2 is nothing but the discord 51 across ρ rq 2 and is

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2 ) is nothing but the discord [51] across ρ RQ 2 and is defined as D ( ρ RQ 2 ) = I parenleftBig ρ RQ 2 parenrightBig J parenleftBig ρ RQ 2 parenrightBig , where I parenleftBig ρ RQ 2 parenrightBig = S ( ρ R ) + S parenleftBig ρ Q 2 parenrightBig S parenleftBig ρ RQ 2 parenrightBig is the mutual information for the state ρ RQ 2 . Now, E f ( AB ) + P min AC = E f ( ρ RQ 1 ) + P min AC E f ( ρ RQ 1 ) + I c ( A ) C ) D ( ρ RQ 2 ) . However, D ( ρ RQ 2 ) S ( ρ R ) = S ( A ) [52, 53]. Hence the proof. This shows that for any tripartite state the amount of entan- glement across one partition restricts the quantum privacy that can be shared across another partition. This is reminiscent of the trade-off relation between the entanglement cost and the distillable common randomness for any tripartite system. One can also physically interpret the relation ( 33 ) as follows. From the compression theorem, we know that one can transfer the information contents of the system R into S ( ρ R ) = S ( A ) qubits per copy. We know that this does retain correlation to other subsystems faithfully in the asymptotically limit. There- fore, the local entropy S ( ρ R ) represents the effective size of the subsystem R measured in qubits. The above trade-off re- lation shows that the entanglement between the subsystem R and the subsystem Q 1 and the privacy between R and subsys- tem Q 2 is limited when the size of system R is limited to S ( A ) qubits. Physically, this implies that the quantum entanglement between one system and the privacy for the other system are mutually exclusive. Thus, the existence of quantum entangle- ment across one partition restricts the quantum privacy across the other partition. V. DISCUSSIONS AND CONCLUSIONS Monogamy of quantum correlation such as quantum en- tanglement and other correlations play an important role in quantum communication. We have shown that the monogamy of quantum privacy exists for tripartite entangled states as well as for multipartite entangled state. In addition to the monogamy of privacy for the multipartite entangled state one can have a monogamy for the square of the privacy for multiqubit entangled states. To prove this, one can use the monogamy relation for the square of the entanglement of formation [44] which is given by E 2 f ( ρ AB ) + E 2 f ( ρ AC ) + · · · + E 2 f ( ρ AN ) E 2 f ( ρ A | BC ··· N ) . For example, if we have a multiqubit entangled state ρ ABC ··· N shared between Alice, Bob, Charlie, · · · , and Nancy and the minimum pri- vacy across AB, AC, · · · , AN over different noisy channels as P min AB , P min AC , · · · , and P min AN , respectively, then by using the Carlen-Lieb inequality [35] we can show that ( P min AB ) 2 + ( P min AC ) 2 + · · · + ( P min AN ) 2 E 2 f ( ρ A | BC ··· N ) . Therefore, the square of the minimal privacy for the single sender and multi- ple receivers respects the monogamy for multiqubit states.
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