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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