# What are the possible and meaningful measurements on a quantum walk on a graph?

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

• whether a measurement is "meaningful" entirely depends on what your goal is. Any measurement can be useful for some application
– glS
Dec 19, 2020 at 19:36
• 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? Dec 19, 2020 at 19:43
• @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 Dec 20, 2020 at 14:40
• @jacob1729 at a first sight the problem is that is degenerate measurement Dec 20, 2020 at 14:41
• @raskolnikov that's not the question you asked though. You should edit the post to reflect what you are actually looking for
– glS
Dec 20, 2020 at 14:44