The new Entropic Information approach

Quantum Entangled Information

notion and definition

Entropy is viewed as a measure of entanglement.

Entropy provides one tool that can be used to quantify entanglement.

If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems.

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n, ...,1/n}.

Therefore, a bipartite pure state ρ ∈ HA ⊗ HB is said to be a maximally entangled state if the reduced state of each subsystem of ρ is the diagonal matrix.

“Where fine grained entropy is the entropy of the density matrix calculated by the standard methods of quantum field theory in curved spacetime. In the literature, this is often simply called the von Neumann entropy”. 

We can say that entropy, is information: indeed, in the micro-canonical language entropy is determined by the number of microstates compatible with a given macrostate.

we can use a more quantum approach of the notion of information

" the informational properties of a system should be dependent only on the number of distinct messages that can be encoded on that system."

Lucien Hardy
from "Probability theories in general and quantum theory in particular "

or

Information is something physical that is encoded in the state of a quantum system.

"Preskill, John.
Quantum Computation (Physics 219/Computer Science 219)

We can now, finally obtain, a valid definition of quantum information :

"information it is a quantum state change due to the modification of one degree of freedom of the considered quantum system"
 

 
Moreover, entanglement is about correlation of information

and the entropy of a mixed state can be view as a measure of quantum entanglement.

The Von Neumann entropy connect to all the other forms on entropy and glimpse the real origin of the second law

as entropy is information..

information as the real origin of the second law

the information increase drives :

the 2nd law of thermodynamics
And
the emergence of the classical world from the  quantum.

Moreover, it also defines the arrow of time

which itself points in the direction of increasing information/entropy and multiplying entanglement viewed as correlation of information.

Dive into Entropic Information Theory