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In the case of a fully correlated scenario of $A$ and $B$, which are descirbed in mutually unbiased bases, why do the other observables have to be maximally uncorrelated. How does this follow from the unbiasedness relation?

I'm quoting the script which we are using (Quantum entaglement at high energies) at university and my questions are going to be written in square bracket []. (I'm in my last year of physics bachelor).

A set of orthonormal bases ${B_1, ..., B_m}$ for a Hilbert space $C^d$ with $B_k = {|i_k\rangle} \: i ={1, ...,d-1}$ is called mutually unbiased if

$$|\langle i_k|j_{k'}\rangle|^{2} = \delta_{kk'} \delta_{ij} + (1 - \delta_{kk'}) \frac{1}{d}$$

and fully correlated: If we can predict with certainty of the outcome of measurement on $A$ when we know the outcome of a measurement on $B$ ( or vice versa).

Let us define a quantity that captures fully correlated and completely uncorrelated mathematically. For example the correlations function

$$C_{A,B} = \sum_{i=0}^{d-1} \Pr(i,i)$$

Alice ($A$) and Bob($B$) choose observables that are mutually unbiased to the each other in a fully correlated scenario. Let us start with the product state $|0_1 0_1\rangle$. This is a representative of the classical fully correlated state.

[Why? I mean it is only a product state. Are product states always correlated?.]

The correlation function gives if the observables' eigenstates are in the basis choice $C_{A,B}=1$

[I'm a bit confused by what this sentence is supposed to mean..Basis choice?.]

Now we choose observables that are mutually unbiased, we obtain

$$C_{A_2, B_2} = \sum_{i=0}^{d} \langle i_2 i_2| 0_1 0_1\rangle \langle 0_1 0_1|i_2 i_2\rangle = \sum_{i=0}^{d} \frac{1}{d}\frac{1}{d} = \frac{1}{d}$$

Consequently, we can optimise the correlation function for any fully classically correlated system for one set of observables $A$, $B$, but then due to the unbiasedness relation it is maximally uncorrelated in the other observables.

[Ok, here is the core of my misunderstood. So

  1. What does it mean to optimise this function? What value do I want to get from the function? Or in other words, what would be the optimal value quantitatively?

  2. And a very basic questions, for which I've never received a good answer: Do all observables describe the same Hilbert space, but just in a different basis? I think this, because the unbiasedness relation describes the relation between different bases... So that most mean, that all observables are in the same Hilbert space but in a different basis representation if I can conclude anything about the observables from the unbiasedness relation.

  3. Why should follow that if $A$ and $B$ start in a mutually unbiased basis which is fully correlated, then the the other observables have to be maximally uncorrelated?]

Inspired by the result we define a quantity

$$I_m^{\textrm{MUB}} = \sum_{i=1}^{m} C_{A_iB_i}$$.

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    $\begingroup$ Your dfn. of unbiasedness is incorrect inasmuch as you’re missing a sqrt. It is the prob. that goes $\sim 1/d$, not the overlap. Easily checked using eigenvectors of Pauli vectors. $\endgroup$ – ZeroTheHero Jan 1 at 18:57
  • $\begingroup$ Mhm. The problem is, that I never the less have to work with the definition above, as this is the definition our professor gave us (we have to strictly stick to the script). But thank you. $\endgroup$ – Yalom Jan 1 at 19:22
  • $\begingroup$ This works if you use $\vert \langle i\vert j\rangle\vert^2$ but not without the modulus squared. See en.m.wikipedia.org/wiki/Mutually_unbiased_bases $\endgroup$ – ZeroTheHero Jan 1 at 19:29
  • $\begingroup$ Thank you! I checked the paper the script is based on, I guess it was a typo :) $\endgroup$ – Yalom Jan 2 at 16:10

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