• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

174 submissions , 137 unreviewed
4,308 questions , 1,640 unanswered
5,089 answers , 21,602 comments
1,470 users with positive rep
635 active unimported users
More ...

  Spatial and polarizing beam splitters in a graphical calculus

+ 4 like - 0 dislike

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?

This post has been migrated from (A51.SE)
asked Feb 9, 2012 in Theoretical Physics by user442920 (90 points) [ no revision ]
retagged Apr 19, 2014 by dimension10

1 Answer

+ 3 like - 0 dislike

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.

This post has been migrated from (A51.SE)
answered Feb 10, 2012 by Urs Schreiber (6,015 points) [ no revision ]

Please log in or register to answer this question.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights