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Abstract: We introduce the concept of mutual independence - correlations sharedbetween distant parties which are independent of the environment. This notionis more general than the standard idea of a secret key - it is a fully quantumand more general form of privacy. The states which possess mutual independencealso generalize the so called private states - those that possess private key.We then show that the problem of distributed compression of quantum informationat distant sources can be solved in terms of mutual independence, if freeentanglement between the senders and the receiver is available. Namely, weobtain a formula for the sum of rates of qubits needed to transmit adistributed state between Alice and Bob to a decoder Charlie. We also show thatmutual independence is bounded from above by the relative entropy modulo aconjecture, saying that if after removal of a single qubit the state becomesproduct, its initial entanglement is bounded by 1. We suspect that mutualindependence is a highly singular quantity, i.e. that it is positive only on aset of measure zero; furthermore, we believe that its presence is seen on thesingle copy level. This appears to be born out in the classical case.

Author: Michal Horodecki, Jonathan Oppenheim, Andreas Winter

Source: https://arxiv.org/

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