## 12.1 Metrics

To begin with, let us work with pure states, and save the problem of dealing with mixed states for a later section.
We will start with the second question: how do we quantify this notion of “close enough”?
The central concept is one with which you are probably already somewhat familiar (we mentioned it in Sections 0.3 and 0.5), namely that of a **metric**, or **distance**.

Given a set **metric** (or **distance**) on

**Identity of indiscernibles:**d(a,b)=0 if and only ifa=b **Symmetry:**d(a,b)=d(b,a) for alla,b\in X **Triangle inequality:**d(a,c)\leqslant d(a,b)+d(b,c) for alla,b,c\in X .

There are four conditions governing metrics (identity of indiscernibles is an “if and only if” statement, so we can separate it into two “if” statements). As is usually the case in mathematics, it is interesting to ask what happens if we drop one or more of these.

- If we drop
d(a,b)=0\implies a=b then we get**pseudometrics**. - If we drop
a=b\implies d(a,b)=0 then we get**metametrics**, or**partial metrics**. - If we drop
d(a,b)=d(b,a) then we get**quasimetrics**. These arise “in real life”, if you think about travelling around a city that has lots of one-way streets, or travelling up or down a big hill. - If we drop
d(a,c)\leqslant d(a,b)+d(b,c) then we get**semimetrics**(though be careful here: lots of authors use “semimetric” to mean almost any one of these generalisations, and the terminology is very non-consistent!).

We can also consider the case of **extended metrics**, where the distance function is allowed to take the value *extended pseudoquasimetric*.

The most common norm is the **Euclidean distance**, that is, distance between two points in Euclidean space.
Given points **normed vector space**, where the **norm**

It turns out that this norm (and thus this metric) actually arises from a more fundamental structure, namely that of the **inner product**.
Returning to the bra-ket notation, we recall that the norm of any vector ** 2-norm**, or the

**(for reasons that we will come back to in Section 12.11.2), and is defined for any finite-dimensional Hilbert space**\ell^2 -norm

Before moving on to talk about state vectors, let us first discuss one other metric space which shows up in information theory (both classical and quantum).
The space^{233} of binary strings (of some fixed length **Hamming distance**.
This is defined quite simply as “the number of positions at which the corresponding bits are different”.
For example,

More formally, if we define the **Hamming weight** of a binary string of length

You can think of this as just a set, but we have already seen that this is actually a vector space over

\mathbb{Z}/2\mathbb{Z} , where addition corresponds to\texttt{XOR} .↩︎