text: In thermodynamics, entropy is related to the amount of energy of a system that can be converted into mechanical work in a concrete process. In quantum mechanics, this translates to the ability to measure and manipulate the system based on the information gathered by measurement. An example is the case of Maxwell’s demon, which has been resolved by Leó Szilárd.
  The entropy of an observable is associated with the complete projective measurement of an observable,  where the operator  has a spectral decomposition:
  : where  are the projection operators of the eigenvalue 
  The probability of outcome  is  The entropy associated with the observable  is the Shannon entropy with respect to the possible outcomes:
  The most significant observable in thermodynamics is the energy represented by the Hamiltonian operator  and its associated energy entropy, 
  John von Neumann suggested to single out the most informative observable to characterize the entropy of the system. This invariant is obtained by minimizing the entropy with respect to all possible observables. The most informative observable operator commutes with the state of the system. The
  entropy of this observable is termed the Von Neumann entropy and is equal to
  As a consequence,  for all observables. At thermal equilibrium the energy entropy is equal to the von Neumann entropy: 
   is invariant to a unitary transformation changing the state. The Von Neumann entropy  is additive only for a system state that is composed of a tensor product of its subsystems: