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Below is a question and an answer.

How does quantum information relate to, diverge from or reduce to Shannon information, which used log probabilities?

What people are more often interested in are averaged quantities such as entropies, conditional entropies and mutual information. These have direct quantum analogues, calculated based on density matrices of qubits rather than classical probability distributions. The density matrix still represents a probability distribution but, rather than using a single fixed basis (i.e. the "0" and "1" of a bit), there's a continuous range of possibilities, which change the actual calculations a bit.

Information theoretic measures are all based on log probabilities. What do log probabilities in classical information theory become in quantum information theory?

What are the quantum analogues of entropy, conditional entropy and mutual information? their formulas, and how are they computed from density matrices of qubits (rather than probability distributions)?

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    $\begingroup$ Here is Mark Wilde's "From Classical to Quantum Shannon Theory". Chapeter 10 introduces all the entropic quantities from information theory you asked about (plus more). Chapter 11 introduces their quantum generalizations. $\endgroup$
    – Rammus
    Oct 29 '20 at 11:01
  • $\begingroup$ are you essentially asking about the von Neumann entropy? You define the entropy of a state as the "standard Shannon entropy" of its eigenvalues $\endgroup$
    – glS
    Nov 3 '20 at 17:19
  • $\begingroup$ Me? or is that what your answer is to the question, because I don't know of any $\endgroup$
    – develarist
    Nov 3 '20 at 17:46

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