Quantum entanglement gives rise to a density matrix analogous to that of a thermodynamic system. One can assign to it entropy by a standard formula. In thermodynamics entropy characterizes the system besides other state variables. Could entanglement entropy have a role similar to that of thermodynamic entropy and allow us to classify entangled systems?

- In thermodynamics there are transformations of thermodynamic state, which in the adiabatic case preserve entropy. There is also a theorem giving an upper bound for the efficiency of a thermal engine transforming heat to work, which is ordered energy. Could one consider a similar theorem? Could thermodynamics generalize to a science of entropy manipulation?
- Could adiabatic transformations preserving entanglement entropy be possible between two systems with the same entanglement entropy? Could the second law stating the increase of entropy generalize and state that transformations decreasing the entanglement entropy are not possible?

- It has been indeed proposed that it is possible to connect any two entangled systems with the same entanglement entropy by a transformation preserving the trace of the density matrix and keeping the eigenvalues positive. This transformation generalizes unitary transformations, which preserve the eigenvalues of the density matrix having probability interpretation. This kind of transformation would consist of basic physical transformations: I must confess that I do not quite understand what this means. If the conjecture is true one could classify entangled systems in terms of the entanglement entropy.
- There are however systems for which the conjecture fails. A weaker condition would be that this kind of transformation exists in a probabilistic sense: the analog of adiabatic transformation would fail for some system pairs. One can apply the transformations of a specified set O of transformations to an ensemble formed by copies of A. Intuitively it would seem that the number of transformations in the set of O transforming A to B adiabatically divided by the total number of transformations gives a measure for the success. I am not quite sure whether this definition is used.
- It is also stated the relative entropy for the probability distributions defined by the density matrices for systems A and B would in turn serve as an entanglement measure. Each fixed system B with fixed density matrix (or A) would define such a measure.

- In TGD inspired theory of consciousness, one can assign to any system a number theoretic negentropy as a sum of p-adic entanglement: here the sum is over the ramified primes of an algebraic extension of rationals considered. The p-adic negentropies obey a formula similar to that for the ordinary entanglement negentropy. p-Adic negentropies can be however positive unlike the ordinary entanglement negentropy as a negative of entanglement entropy. The sum of p-adic negentropies measures the information of entanglement unlike ordinary entropy which measures the loss of information about either entangled state caused by the entanglement.
- The number theoretic vision of TGD predicts that the entanglement negentropy is bound to increase in statistical sense in the number theoretic evolution and that this increase forces the increase of the ordinary entanglement entropy. Could the basic theorems of thermodynamics generalize also to this case and provide a mathematical way to understand the evolution of intelligent systems?

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

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