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.

中文翻译：

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基于Pauli组的一类对称状态最大密集编码的operator算子构造解

量子密集编码在量子密码通信中起着重要的作用，如何选择一组合适的unit运算符对消息进行编码是量子通信协议设计的主要工作。Shukla等。提出了一种基于泡利群乘积的construction算子构造的初步方法，用于量子对话中的密集编码。但是，该方法缺乏可行的步骤或条件，并且不能构造所有可能的unit算子集。在这项研究中，提出了构造用于量子最大密集编码的ary算子集的可行解决方案，其目的是使用最小量子位最大地编码一类t-量子位对称状态。这些状态具有偶数个叠加项，并且至少有一组\（\ left \ lceil {{t \ over 2}} \ right \ rceil \）量子位，其叠加项彼此正交。首先，我们提出了构造\（{2 ^ {t}} \）-阶乘积修正广义Pauli子群（乘积MGP子群）的过程和相应的算法。然后，给出用于t -qubit对称状态的两个条件，以从上述子组中选择适当的unit算子集。最后，我们以3量子比特GHZ，4量子比特W，4量子比特簇和5量子比特簇状态为例，并演示了如何通过我们的构造解决方案找到用于最大密集编码的所有operator运算符集，这表明我们的解决方案是可行且方便。