⊗ ... ⊗ HAn → HB1 ⊗ ... ⊗ HBk, let the diagram be causally faithful if the following holds: there is no path in the diagram from Ai to Bj if and only if Ai Bj in U
Summary
Understanding causal structure in quantum theory is important to the foundations of quantum theory, as well as to applications in the field of quantum information science. Circuit diagrams built from sequential and tensor product composition of unitary transformations do not suffice to understand causal structure. Referring to a unitary transformation with n input and k output subsystems as being of type (n, k), we provide causally faithful extended circuit decompositions for all unitaries of type (n, k) with n ≤ 3, all of type (n, k) with k ≤ 3, and for a range of type (4, 4) cases. That a unitary can have, and the application of the results to broken unitary circuits (or unitary combs)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
