# Using the carlen lieb inequality e f parenleftbig ρ

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. Using the Carlen-Lieb inequality E f parenleftBig ρ RQ parenrightBig max { S ( R ) S ( RQ ) , S ( Q ) S ( RQ ) , 0 } [35] and assum- ing that S ( Q ) S ( RQ ) is the maximum of right-hand side, we have P min AB E f ( RQ ) . Next, we ask if there can be a trade-off relation between the quantum privacy and the disturbance caused by Eve to the quantum state that is sent through a noisy channel. We show that there is indeed a trade-off relation between them. Intu- itively, one may say that a system is disturbed when the initial state is different than the final state and it is not possible to go back to the initial state in a reversible manner. The dis- turbance is usually an irreversible change in the state of the system, caused by a quantum channel. The disturbance mea- sure should satisfy the following conditions: ( i ) D should be a function of the initial state ρ Q and the CPTP E Q only, ( ii ) D should be null iff the CPTP map is invertible on the initial

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4 state ρ Q , because in this case the change in state can be re- versed hence the system is not disturbed, ( iii ) D should be monotonically non-decreasing under successive application of CPTP maps and ( iv ) D should be continuous for maps and initial states which do not differ too much. Keeping in mind these desirable properties, Maccone [36] has defined a mea- sure of disturbance that satisfies the above conditions. This is given by D ( ρ Q , E Q ) S ( ρ Q ) I c ( R ) Q ) = S ( ρ Q ) S ( E ( ρ Q )) + S ( I ( E )( | Ψ ) RQ ( Ψ | )) , (14) where I c ( R ) Q ) is the coherent information for ρ Q through a channel E Q . Since the coherent information I c is non- increasing under successive application of quantum opera- tion, the disturbance measure is indeed monotonically non- decreasing under CPTP map. Note that D ( ρ Q , E Q ) satisfies 0 D ( ρ Q , E Q ) 2 log 2 (dim H Q ) . The trade-off relation between the disturbance and the min- imal guaranteed privacy is given by D ( ρ Q , E Q ) + P min AB S ( ρ Q ) . (15) Thus, given the limited amount of local entropy S ( A ) = S ( ρ Q ) , the amount of minimal guaranteed privacy cannot be more if the disturbance caused by Eve is large. Since S ( A ) = E (vextendsingle vextendsingle Ψ RQ )big) = Tr ( ρ Q log ρ Q ) is the initial entan- glement between R and Q , the disturbance and the minimal privacy is also bounded by the initial entanglement. IV. MONOGAMY OF PRIVACY We now come to the main part of the paper where we ana- lyze the private communication between a single sender (Al- ice) and two receivers (Bob and Charlie) as shown in Fig. 2 , where Alice shares the entangled state with Bob and Charlie by sending parts of the system over two separate noisy chan- nels to them. In the sequel, we prove the monogamy relation for quantum privacy for the tripartite quantum system and explore how en- tanglement across one partition affects the privacy across an- other partition. Imagine that Alice prepares a pure entangled state ρ RQ 1 Q 2 = vextendsingle vextendsingle Ψ RQ 1 Q 2 )big(big Ψ RQ 1 Q 2 vextendsingle vextendsingle and sends Q 1 part of the system to Bob and Q 2 part to Charlie over noisy channels.
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