Qantum estimation/metrology is a research field closely tied with quantum information theory. Everything was born with the Heisenberg uncertainty principle, when we realized that in general measurements can be non-commutative in contrast with the classical world where everything is commutative i.e. the order of the measurements is irrelevant. The consequence of the non-commutativity is that two given observables may be incompatible for simultaneous arbitrarily precise measurements.

The next step was done by Helstrom in the 70's who introduced the Symmetric Logarithmic Derivative (SLD) operator to obtain the Quantum Fisher information. It was later in the 80's when Wootters came with the refined and elegant concept of quantum statistical distinguishability, which was tied with the work of Helstrom by Braunstein and Caves (1994) in their famous paper Statistical distance and the geometry of quantum states. This paper is remarkable because it also established the connection with the completely independent work of Uhlmann about the Bures metric, which was born in the generalization of the Berry phase for mixed states.

More recently, we witnessed many important developments in the quantum single parameter estimation with one unexpected twist. Now we know that quantum mechanics and in particular quantum entanglement have the potential to defeat their classical estimation methods in the quest for higher precision. For example, you can see my older post about NOON states and quantum interferometry.

However, this new knowledge cannot be applied in the real world until we learn how to generate a considerable number of entangled states in a reliable way.

You can also see my wikipedia article about the Bures metric

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