A Unitary Operator Construction Solution Based on Pauli Group for Maximal Dense Coding

TL;DR

This work tackles the problem of constructing unitary operator sets for maximal dense coding on $t$-qubit symmetric states using minimal qubits. It introduces a two-phase approach: (i) systematic construction of multiplicative generalized Pauli subgroups (MGP) of order $2^t$ and (ii) selection of operator sets that satisfy two state-specific criteria to ensure the encoded states are mutually orthogonal. Compared with Shukla et al., the authors provide explicit construction steps, conditions, and a broader set of admissible operator sets, demonstrated on $|GHZ\rangle_3$, $|W\rangle_4$, and cluster states up to $|\phi\rangle_5$. The approach leverages the modified Pauli group and Sylow-subgroup structure to reduce search space, enabling maximal dense coding with fewer qubits for a class of symmetric states. While the solution is not universal to all states, it offers a concrete framework and improves encoding potential, with future work aimed at generalizing beyond the symmetric-state class.

Abstract

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 t/2 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.

Figures (5)

• Figure 1: Two types of quantum communication process based on dense coding: (a) is one-way communication and (b) is two-way communication. In the one-way communication, the sender firstly encodes the classical information into quantum states using dense coding, and transmits them to the receiver through the quantum channel. Finally, the receiver gets the transmitted states and measures them to get the classical information (i.e., quantum decoding). In the two-way communication, the receiver firstly send message qubits (which are used for massage encoding) to the sender, and the remaining steps are the same as the one-way communication process.
• Figure 2: The structure of a $2^t$-order multiplicative MGP subgroup
• Figure 3: The structure division of $G_n^i$
• Figure 4: The structure of a 2-qubit multiplicative MGP subgroup
• Figure 5: The structure of a 3-qubit multiplicative MGP subgroup