Quantum communication channel has a distinct

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Quantum communication channel has a distinct advantage over the classical communication channel. If there is informa- tion leakage to an eavesdropper (Eve) that is trying to infer the communication happening between the sender (Alice) and the receiver (Bob), then her presence can be detected. The secrecy of quantum communication against copying of the signal by Eve is guaranteed by laws of physics as it was shown in the [email protected] [email protected] quantum key distribution (QKD) protocol independently by Bennet and Brassard, and by Ekert [12, 13]. Sharing of the secret key is important for providing privacy as one can use it in cryptographic protocols such as the one-time pad [14]. Although quantum cryptography provides the best security available at present, it is not immune to attack and exploits by Eve due to leakage of information or due to botched im- plementations [15–17]. The existing systems may be prone to side-channel attacks that rely on imperfect experimental im- plementation and hence side-channel free QKD was proposed that replaces real channels with virtual channels in QKD pro- tocol to eliminate the attack [18]. Further study on absolute limits of privacy has been done by Ekert and Renner [19, 20] and a recent security proof for quantum key distribution has been carried out by Tomamichel and Leverrier [21]. In realistic quantum communication, the channel is always noisy and the information that is leaked to the environment reveals the activity between the sender and the receiver to the eavesdropper. For a single sender and a single receiver, it was initially understood that the critical part of such communica- tion task is to securely share entanglement between two par- ties. The amount of information that can be securely shared is proportional to the amount of entanglement that can be shared between the two parties. It was shown that secret key sharing between two parties is equivalent to “entanglement purifica- tion” [22]. Later it was found that bound entangled states can also be used to share secret key without third-party sharing it. Security of such scheme was further investigated in Ref.[23– 25]. Schumacher and Westmoreland quantified the privacy of a channel that is measured by the information available to the receiver and not available to any eavesdropper and showed that it can be made as large as the channel’s coherent informa- tion [26, 27]. In this paper, we ask the question if quantum mechanics can ensure privacy for more than two parties? To answer this ques- tion we generalize the result of Schumacher and Westmore- land [26] and analyze the case where a sender (Alice) shares
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2 the entangled state with two receivers (Bob and Charlie) over a noisy quantum channel. First, we prove 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. Next, we show that the minimal guaranteed quantum privacy obeys a strict monogamy relation for a single sender and two receivers. For a tripartite entangled
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