How should I understand the link between $x$ and $| x \rangle $?

Question

In the paper "Quantum Expectation-Maximization for Gaussian Mixture Models" I encountered the following proposition :

Consider two vectors x,y and $\theta$ the angle between x and y, $\theta < \frac{\pi}{2}$. Then $||x - y|| \leq \epsilon$ implies $|| |x\rangle - |y\rangle || \leq \frac{\sqrt 2 \epsilon}{||x||}$.

I'm not sure how to interpret the link between $x$ and $| x \rangle $ ? In the case x is an integer, for example $x = 4$ then I'd say that $| x \rangle = | 100 \rangle $. But what about $x = 3.5$ or $x = (2.5, 2.5)$ ? In those cases what $| x \rangle$ would be ?

Answer 1

Generally it's context-specific what the label means. For example, if $x$ and $y$ are integers, then if $x\neq y$ then $|x\rangle$ and $|y\rangle$ are orthogonal (since they are different binary strings).

I haven't read the paper you mentioned but in the "Preliminaries" section of 3.1, they describe what they mean by $| x\rangle$. Specifically: They assume $x\in \mathbb{R}^d$, and they define $$|x\rangle = \frac{1}{\Vert x\Vert}\sum_{j=1}^d x_j|j\rangle$$

So the $|j\rangle$ will be computational basis vectors, i.e. for $j=1$, $|j\rangle = |0\dots 01\rangle$, etc. The components of $x$ give the amplitudes of each of those basis vectors in some superposition.

With this definition of $|x\rangle$, the "Claim 3.4" should follow.