Sequential quantum secret sharing in a noisy environment aided with weak measurements

In this work we give a (n, n)-threshold protocol for sequential secret sharing of quantum information for the first time. By sequential secret sharing we refer to a situation where the dealer is not having all the secrets at the same time, at the beginning of the protocol; however if the dealer wishes to share secrets at subsequent phases she/he can realize it with the help of our protocol. First of all we present our protocol for three parties and later we generalize it for the situation where we have more (n > 3) parties. Interestingly, we show that our protocol of sequential secret sharing requires less amount of quantum as well as classical resource as compared to the situation wherein existing protocols are repeatedly used. Further in a much more realistic situation, we consider the sharing of qubits through two kinds of noisy channels, namely the phase damping channel (PDC) and the amplitude damping channel (ADC). When we carry out the sequential secret sharing in the presence of noise we observe that the fidelity of secret sharing at the kth iteration is independent of the effect of noise at the (k - 1)th iteration. In case of ADC we have seen that the average fidelity of secret sharing drops down to 1/2 which is equivalent to a random guess of the quantum secret. Interestingly, we find that by applying weak measurements one can enhance the average fidelity. This increase of the average fidelity can be achieved with certain trade off with the success probability of the weak measurements.