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In the contest of quantum walks, the graph is defined as $G=\{V,E\}$ with $V$ the set of vertices and $E$ the set of edges. Thus, the Hilbert space is defined as the $\mathcal{H}=\operatorname{span}\{\vert i \rangle, i \in V\}$.

The usual measurement one can consider on quantum states that live on the graph is the position measurement $$ \hat{M} = \sum_i i \vert i \rangle \langle i \vert $$ My question is: are there other possible and meaningful measurements, considering only the position degree of freedom $\vert i \rangle$?

Just to make an example of what I mean, a possible meaningful measurement could be the position along the $x$ and $y$ axis when the graph is projected on the plane, that is ($N=\deg(V)$)

$$ \hat{M}_x = \sum_i \cos\left(\frac{\pi}{N}i\right) \vert i\rangle \langle i \vert \\ \hat{M}_y = \sum_i \sin\left(\frac{\pi}{N}i\right) \vert i\rangle \langle i \vert $$

What I’d like to find is a single (or multiple) measurement which can distinguish the nodes in the graph. The problem with, for instance, a measurement of the index is that the variance of this measurement is index dependent and this is actually a problem in graph like the complete graph, where my intuition fails with a variance index dependent

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    $\begingroup$ whether a measurement is "meaningful" entirely depends on what your goal is. Any measurement can be useful for some application $\endgroup$
    – glS
    Dec 19, 2020 at 19:36
  • $\begingroup$ A comment from someone that has no particular knowledge of quantum walks of graphs - is there a reason you don't consider $\hat{Q} = |i\rangle\langle j | + |j\rangle \langle i |$ to be meaningful? $\endgroup$
    – jacob1729
    Dec 19, 2020 at 19:43
  • $\begingroup$ @glS actually the main reason I’m asking is that i’d like to find a single (or multiple) measurement which can distinguish the nodes in the graph. The problem with, for instance, a measurement of the index is that the variance of this measurement is index dependent and this is actually a problem in graph like the complete graph, where my intuition fails with a variance index dependent $\endgroup$ Dec 20, 2020 at 14:40
  • $\begingroup$ @jacob1729 at a first sight the problem is that is degenerate measurement $\endgroup$ Dec 20, 2020 at 14:41
  • $\begingroup$ @raskolnikov that's not the question you asked though. You should edit the post to reflect what you are actually looking for $\endgroup$
    – glS
    Dec 20, 2020 at 14:44

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