12.3 Fidelity

Sometimes, when quantifying closeness of states, the inner product is a more convenient tool than the distance/norm. Analogous to how we define the distance between states |u\rangle and |v\rangle as d(u,v)=\|u-v\|, we define the fidelity between them as F(u,v)\coloneqq |\langle u|v\rangle|^2. This is not a metric, but it does have some similarly nice properties: for example, F(u,v)=1 when the two states are identical, and F(u,v)=0 when the two states are orthogonal (which means that they are “as different as possible”). Intuitively, we can understand fidelity as the probability that the state |u\rangle (resp. |v\rangle) would pass a test for being in state |v\rangle (resp. |u\rangle). In other words, if we perform an orthogonal measurement on |u\rangle that has two outcomes (\texttt{true} if the state is |v\rangle; \texttt{false} if the state is orthogonal to |v\rangle), then the fidelity F(u,v)=|\langle u|v\rangle|^2 is exactly the probability that we measure the outcome \texttt{true}.

Recall our definition of state distance: d(u,v) = \sqrt{2(1-|\langle u|v\rangle|)} This gives us a relation between distance and fidelity: once we know one, we can easily calculate the other. However, everything we have said so far applies only to pure states — we will see how the mixed state case is slightly more complicated shortly.

One final remark: as another example of the many inconsistencies in the literature, some authors define F(u,v) to be |\langle u|v\rangle| instead of |\langle u|v\rangle|^2. Whenever we say fidelity, we mean the latter: |\langle u|v\rangle|^2.