  Quantum mechanics ensures that the information stored in a quantum state is
secure and the ability to send private information through a quantum channel is
at least as great as the coherent information. We derive trade-off relations
between quantum privacy, information gain by Eve and the disturbance caused by
Eve to the quantum state that is being sent through a noisy channel. For
tripartite quantum states, we show that monogamy of privacy exists in the case
of a single sender and multiple receivers. When Alice prepares a tripartite
entangled state and shares it with Bob and Charlie through two different noisy
quantum channels, we prove that if the minimally guaranteed quantum privacy
between Alice and Bob is positive, then the privacy of information between
Alice and Charlie has to be negative. Thus, quantum privacy for more than two
parties respects mutual exclusiveness. Then, we prove a monogamy relation for
the minimally guaranteed quantum privacy for tripartite systems. We also prove
a trade-off relation between the entanglement of formation across one partition
and the quantum privacy along another partition. Our results show that quantum
privacy cannot be freely shared among multiple parties and can have implication
in future quantum networks.
