In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises, and assignments if you are ready for them. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

Week 4

Lectures

You should need about two hours to watch these

Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. You might want to complete Assignment 1, or submit more Chapter 1 exercises for feedback, before doing Chapter 2 exercises.

  • p.62, Q. 1-2
  • p.70, Q. 1-7

Submit work for Canvas feedback by Monday 22 February for video feedback after Tuesday 23 February. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

Week 5

We will finish Chapter 2.

15% Assignment 2

Based on Chapter 2, should be available some time in Week 4, for submission in Week 7.

Study

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

As soon as I send this I am putting the finishing touches to your first assessment — it might be here by tomorrow, Wednesday 3 February.

Week 3

Lectures

You should need about two hours to watch these

Temperature of a Plate: Jacobi Method (34 minutes)

If you want a deep dive into the temperature of a plate stuff there is a post here which explains — in great detail — how the ‘iterations’ are actually physically meaningful.

Intro to Method of Undetermined Coefficients (24 minutes)

Method of Undetermined Coefficients: Theory (24 minutes)

Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises: you may want to put time into the Week 1 and 2 exercises if you did not do so previously.

  • p.55, Q. 1-4

Submit work for Canvas feedback by Monday 15 February for video feedback after Tuesday 16 February. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

Week 4

We will go deeper into Chapter 2

35%  Assignment 1

Should be released Wednesday 3 February.

Study

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

In theory you are supposed to spend seven hours per week on MATH6040: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH6040 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are three hours of MATH7021 slots on your timetable that you can use if you want. Even with just three hours you should have at least one hour to try and submit exercises for video feedback.

Week 2

Lectures

You should need about two hours to watch these (I recommend 50% extra time for pausing/rewinding)

Vectors Example (20 minutes)

Vector/Cross Product I (36 minutes)

Vector/Cross Product II (29 minutes)

Latest annotated notes here.

Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises. 

  • p.27, Q. 1-13 (you might have some of these done in Week 1)
  • p.38, Q.1-5
  • Additional Exercises, p.38, Q. 6-1

You may need the formula that the area of a triangle is equal to \displaystyle \frac12 ab\sin C.

Submit work for Canvas feedback by Sunday 7 February for video feedback after Monday 8 February. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

Week 3

We will look at the applications of vectors to work and moments

25% Test 1 in Week 5

I will probably design student-number-personalised assessments. Watch this space: official notice in Week 3.

Study

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7016. My recommendation is that you watch the lecture material, then complete the lab, and if there is any time left over do the suggested theory exercises.

It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7016 slots on your timetable but remember that half an hour is taken up with the weekly Q&A. Learners should decide for themselves whether the weekly Q&A is helping their learning.

Week 2

Lectures

There are about 82 minutes of lectures. You could schedule about 2 hours and 10 minutes to watch them and take the notes in your manual. You need this extra time above 82 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A (starting Tuesday 2 February)

(Last year with the strike I recorded the same stuff in a classroom)

The Q&A is dedicated to answering questions but do not hesitate to contact me with questions at any time. My usual modus operandi is to answer emails every morning.

Lab

Once you have watched the lectures you should attempt VBA Lab 2.

Theory Exercises and Q & A

p.34, Q.1-4

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

Week 3

We will do more on Taylor Series and the Euler Method. When we have that done we will look at Huen’s Method.

In VBA we will finish off our Euler Method Lab.

Assessment

I really have not yet thought about assessment but this is what I am thinking at the moment. This is provisional and subject to change.

  1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
  2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
  3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
  4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

Study

Study should consist of

  • doing exercises from the notes
  • completing VBA exercises

Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

Week 2

Lectures

You should need about two and half hours to watch these

Gaussian Elimination IV (34 minutes)

Gaussian Elimination V (14 minutes)

Linear Systems: Traffic Flow (30 minutes)

Linear Systems: Pipe Network (22 minutes)

Latest annotated notes here.

Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises. 

  • p.30, Q. 1-11 (you might have some of these done in Week 1)
  • p.42, Q. 1-3
  • p.48, Q. 1-3
  • Additional Exercises, p.33, Q. 12-22

Submit work for Canvas feedback by Monday 8 February for video feedback after Tuesday 9 February. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

Week 3

We will finish Chapter 1 and make a start to Chapter 2.

35%  Assignment 1

Watch this space — I might release it before I have a definite due date.

Study

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021. We have voted for the OFFLINE model: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

Week 1

Lectures

We had one hour of Live Zoom. This should be recorded in “Zoom” on the left there. Updated annotated notes.here 

You should watch this hour and 23 minutes of lecture video this week:

Gaussian Elimination I (33 minutes)

Gaussian Elimination II (23 minutes)

Gaussian Elimination III (27 minutes)

Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises. 

It might be the case that you need to see more examples (in Week 2 lectures… probably available by Friday evening) before attempting these. On the other hand, if you can do these already you are in a great position in MATH7021.

  • p.30, Q. 1-11
  • Additional Exercises, p.33, Q. 12-22

Submit work for Canvas feedback by Monday 1 February for video feedback after Tuesday 2 February. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

Week 2

We will finish the Gaussian Elimination examples and begin to look at applications of linear systems to traffic and pipe flow.

35%  Assignment 1

I may release Assignment 1 before I know the hand-in date… hopefully there will be some coordination on this quite soon.

Study

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

Student Resources

Please see  Student Resources for information on the Academic Learning Centre, etc.

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH6040. I know this might not be possible for everyone. My recommendation is that you watch the lecture material (n/a in Week 1 if you attended the live lecture), then spend time doing exercises — how long will depend on the…

Delivery Model Vote

Please read here for more on this. Vote here BEFORE 12:00 Thursday 28 January.

Read the rest of this entry »

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7016. My recommendation is that you watch the lecture material (n/a in Week 1 if you attended the live lectures), then complete the lab, and if there is any time left over do the suggested theory exercises.

Week 1

Lectures

We had two hours of Live Zoom. These should be recorded in “Zoom” on the left there.

By briefly looking at a number of examples (many of which we have seen before), we had a review of some central ideas from approximation theory such as approximation, measurement erroraccuracy & precisioniterationconvergencemeshingerror, etc.

We looked at where ordinary differential equations come into Engineering, and we started talking about Euler’s Method.

Lab

Where you find the time to do your VBA Lab 1 is up to you but I recommend using the two hours on your timetable. If you are serious about MATH7016 you might need to spend more than two hours.

Theory Exercises and Q & A

p.17, Q.1-2

Q & A on these and any other MATH7016 topic next Tuesday 12.30 (waiting room open 12.25).

Week 2

We will look at the Euler Method, and then start looking at big \mathcal{O} notation, and Taylor Series.

Once I record the lectures I will put together a Week 2 announcement like this one.

Assessment

I really have not yet thought about assessment but this is what I am thinking at the moment. This is provisional and subject to change.

  1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
  2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
  3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
  4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

Study

Study should consist of

  • doing exercises from the notes
  • completing VBA exercises

Student Resources

Please see the Student Resources  for information on the Academic Learning Centre, etc..

I am currently (slowly) working on an essay/paper where I expand upon the ideas in this talk. In this post I will try and explain in this framework why there is no quantum cyclic group, no quantum S_3, and ask why there is no quantum alternating group.

Quantum Permutations Basics

Let A be a unital \mathrm{C}^*-algebra. We say that a matrix u\in M_N(A) is a magic unitary if each entry is a projection u_{ij}=u_{ij}^2=u_{ij}^*, and each row and column of u is a partition of unity, that is:

\displaystyle \sum_ku_{ik}=\sum_k u_{kj}=1_A.

It is necessarily the case (but not for *-algebras) that elements along the same row or column are orthogonal:

u_{ij}u_{ik}=\delta_{j,k}u_{ij} and u_{ij}u_{k j}=\delta_{i,k}u_{ij}.

Shuzou Wang defined the algebra of continuous functions on the quantum permutation group on N symbols to be the universal \mathrm{C}^*-algebra C(S_N^+) generated by an N\times N magic unitary u. Together with (leaning heavily on the universal property) the *-homomorphism:

\displaystyle \Delta:C(S_N^+)\rightarrow C(S_N^+)\underset{\min}{\otimes}C(S_N^+), u_{ij}\mapsto \sum_{k=1}^N u_{ik}\otimes u_{kj},

and the fact that u and (u)^t are invertible (u^{-1}=u^t)), the quantum permutation group S_N^+ is a compact matrix quantum group.

Any compact matrix quantum group generated by a magic unitary is a quantum permutation group in that it is a quantum subgroup of the quantum permutation group. There are finite quantum groups (finite dimensional algebra of functions) which are not quantum permutation groups and so Cayley’s Theorem does not hold for quantum groups. I think this is because we can have quantum groups which act on algebras such as M_N(\mathbb{C}) rather than \mathbb{C}^N — the algebra of functions equivalent of the finite set \{1,2,\dots,N\}.

This is all basic for quantum group theorists and probably unmotivated for everyone else. There are traditional motivations as to why such objects should be considered algebras of functions on quantum groups:

  • find a presentation of an algebra of continuous functions on a group, C(G), as a commutative universal \mathrm{C}^*-algebra. Study the the same object liberated by dropping commutativity. Call this the quantum or free version of G, G^+.
  • quotient C(S_N^+) by the commutator ideal, that is we look at the commutative \mathrm{C}^*-algebra generated by an N\times N magic unitary. It is isomorphic to F(S_N), the algebra of functions on (classical) S_N.
  • every commutative algebra of continuous functions on a compact matrix quantum group is the algebra of functions on a (classical) compact matrix group, etc.

Here I want to take a very different direction which while motivationally rich might be mathematically poor.

Weaver Philosophy

Take a quantum permutation group \mathbb{G} and represent the algebra of functions as bounded operators on a Hilbert space \mathsf{H}. Consider a norm-one element \varsigma\in P(\mathsf{H}) as a quantum permutation. We study the properties of the quantum permutation by making a series of measurements using self-adjoint elements of C(\mathbb{G}).

Suppose we have a finite-spectrum, self-adjoint measurement f\in C(\mathbb{G})\subset B(\mathsf{H}). It’s spectral decomposition gives a partition of unity (p^{f_i})_{i=1}^{|\sigma(f)|}. The measurement of \varsigma with f gives the value f_i with probability:

\displaystyle \mathbb{P}[f=f_i\,|\,\varsigma]=\langle\varsigma,p^{f_i}\varsigma\rangle=\|p^{f_i}\varsigma\|^2,

and we have the expectation:

\displaystyle \mathbb{E}[f\,|\,\varsigma]=\langle\varsigma,f\varsigma\rangle.

What happens if the measurement of \varsigma with f yields f=f_i (which can only happen if p^{f_i}\varsigma\neq 0)? Then we have some wavefunction collapse of

\displaystyle \varsigma\mapsto p^{f_i}\varsigma\equiv \frac{p^{f_i}\varsigma}{\|p^{f_i}\varsigma\|}\in P(\mathsf{H}).

Now we can keep playing the game by taking further measurements. Notationally it is easier to describe what is happening if we work with projections (but straightforward to see what happens with finite-spectrum measurements). At this point let me quote from the essay/paper under preparation:

Suppose that the “event” p=\theta_1 has been observed so that the state is now p^{\theta_1}(\psi)\in P(\mathsf{H}). Note this is only possible if p=\theta_1 is non-null in the sense that

\displaystyle \mathbb{P}[p=\theta_1\,|\,\psi]=\langle\psi,p^\theta(\psi)\rangle\neq 0.

The probability that measurement produces q=\theta_2, and p^{\theta_1}(\psi)\mapsto q^{\theta_2}p^{\theta_1}(\psi)\in P(\mathsf{H}), is:

\displaystyle \mathbb{P}\left[q=\theta_2\,|\,p^{\theta_1}(\psi)\right]:=\left\langle \frac{p^{\theta_1}(\psi)}{\|p^{\theta_1}(\psi)\|},q^{\theta_2}\left(\frac{p^{\theta_1}(\psi)}{\|p^{\theta_1}(\psi)\|}\right)\right\rangle=\frac{\langle p^{\theta_1}(\psi),q^{\theta_2}(p^{\theta_1}(\psi))\rangle}{\|p^{\theta^1}(\psi)\|^2}.

Define now the event \left((q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right), said “given the state \psi, q is measured to be \theta_2 after p is measured to be \theta_1“. Assuming that p=\theta_1 is non-null, using the expression above a probability can be ascribed to this event:

\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]:=\mathbb{P}[p=\theta_1\,|,\psi]\cdot \mathbb{P}[q=\theta_2\,|\,p^{\theta_1}(\psi)]
\displaystyle =\langle\psi,p^{\theta_1}(\psi)\rangle\frac{\langle p^{\theta_1}(\psi),q^{\theta_2}(p^{\theta_1}(\psi))\rangle}{\|p^{\theta^1}(\psi)\|^2}
=\|q^{\theta_2}p^{\theta_1}\psi\|^2.

Inductively, for a finite number of projections \{p_i\}_{i=1}^n, and \theta_i\in{0,1}:

\displaystyle \mathbb{P}\left[(p_n=\theta_n)\succ\cdots \succ(p_1=\theta_1)\,|\,\psi\right]=\|p_n^{\theta_n}\cdots p_1^{\theta_1}\psi\|^2.

In general, pq\neq qp and so

\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]\neq \mathbb{P}\left[(p=\theta_1)\succ (q=\theta_1)\,|\,\psi\right],

and this helps interpret that q and p are not simultaneously observable. However the sequential projection measurement q\succ p is “observable” in the sense that it resembles random variables with values in \{0,1\}^2. Inductively the sequential projection measurement p_n\succ \cdots\succ p_1 resembles a \{0,1\}^n-valued random variable, and

\displaystyle \mathbb{P}[p_n\succ \cdots\succ p_1=(\theta_n,\dots,\theta_1)\,|\,\psi]=\|p_n\cdots p_1(\psi)\|^2.

If p and q do commute, they share an orthonormal eigenbasis, and it can be interpreted that they can “agree” on what they “see” when they “look” at \mathsf{H}, and can thus be determined simultaneously. Alternatively, if they commute then the distributions of q\succ p and p\succ q are equal in the sense that

\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]= \mathbb{P}\left[(p=\theta_1)\succ (q=\theta_1)\,|\,\psi\right],

it doesn’t matter what order they are measured in, the outputs of the measurements can be multiplied together, and this observable can be called pq=qp.

Consider the (classical) permutation group S_N or moreover its algebra of functions F(S_N). The elements of F(S_N) can be represented as bounded operators on \ell^2(S_N), and the algebra is generated by a magic unitary u^{S_N}\in M_N(B(\ell^2(S_N))) where:

u_{ij}^{S_N}(e_\sigma)=\mathbf{1}_{j\rightarrow i}(e_\sigma)e_{\sigma}.

Here \mathbf{1}_{j\rightarrow i}\in F(S_N) (‘unrepresented’) that asks of \sigma… do you send j\rightarrow i? One for yes, zero for no.

Recall that the product of commuting projections is a projection, and so as F(S_N) is commutative, products such as:

\displaystyle p_\sigma:=\prod_{j=1}^Nu_{\sigma(j)j}^{S_N},

There are, of, course, N! such projections, they form a partition of unity themselves, and thus we can build a measurement that will identify a random permutation \varsigma\in P(\ell^2(S_N)) and leave it equal to some e_\sigma after measurement. This is the essence of classical… all we have to do is enumerate n:S_N\rightarrow \{1,\dots,N!\} and measure using:

\displaystyle f=\sum_{\sigma\in S_N}n(\sigma)p_{\sigma}.

A quantum permutation meanwhile is impossible to pin down in such a way. As an example, consider the Kac-Paljutkin quantum group of order eight which can be represented as F(\mathfrak{G}_0)\subset B(\mathbb{C}^6). Take \varsigma=e_5\in \mathbb{C}^6. Then

\displaystyle\mathbb{P}[(\varsigma(1)=4)\succ(\varsigma(3)=1)\succ(\varsigma(1)=3)]=\frac{1}{8}.

If you think for a moment this cannot happen classically, and the issue is that we cannot know simultaneously if \varsigma(1)=3 and \varsigma(3)=1… and if we cannot know this simultaneously we cannot pin down \varsigma to a single element of S_N.

No Quantum Cyclic Group

Suppose that \varsigma\in \mathsf{H} is a quantum permutation (in S_N^+). We can measure where the quantum permutation sends, say, one to. We simply form the self-adjoint element:

\displaystyle x(1)=\sum_{k=1}^Nku_{k1}.

The measurement will produce some k\in \{1,\dots,N\}… but if \varsigma is supposed to represent some “quantum cyclic permutation” then we already know the values of \varsigma(2),\dots,\varsigma(N) from \varsigma(1)=k, and so, after measurement,

u_{k1}\varsigma \in \bigcap_{m=1}^N \text{ran}(u_{m+k-1,m}), u_{k1}\varsigma\equiv k-1\in\mathbb{Z}_N.

The significance of the intersection is that whatever representation of C(S_N^+) we have, we find these subspaces to be C(S_N^+)-invariant, and can be taken to be one-dimensional.

I believe this explains why there is no quantum cyclic group.

Question 1

Can we use a similar argument to show that there is no quantum version of any abelian group? Perhaps using F(G\times H)=F(G)\otimes F(H) together with the structure theorem for finite abelian groups?

No Quantum S_3

Let C(S_3^+) be represented as bounded operators on a Hilbert space \mathsf{H}. Let \varsigma\in P(\mathsf{H}). Consider the random variable

x(1)=u_{11}+2u_{21}+3u_{31}.

Assume without loss of generality that u_{31}\varsigma\neq0 then measuring \varsigma with x(1) gives x(1)\varsigma=3 with probability \langle\varsigma,u_{31}\varsigma\rangle, and the quantum permutation projects to:

\displaystyle \frac{u_{31}\varsigma}{\|u_{31}\varsigma\|}\in P(\mathsf{H}).

Now consider (for any \varsigma\in P(\mathsf{H}), using the fact that u_{21}u_{31}=0=u_{32}u_{31} and the rows and columns of u are partitions of unity:

u_{31}\varsigma=(u_{12}+u_{22}+u_{32})u_{31}\varsigma=(u_{21}+u_{22}+u_{23})u_{31}\varsigma

\Rightarrow u_{12}u_{31}\varsigma=u_{23}u_{31}\varsigma (*)

Now suppose, again without loss of generality, that measurement of u_{31}\varsigma\in P(\mathsf{H}) with x(2)=u_{12}+2u_{22}+3u_{33} produces x(2)u_{31}\varsigma=1, then we have projection to u_{12}u_{31}\varsigma\in P(\mathsf{H}). Now let us find the Birkhoff slice of this. First of all, as x(2)=1 has just been observed it looks like:

\Phi(u_{12}u_{31}\varsigma)=\left[\begin{array}{ccc}0 & 1 & 0 \\ \ast & 0 & \ast \\ \ast & 0 & \ast \end{array}\right]

In light of (*), let us find \Phi(u_{12}u_{31}\varsigma)_{23}. First let us normalise correctly to

\displaystyle \frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|}

So

\displaystyle\Phi(u_{12}u_{31}\varsigma)_{23}=\left\langle\frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|},u_{23}\frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|}\right\rangle

Now use (*):

\displaystyle\Phi(u_{12}u_{31}\varsigma)_{23}=\left\langle\frac{u_{23}u_{31}\varsigma}{\|u_{23}u_{31}\varsigma\|},u_{23}\frac{u_{23}u_{31}\varsigma}{\|u_{23}u_{31}\varsigma\|}\right\rangle=1

\displaystyle \Rightarrow \Phi(u_{12}u_{31}\varsigma)=\left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ \Phi(u_{12}u_{31}\varsigma)_{31} & 0 & 0 \end{array}\right],

and as \Phi maps to doubly stochastic matrices we find that \Phi(u_{12}u_{31}\varsigma) is equal to the permutation matrix (132).

Not convincing? Fair enough, here is proper proof inspired by the above:

Let us show u_{11}u_{22}=u_{22}u_{11}. Fix a Hilbert space representation C(S_3^+)\subset B(\mathsf{H}) and let \varsigma\in\mathsf{H}.

The basic idea of the proof is, as above, to realise that once a quantum permutation \varsigma is observed sending, say, 3\rightarrow 2, the fates of 2 and 1 are entangled: if you see 2\rightarrow 3 you know that 1\rightarrow 1.

This is the conceptional side of the proof.

Consider u_{23}\varsigma which is equal to both:

(u_{11}+u_{21}+u_{31})u_{23}\varsigma=(u_{31}+u_{32}+u_{33})u_{23}\varsigma\Rightarrow u_{11}u_{23}\varsigma=u_{32}u_{23}\varsigma.

This is the manifestation of, if you know 3\rightarrow 2, then two and one are entangled. Similarly we can show that u_{22}u_{13}\varsigma=u_{31}u_{13}\varsigma and u_{22}u_{33}=u_{11}u_{33}.

Now write

\varsigma=u_{13}\varsigma+u_{23}\varsigma+u_{33}\varsigma

\Rightarrow u_{11}\varsigma=u_{11}u_{23}\varsigma+u_{11}u_{33}\varsigma=u_{32}u_{23}\varsigma+u_{22}u_{33}\varsigma

\Rightarrow u_{22}u_{11}\varsigma=u_{22}u_{33}\varsigma.

Similarly,

u_{22}\varsigma=u_{22}u_{13}\varsigma+u_{22}u_{33}\varsigma=u_{31}u_{13}\varsigma+u_{22}u_{33}\varsigma

\Rightarrow u_{11}u_{22}\varsigma=u_{11}u_{22}u_{33}\varsigma=u_{11}u_{11}u_{33}\varsigma=u_{11}u_{33}\varsigma=u_{22}u_{33}\varsigma

Which is equal to u_{22}u_{11}x, that is u_{11} and u_{22} commute.

Question 2

Is it true that if every quantum permutation in a \mathsf{H} can be fully described using some combination of u_{ij}-measurements, then the quantum permutation group is classical? I believe this to be true.

Quantum Alternating Group

Freslon, Teyssier, and Wang state that there is no quantum alternating group. Can we use the ideas from above to explain why this is so? Perhaps for A_4.

A possible plan of attack is to use the number of fixed points, \text{tr}(u), and perhaps show that \text{tr}(u) commutes with x(1). If you know these two simultaneously you nearly know the permutation. Just for completeness let us do this with (\text{tr}(u),x(1)):

tr(u)\x(1)1234
0(12)(34)(13)(24)(14)(23)
1(234),(243)(134),(143)(124),(142)(123),(132)
4e

The problem is that we cannot assume that that the spectrum of \text{tr}(u) is \{0,1,4\}, and, euh, the obvious fact that it doesn’t actually work.

What is more promising is

x(1)\x(2)1234
1e(234)(243)
2(12)(34)(123)(124)
3(132)(134)(13)(24)
4(142)(143)(14)(23)

However while the spectrums of x(1) and x(2) are cool (both in \{1,2,3,4\}), they do not commute.

Question 3

Are there some measurements that can identify an element of A_4 and via a positive answer to Question 3 explain why there is no quantum A_4? Can this be generalised to A_n.

Outlook

Week 9

There is only a mini-chapter on examples of (real) functions. You should spend the rest of the time revising Chapters 3 and 4 (i.e. doing exercises). Some of you might have to catch up on the Week 8 lectures.

Week 10: 14 to 20 December.

There will be feedback on submitted exercises Wednesday, Thursday, and Friday.

I plan that we will have a number of Zoom tutorials (these are provisional):

  • Tuesday 14:00
  • Tuesday 15:00 C-X only
  • Wednesday 12:00 C-Y only
  • Thursday 12:00

There is a small chance that we will also have Monday 14:00 Zoom but will not know until closer to the day.

Week 11 – 40% Test 3

The final, 40% Test 3 on Functions and Algebra (chapters 3-5) will take place (provisionally) at 18:00-19:30 (perhaps with additional upload time) on Monday 21 December. There will be no more feedback on submitted exercises but there might be a 14:00 Zoom before the final Test.

30% Test 2 – Results

I had said that my aim was to have these to you before Wednesday 9 December. However there is a possibility now that the Test 2 results will not be released until after you sit Test 3. This is outside my control (watch this space).

Week 9

Lectures

Schedule about an hour and 20 minutes for these:

Exercises

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