Monogamy of Quantum Privacy

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.