MS Quantum Katas - Measurements Task 2.3 - problems in understanding the solution

3

I am referring to the MS Quantum Katas, Measurements, 2.3 Peres/Wooters game I have big problems to understand the solution. The task is defined as:

--begin

Input: A qubit which is guaranteed to be in one of the three states:

  • $|A\rangle = \frac{1}{\sqrt{2}} \big( |0\rangle + |1\rangle \big)$,
  • $|B\rangle = \frac{1}{\sqrt{2}} \big( |0\rangle + \omega |1\rangle \big)$,
  • $|C\rangle = \frac{1}{\sqrt{2}} \big( |0\rangle + \omega^2 |1\rangle \big)$,

Here $\omega = e^{2i \pi/ 3}$.

Output:

  • 1 or 2 if the qubit was in the $|A\rangle$ state,
  • 0 or 2 if the qubit was in the $|B\rangle$ state,
  • 0 or 1 if the qubit was in the $|C\rangle$ state.

You are never allowed to give an incorrect answer. Your solution will be called multiple times, with one of the states picked with equal probability every time.

The state of the qubit at the end of the operation does not matter.

--end

The solution is described as:

Solution

The task is a simple game inspired by a quantum detection problem due to Holevo[1] and Peres/Wootters[2]. In the game, a player A thinks of a number (0,1 or 2) and the opponent, player B, tries to guess any number but the one chosen by player A.

Classically, if you just made a guess, you'd have to ask two questions to be right $100\%$ of the time. If instead, player A prepares a qubit with 0, 1, or 2 encoded into three single qubit states that are at an angle of 120 degrees with respect to each other and then hands the state to the opponent, then player B can apply a Positive Operator Valued Measure (POVM) consisting of 3 states that are perpendicular to the states chosen by player A. It can be shown that this allows B to be right $100\%$ of the time with only 1 measurement, which is something that is not achievable with a von Neumann measurement on 1 qubit. See also Peres[3, chapter 9.6] for a nice description of the optimal POVM.

Next, we address how we can implement the mentioned POVM by way of a von Neumann measurement, and then how to implement said von Neumann measurement in Q#. First, we note that the POVM elements are given by the columns of the following matrix:

$$M = \frac{1}{\sqrt{2}}\left(\begin{array}{rrr} 1 & 1 & 1 \\ 1 & \omega & \omega^2 \end{array} \right)$$

where $\omega = e^{2 \pi i/3}$ denotes a primitive $3$rd root of unity. Our task will be to implement the rank 1 POVM given by the columns of $M$ via a von Neumann measurement. This can be done by "embedding" $M$ into a larger unitary matrix (taking complex conjugates and transposed):

$$M' = \frac{1}{\sqrt{2}}\left(\begin{array}{cccc} 1 & -1 & 1 & 0 \\ 1 & -\omega^2 & \omega & 0 \\ 1 & -\omega & \omega^2 & 0 \\ 0 & 0 & 0 & -i \end{array} \right)$$

Notice that applying $M'$ to input states given by column $i$ of $M$ (padded with two zeros to make it a vector of length $4$), where $i=0, 1, 2$ will never return the label $i$ as the corresponding vectors are perpendicular.

We are therefore left with the problem of implementing $M'$ as a sequence of elementary quantum gates. Notice that

$$M' \cdot {\rm diag}(1,-1,1,-1) = M' \cdot (\mathbf{1}_2 \otimes Z) = \frac{1}{\sqrt{2}}\left(\begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & \omega^2 & \omega & 0 \\ 1 & \omega & \omega^2 & 0 \\ 0 & 0 & 0 & i \end{array} \right)$$

...
--end

I have several questions / problems here.

  1. What exactly is a "von-Neumann" measurement? Is it just a measurement in the standard basis?

  2. I understand how to get M:
    A POVM M with ${E_0, E_1, E_2}, E_k = |\psi_k\rangle\langle\psi_k|, |\psi_k\rangle=1/sqrt(2)(|0\rangle+\omega^k|1\rangle)$ results in $M=(\psi_0 \psi_1 \psi_2) = \frac{1}{\sqrt{2}}\left(\begin{array}{rrr} 1 & 1 & 1 \\ 1 & \omega & \omega^2 \end{array}\right)$

Next step is

Our task will be to implement the rank 1 POVM given by the columns of M via a von Neumann measurement. This can be done by "embedding" M into a larger unitary matrix (taking complex conjugates and transposed)

What does this embedding mean exactly, what is the math behind? I only found one article about "Emdedded Transforms" (https://books.google.de/books?id=pefvCAAAQBAJ&pg=PA104&lpg=PA104&dq=embedding+matrix+into+larger+unitary+matrix&source=bl&ots=2apFdXFYNd&sig=ACfU3U3JCtj7GgJimx5cgCcKE_FrMUx_4Q&hl=de&sa=X&ved=2ahUKEwjo-K3EipHpAhW8UxUIHVuyDtYQ6AEwCXoECAgQAQ#v=onepage&q=embedding%20matrix%20into%20larger%20unitary%20matrix&f=false), but this is valid only for square matrices.

  1. The next step in the solution is:

Notice that applying $M'$ to input states given by column $i$ of $M$ (padded with two zeros to make it a vector of length $4$), where $i=0, 1, 2$ will never return the label $i$ as the corresponding vectors are perpendicular. We are therefore left with the problem of implementing $M'$ as a sequence of elementary quantum gates.

What does this mean? I don't understand the sense of this.

Thanks in advance, Markus

mbuchberger1967

Posted 2020-05-02T17:21:13.267

Reputation: 105

This is probably not an official solution because the details have mistakes. For example, $M'$ isn't unitary and $M$ can't be embedded to a unitary because the first row has norm >1. – Danylo Y – 2020-05-04T07:09:39.247

Ah, It looks like this is what you were trying to retype here (problem B2) https://assets.codeforces.com/rounds/1116/contest-editorial.pdf You've missed some essential parts.

– Danylo Y – 2020-05-04T09:22:49.363

Actually I was referring to (https://render.githubusercontent.com/view/Workbook_Measurements_Part2.ipynb#peres-wooters-game), but this is very similar to your link (assets.codeforces.com/rounds/1116/contest-editorial.pdf) except for the normalization factor. Can you outline what I missed?

– mbuchberger1967 – 2020-05-04T22:33:56.597

Your link isn't working for me. Just read carefully my link. That solution is correct and has all the details. – Danylo Y – 2020-05-04T23:25:21.910

Answers

1

Von Neumann measurement is a measurement that corresponds to an orthonormal basis, not necessary standard. Though it's related to the standard by a unitary transformation.

As for the solution, at first you need to construct a POVM from the vectors orthogonal to those $|A\rangle, |B\rangle, |C\rangle$. For example, you can take $$ |E_0\rangle = \frac{1}{\sqrt{2}} \big( |0\rangle - |1\rangle \big), $$ $$ |E_1\rangle = \frac{1}{\sqrt{2}} \big( |0\rangle - \omega |1\rangle \big), $$ $$ |E_2\rangle = \frac{1}{\sqrt{2}} \big( |0\rangle - \omega^2 |1\rangle \big). $$

You can verify that it's indeed a POVM, i.e. $\sum_i |E_i\rangle\langle E_i| = \frac{3}{2}I$. Not any set of vectors is suitable for a POVM.

So, if the result of the corresponding POVM measurement is $E_1$ (label 1), then the input state certainly wasn't $|B\rangle$ and you return 1 as an answer $-$ this will satisfy the output requirement.

The corresponding POVM matrix is
$$ M = \frac{1}{\sqrt{3}}\left(\begin{array}{rrr} 1 & 1 & 1 \\ -1 & -\omega & -\omega^2 \end{array} \right) $$ Notice that we've renormalized vectors, i.e. $|E_i'\rangle = \sqrt{\frac{2}{3}}|E_i\rangle$, to satisfy $\sum_i |E_i'\rangle\langle E_i'| = I$.

To construct von Neumann measurement we must find "embedding" of this $M$ into some unitary matrix $M^\prime$. Embedding means that $M$ is just a submatrix of the matrix $M^\prime$.

You can take $$ M' = \frac{1}{\sqrt{3}}\left(\begin{array}{cccc} 1 & 1 & 1 & 0 \\ -1 & -\omega & -\omega^2 & 0 \\ 1 & \omega^2 & \omega & 0 \\ 0 & 0 & 0 & \sqrt{3} \end{array} \right) $$ Here the embedding is obvious - $M$ is the top left corner.

In a unitary matrix columns form an orthonormal basis - this is our von Neumann measurement. Our embedding means that the input state to this measurement is $|\psi\rangle \otimes |0\rangle$, where $|\psi\rangle$ is the given state which is either $|A\rangle,|B\rangle$ or $|C\rangle$.

How it corresponds to a POVM measurement $M$? In this case trivially $-$ label $i$ of $M'$ corresponds to the label $i$ of $M$ if $i=0,1,2$. The result labelled $i=3$ can't happen because $|\psi\rangle \otimes |0\rangle$ is orthogonal to $|11\rangle$.

Danylo Y

Posted 2020-05-02T17:21:13.267

Reputation: 3 940

Thank you so much for your answer and for taking the time! von_Neumann measurement is clarified now :) Your aswer makes a lot of sense for me, but it raises also some questions:

  1. You are choosing $$M = \frac{1}{\sqrt{3}}\left(\begin{array}{rrr}1 & 1 & 1 \ -1 & -\omega & -\omega^2 \end{array}\right)$$.

But in the solution of the MS Kata the POVM is $$ M = \frac{1}{\sqrt{2}}left(\begin{array}{rrr}1 & 1 & 1 \ 1 & \omega & \omega^2 \end{array}\right) $$ which is quite different. Can both solutions be correct or is only one of them correct? – mbuchberger1967 – 2020-05-03T20:56:07.413

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  • You embed M into M' by putting it in the top left. How do you determine the remaining values? Just somehow so that the column vectors are orthogonal?
  • < – mbuchberger1967 – 2020-05-03T21:00:30.467

    start="3">

  • The embedding from the solution is different. They dont use M as the top left, it is somehow transposed and conjugated. Why is this embedding different from your proposal? what kind of algorithm is it following? Do you understand the solution?
  • BTW, in case you wanna look it up: (https://render.githubusercontent.com/view/Workbook_Measurements_Part2.ipynb#peres-wooters-game)

    – mbuchberger1967 – 2020-05-03T21:03:24.613

    1The solution you've presented is correct only in the quoted part. The rest has mistakes. For example, that $M'$ is not unitary at all. The norms of column vectors must be equal to 1. – Danylo Y – 2020-05-03T21:18:01.693

    1Yes, any remaining values in $M'$ are good if columns are orthogonal and have norms 1. – Danylo Y – 2020-05-03T21:19:49.173

    1Bra is a conjugate transpose of Ket, not just transpose. E.g. $\langle E_1| = \begin{bmatrix} 1 & -\omega^2\end{bmatrix}$. – Danylo Y – 2020-05-04T23:20:45.340

    Why do we embed M in to a 4x4 matrix M', why not into a 3x3 matrix? – mbuchberger1967 – 2020-05-10T00:44:10.473

    1We did embed it in a 3x3 unitary matrix, but we have to work with qubits, so the dimension must be power of 2. That's why we embed it further into 4x4. – Danylo Y – 2020-05-10T07:52:22.270

    yes, stupid question, i found out myself as I wanted to do the maths. If I like to do the maths for the measurement myself, I need to determine the eigenvalues next. So I need to solve det(M′−λI)=0. And knowing the eigenvalues I could determine the eigenvectors $v_i$ by solving the linear equations: M′v=λv is that correct so far? Do you have a tip for me regarding determining the eigenvectors? the calculation of det() for 4x4 or even 3x3 matrices is quite complex, and i could not easily determine the eigenvalues except for λ=1. – mbuchberger1967 – 2020-05-10T21:27:53.190

    Calculation of the eigenvalues is not enough to construct a circuit. Constructing a circuit for a given unitary is a very hard task in general. – Danylo Y – 2020-05-11T06:13:54.557

    After determining M' as we did it, and thanks to you i understood that, how can i go on? I would like to understand measurement now. When I get it right, I want to do a von Neumann measurement on M' now. my knowledge here: determine the eigenvalues, dtermine the eigenvectors, from the eigenvectors I can derive the projection operators... is this right so far? Do you see any chance to do this calculation for that problem? or is this scenario already too complex to calculate it by hand? – mbuchberger1967 – 2020-05-11T20:36:38.573

    Just follow the explanation from the link I've posted under the question (problem B2). – Danylo Y – 2020-05-12T09:21:16.557