# Spatial and polarizing beam splitters in a graphical calculus

Suppose I have four wires, and I tensor product them together

$A \otimes B \otimes C \otimes D$

I pass $A \otimes B$ through a spatial beam splitter

$Spl: A \otimes B \rightarrow A^\prime \otimes B^\prime$

and I pass $C \otimes D$ through a polarizing beam splitter

$Pspl : C \otimes D \rightarrow C^\prime \otimes D^\prime$.

What kind of product do I use to combine $Pspl$ and $Spl$? For instance, can I just tensor them and get

$Spl \otimes Pspl : A \otimes B \otimes C \otimes D \rightarrow A^\prime \otimes B^\prime \otimes C^\prime \otimes D^\prime$?

I guess this doesn't make perfect sense yet as there is no notion of a "wire". In my calculations so far, I am seeing 4 port devices as taking a state on two wires "1,2" and sending it to a state on two other wires "3,4". I recall someone (Phill Scott, Abramsky?) doing something with tensors where the tensor indices were labelled wire inputs/outputs. Upper indices were input and lower indices were outputs. Has anyone else seen that?

I want to do everything in the string diagrams, so I want rules for rewriting diagrams with polarization beam splitters (call it a "P" box) and also regular beam splitters (call it an "S" box). Can anyone help?

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You are looking for the formalism described in the references listed here.

The original article that got this line of research started is

• Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) (arXiv:quant-ph/0402130)

Bob Coecke has been writing several expositions since, each with many further references. For instance

A fairly comprehensive account also of recent developments is in

• Bob Coecke, Ross Duncan, Interacting Quantum Observables: Categorical Algebra and Diagrammatics (arXiv:0906.4725)

More formal discussion of the underlying mathematics is in

• Peter Selinger, Dagger compact closed categories and completely positive maps (web, pdf)

Concerning your question: you can certainly tensor $PSpl$ and $Spl$ as you indicate. That corresponds to combining the two "systems" and the operations on them. It's not clear to me from your question if this is or is not what you want to model.

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