Say you are given two quantum states and , and you are asked the following questions:

* How similar are they? Can you distinguish them?*

In practice, this situation arises in many scenarios. For instance if you have an imperfect experiment (due to noise) that produces a quantum state , and you wish to find out how close it is to the state that you originally wanted to create.

While there are many ways to **quantify the distinguishability (i.e., the degree of similarity)** between two quantum states, we will focus here on one quantity of particular interest in Quantum Information theory: the **Quantum Fidelity **(or the so-called **Uhlmann-Jozsa Fidelity**). The Fidelity is defined as

(1)

where is the 1-norm.

### A brief History

Let us first try to understand **why the Fidelity is defined the way it is**.

The history of the Quantum Fidelity can be traced back to the late 70’s and early 80’s to the works of Uhlmann [1] and Alberti [2], who were studying generalizations of the quantum mechanical **transition probability** between two states. In later years the Fidelity was studied by Richard Jozsa (who coined the term “*Fidelity*“) [3] and then by Benjamin Schumacher [4] in the context of quantum communications.