Authentication protocol based on polygamous nature of quantum steering

    It is well known that certain quantum correlations like quantum steering exhibit a monogamous relationship. In this paper, we exploit the asymmetric nature of quantum steering and show that there exist states which... [ view full abstract ]

    It is well known that certain quantum correlations like quantum steering exhibit a monogamous relationship. In this paper, we exploit the asymmetric nature of quantum steering and show that there exist states which exhibit a kind of polygamous correlation, where the state of one party, Alice, can be steered only by the joint effort of the other two parties, Bob and Charlie. 

     Since we are solely interested in a subset of states for which Alice cannot be steered individually by Bob or Charlie but only by their joint efforts in a tripartite scenario, we start with a tripartie state $\rho_{abc}$ prepared by Bob (or Charlie). Bob sends the subsystem $A$ to Alice and $C$ to Charlie. Since Alice does not believe Bob or Charlie, she asks them to perform a set of measurements and send her the outcomes. Based on the measurement outcomes, she computes the coherence of her conditional states. We show that there exist states $\rho_{abc}$ for which Alice is steerable if and only if Bob and Charlie make an effort together but not otherwise. To find such a set of states $\{S(A\leftarrow B:C)\}$, we first find out a set of states $\{S(A\leftarrow B, C)\}$ for which Alice is steerable by Bob and Charlie together as well as individually. We then compute the union of set of states $\{S(A\leftarrow B)\}\cup \{S(A\leftarrow C)\}$ for which Alice is steerable by Bob and Charlie individually. Our set of interest is the difference of the above two sets, i.e., $S_{i}\equiv S(A\leftarrow B, C)\setminus \{S(A\leftarrow B)\cup S(A\leftarrow C)\}$. 

   To single out such states, we find states such that an LHS model of Alice for the state $\rho_abc$ doesn't exist but an LHS model of Alice for the bipartite states $\rho_ab$ and $\rho_ac$ both exists.

    As an example, we explicitly single out a particular set of $3$ qubit states which exhibit polygamous relationship and also provide a recipe to identify the complete set of such states. We also provide a possible application of such states to an information theoretic task, we term as quantum key authentication (QKA). QKA can also be used in conjunction with other well known cryptography protocols to improve their security and we provide one such example with quantum private comparison (QPC).