Editing Quantum entanglement (section)

==== Definition ====
In classical [[information theory]] {{mvar|H}}, the [[Shannon entropy]], is associated to a probability distribution, <math>p_1, \cdots, p_n</math>, in the following way:<ref name="SE">{{cite journal |url=http://authors.library.caltech.edu/5516/1/CERpra97b.pdf#page=10 |title=Information-theoretic interpretation of quantum error-correcting codes |journal=Physical Review A |date=September 1997 |volume=56 |number=3 |pages=1721–1732 |arxiv=quant-ph/9702031 |doi=10.1103/PhysRevA.56.1721 |first1=Nicolas J. |last1=Cerf |first2=Richard |last2=Cleve |bibcode=1997PhRvA..56.1721C }}</ref>
: <math>H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i.</math>

Since a mixed state {{mvar|ρ}} is a probability distribution over an ensemble, this leads naturally to the definition of the [[von Neumann entropy]]:<ref name="Peres1993"/>{{rp|264}}
: <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right),</math>
which can be expressed in terms of the [[eigenvalue]]s of {{mvar|ρ}}: 
: <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i</math>.

Since an event of probability 0 should not contribute to the entropy, and given that
: <math> \lim_{p \to 0} p \log p = 0,</math>
the convention {{math|0 log(0) {{=}} 0}} is adopted. When a pair of particles is described by the spin singlet state discussed above, the von Neumann entropy of either particle is {{math|log(2)}}, which can be shown to be the maximum entropy for {{math|2 × 2}} mixed states.<ref name="Holevo2001"/>{{rp|15}}