Quantum Entanglement and Entropy

Filippo Giraldi, Paolo Grigolini

Published 2001-01-31Version 1

Entanglement is the fundamental quantum property behind the now popular field of quantum transport of information. This quantum property is incompatible with the separation of a single system into two uncorrelated subsystems. Consequently, it does not require the use of an additive form of entropy. We discuss the problem of the choice of the most convenient entropy indicator, focusing our attention on a system of 2 qubits, and on a special set, denoted by $\Im$. This set contains both the maximally and the partially entangled states that are described by density matrices diagonal in the Bell basis set. We select this set for the main purpose of making more straightforward our work of analysis. As a matter of fact, we find that in general the conventional von Neumann entropy is not a monotonic function of the entanglement strength. This means that the von Neumann entropy is not a reliable indicator of the departure from the condition of maximum entanglement. We study the behavior of a form of non-additive entropy, made popular by the 1988 work by Tsallis. We show that in the set $\Im$, implying the key condition of non-vanishing entanglement, this non-additive entropy indicator turns out to be a strictly monotonic function of the strength of the entanglement, if entropy indexes $q$ larger than a critical value $Q$ are adopted. We argue that this might be a consequence of the non-additive nature of the Tsallis entropy, implying that the world is quantum and that uncorrelated subsystems do not exist.