Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

Question

I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following:

$$ F(\rho, \sigma) := \operatorname{tr} \sqrt{\rho^{\frac{1}{2}} \sigma \rho^{\frac{1}{2}}} $$

An implementation of this definition is also present in this QETLAB library.

Another definition that I have found is: $$ F(\rho, \sigma) := \Big(\operatorname{tr} \sqrt{\rho^{\frac{1}{2}} \sigma \rho^{\frac{1}{2}}}\Big)^2 $$ This def is present in the highly regarded Preskill notes and most other places like wiki, quantiki.

Was this definition changed sometime?

Answer 1

Both definitions are used and authors usually make it clear which one they mean.

Wikipedia also points this out under the Alternative Defintion section.