The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum algorithms or texts on quantum computing?


That quantity appears to be identical to Holevo information, which turns out to be the upper bound on how much classical information you can transmit using a quantum channel [1].

More generally the Holevo information is an upper bound for a quantity called "accessible information" which is (roughly speaking) the maximum information you can learn from an optimal measurement performed on an ensemble of quantum states.

[1] Holevo, A. S. (1973). Bounds for the quantity of information transmitted by a quantum communication channel. Problemy Peredachi Informatsii, 9(3), 3-11.

  • $\begingroup$ what is the classical analogue of Holevo information? $\endgroup$ – develarist Nov 4 '20 at 6:01
  • $\begingroup$ It looks/behaves like a mutual information defined between the classical and quantum parts of an ensemble. If I "tag" each member of a quantum ensemble with a classical bit of information - $\rho = \sum_i p_i |i \rangle \langle i |_C \otimes \rho_{i, Q}$ I can write the joint entropy of this classical-quantum system as $H(C, Q) = H(C) + \sum_i p_i S(\rho_{i, Q})$, the first term being entropy over classical distribution $(p_i, |i\rangle \langle i|)$. Then define $\chi$ as $S(C:Q) \equiv H(C) + S(\rho) - H(C, Q)$ to recover Holevo information. $\endgroup$ – forky40 Nov 4 '20 at 20:22
  • $\begingroup$ but its more intuitive to think of $\chi$ as the upper bound to accessible information, which really is just mutual information between an (optimal) measurement and the input state $\rho$. $\endgroup$ – forky40 Nov 4 '20 at 20:24

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