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We know that in classical information theory the relation between different entropies can be depicted by Venn Diagram as given below.

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Can we create such Venn-diagrams for the quantum information theory case, where we consider Von-Neumann Entropy? I feel such Venn diagrams are not possible because in Quantum Information, some entropies can be negative. What can we do in such a scenario. Is there any research being performed in this direction, to create a geometrical picture of quantum information theory?

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    $\begingroup$ Even for three discrete random variables, the central region in the Venn diagram may represent a negative entropy. See for instance 8.31 in MacKay's book: inference.org.uk/itprnn/book.pdf. $\endgroup$ Feb 18, 2020 at 11:13
  • $\begingroup$ @Mark Wildon Thanks, I did not know that the representation of entropies in terms of Venn diagrams is misleading. A lot of books on Quantum Information follow this approach of using Venn Diagrams, also a lot of them do not consider 3 random variables. I had not read this book so I did not know about this. $\endgroup$ Feb 18, 2020 at 11:52

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Conditional entropies can be negative, so the sign is not an issue here. An introduction to quantum Venn diagrams that you may like is given by Chris Adami. Here is an example of a quantum system Q measured by a classical measurement device A that consists of two pieces $A_1$ and $A_2$. The shared information (central region) is always zero, which is what prevents cloning of a quantum state.

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