To summarize we have proved a trade off relation

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To summarize, we have proved a trade-off relation between the quantum privacy and the information gain by Eve. Similar trade-off relation holds for the disturbance caused by Eve and the minimal guaranteed quantum privacy. We have derived the exclusive monogamy relation for quantum privacy in the case of a single sender and multiple receivers and shown that if the sender has a positive privacy of shared information with the first receiver then the privacy with other receiver has to be negative. This means that both the parties cannot reliably generate a secret key due to the restrictions imposed by the exclusive monogamy inequality. Similarly, we proved another monogamy relation for the quantum privacy for any tripartite system which shows that the sum of minimal guaranteed pri- vacies across Alice-Bob and Alice-Charlie are bounded by the optimal guaranteed privacy across Alice-(Bob and Charlie). Thus, quantum privacy cannot be freely shared across many subsystems. Furthermore, we have proved a trade-off relation between the entanglement of formation and the quantum pri- vacy across different partitions of a tripartite system. We hope that our results will have a significant impact in multiparty quantum key distribution and the quantum network when one is trying to share secret key simultaneously with multiple re- ceivers. From our result, it follows that if Alice wants to share privacy with Bob and Charlie together, it is better to share two separate bipartite entangled states with both the receivers. On the other hand, if Alice wants to share privacy with one of the two receivers at a later point in time, she can share a tripar- tite entangled state and has the option to choose with whom to share the privacy. In future, it may be worth exploring if the monogamy relation holds for the optimal guaranteed privacy. ACKNOWLEDGMENTS KS acknowledges the support from HRI, Allahabad for summer visiting program.
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