# Definition

Consider a complete set of orthogonal projectors, which represent a projective quantum measurement

The probability of obtaining measurement outcome *i* is given by

We define the corresponding volume element as

(This is the same as the dimension of the subspace corresponding to the measurement outcome* i.*)

Observational entropy (also known as coarse-grained entropy) is defined as the sum the Shannon entropy of measurement and the mean Boltzmann entropy

This definition can be further generalized. Consider a quantum instrument, which is a complete set of quantum channels, representing the most general quantum measurement.

Each quantum instrument has its corresponding complete set of POVM elements

The probability of obtaining measurement outcome *i* is given by

The corresponding volume element is defined by

The definition of observational entropy is the same as above. Advantage of this general approach is that it also includes indirect measurements, such as those using a probe. It also generalizes observational entropy to multiple coarse-grainings, because any sequence of coarse-grainings can be written as a single quantum instrument with a vector of outcomes.