LOG#138. Non-extensive entanglement?

quantum_entanglement

Entanglement is one of the most surprising quantum phenomena.

We quantify entanglement using information measures or entropies. The most used entropies till now (likely it is going to change in the near future) are Boltzmann’s entropy and Von Neumann entropy. The Von Neumann entropy is defined as:

S(VN)=-\mbox{Tr}\left(\rho\ln \rho\right)

with

\mbox{Tr}(\rho)=1

Features of this entropy:

1st. S(VN) does NOT depend on ANY of the observables of the system.

2nd. If the system is at a pure state, this entropy vanishes.

3rd. After unitary evolution, this entropy remains the same.

4th. For statistical mixing of pure states, S(VN)>0, i.e., classical uncertainty increase the entropy of the state.

Recently, several generalizations of Boltzmann’s entropy arised. For instance, Tsallis entropy reads

S_q=-\dfrac{1-\mbox{Tr}(\rho^q)}{1-q}

and

\displaystyle{\lim_{q\rightarrow 1}S_q=S(VN)}

where q is generally a real numbe/parameter related to the non-extensive properties of the relevant underlying physical system. It has been speculated that the role of generalized statistics in physics could be very important, specially when fractal or multifractal structures do appear. The answer is not clear or definitive yet.

The quantum VN mutual information reads

I=S_A+S_B-S_{AB}

relative to two subsystems is also used to compare distributions and quantum states. Even generalized entropies like the above Tsallis entropy or the Renyi entropy (a subcase of Tsallis entropy itself) are becoming more and more important in quantum information theory.

The reduced density operators are defined

\rho_A=\mbox{Tr}_B\rho_{AB}

\rho_B=\mbox{Tr}_A\rho_{AB}

If \rho_{AB} is a pure state, then S_{AB}=0 and

S_A-S_B\leq S_{AB}\leq S_A+S_B

The Tsallis entropy generalization of VN mutual information provides

I_q=S_{qA}+S_{qB}-S_{qAB}

The main use and consequence of this treatment is to understand how an why entanglement emerges/arise/is enhanced or destroyed in some particular systems. It is reasonable to imagine that entanglement arises depending on the microscopic degrees of freedom of some system and its general properties (like non-extensivity). It can be useful to introduce some entropic index measuring the departure from extensivity, but it is generally obscure (yet) the meaning of this entropic parameter q in most of the known examples.

However, the study of generalized entropies and entanglement with these new information measurements could provide:

1st. New measures of information and its generalizations.

2nd. Hints of the physical reason and origin of entanglement at the subatomic realm (remember that entanglement itself is quantum, a highly non-trivial nonlocal feature of current Quantum Mechanics).

3rd. Insights on the informational foundations of Quantum Physics. Specially, a non-extensive origin of decoherence could be a reasonable target!

4th. Solutions to a wider class of problems where S(VN) is not useful as informational tool. Note that non-extensive entropies like the Tsallis entropy S_q arise from complex systems, in general, or systems with some inner microstructure (composite structure).

See you in my next TSOR post!

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LOG#138. Non-extensive entanglement? — 3 Comments

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