# The noisy channels act on the state and their action

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part to Charlie over noisy channels. The noisy channels act on the state and their action may be de- scribed by the superoperators E Q 1 and E Q 2 , respectively. The superoperators E Q 1 and E Q 2 can be represented as unitary evo- lutions of larger quantum systems that include environments E 1 and E 2 , respectively. The environments can be assumed to be initially in pure state | E 1 ) and | E 2 ) , respectively. The evolution can be shown to be ρ RQ 1 Q 2 ⊗ | E 1 )( E 1 | U Q 1 E 1 −−−−→ ρ RQ 1 E 1 Q 2 (16) ρ RQ 1 E 1 Q 2 ⊗ | E 2 )( E 2 | U Q 2 E 2 −−−−→ ρ RQ 1 E 1 Q 2 E 2 . (17) Alice R | Ψ RQ 1 Q 2 ) U N 1 B Bob | Ψ RQ 1 E 1 Q 2 ) Eve E 1 B C U N 2 C Charlie | Ψ RQ 1 E 1 Q 2 E 2 ) Eve E 2 FIG. 2. Alice, Bob, and Charlie share the entanglement through the noisy channels. The joint state after passing through two different noisy channels are | Ψ RQ 1 E 1 Q 2 ) and | Ψ RQ 1 E 1 Q 2 E 2 ) . U N 1 and U N 2 are the unitary realization of the noisy channels E Q 1 and E Q 2 , respec- tively. Since local operations do not affect entanglement of the parties involved, hence RQ 2 acts as a reference system dur- ing the communication between Alice to Bob, and remains unchanged. Hence, we can define the privacy of the channel between Alice to Bob as P AB = H Bob H Eve 1 . (18) Similarly, RQ 1 E 1 acts as the reference system during the communication between Alice to Charlie, and remains un- changed. Hence, we can define the privacy of channel be- tween Alice to Charlie as P AC = H Charlie H Eve 2 . (19) Now, we ask if we give all the computational power to Eve, then how much privacy can be maintained between Alice to Bob and Alice to Charlie. In a competitive scenario, Bob will want to make H Bob as close as possible to χ Q 1 by a suitable choice of coding and decoding observable. Similarly, Eve will try to make H Eve 1 as close as possible to χ E 1 by quantum technological arsenal at her disposal. Therefore, we can define “minimal guaranteed privacy” between Alice and Bob as P min AB = H Bob max H Eve 1 = H Bob χ E 1 . (20) Since H Bob χ Q 1 , we have P min AB χ Q 1 χ E 1 and hence P min AB I c ( A ) B ) . Thus, the minimal guaranteed privacy and the optimal guaranteed privacy across Alice and Bob obey the inequality P min AB I c ( A ) B ) ≤ P AB . Similarly, we can define “minimal guaranteed privacy” be- tween Alice and Charlie as P min AC = H Bob max H Eve 2 = H Bob χ E 2 . (21) The minimal guaranteed privacy and the optimal guaranteed privacy across Alice and Charlie obey the inequality P min AC I c ( A ) C ) ≤ P AC . Having defined the minimal quantum privacy for the com- munication channel between Alice to Bob and between Alice to Charlie, we prove that they obey the exclusive monogamy inequality.

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5 Theorem 1 If P min AB and P min AC are the minimal privacy of in- formation between Alice to Bob and Alice to Charlie, respec- tively, then the following mutually exclusive relation holds P min AB + P min AC 0 . (22) Proof.
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