A unitary operator construction solution based on Pauli group for maximal dense coding with a class of symmetric states

Quantum dense coding plays an important role in quantum cryptography communication, and how to select a set of appropriate unitary operators to encode message is the primary work in the design of quantum communication protocols. Shukla et al. proposed a preliminary method for unitary operator construction based on Pauli group under multiplication, which is used for dense coding in quantum dialogue. However, this method lacks feasible steps or conditions and cannot construct all the possible unitary operator sets. In this study, a feasible solution of constructing unitary operator sets for quantum maximal dense coding is proposed, which aims to use minimum qubits to maximally encode a class of t-qubit symmetric states. These states have an even number of superposition items, and there is at least one set of \(\left\lceil {{t \over 2}} \right\rceil \) qubits whose superposition items are orthogonal to each other. Firstly, we propose the procedure and the corresponding algorithm for constructing \({2^{t}}\)-order multiplicative modified generalized Pauli subgroups (multiplicative MGP subgroups). Then, two conditions for t-qubit symmetric states are given to select appropriate unitary operator sets from the above subgroups. Finally, we take 3-qubit GHZ, 4-qubit W, 4-qubit cluster and 5-qubit cluster states as examples and demonstrate how to find all unitary operator sets for maximal dense coding through our construction solution, which shows that our solution is feasible and convenient.