Tunable M-Ary Quantum Secure Direct Communication via Correlation-Histogram Modulation

We propose a two-way entanglement-based quantum secure direct communication (QSDC) protocol in which information is encoded as the value of a continuously tunable parameter and decoded from empirically estimated joint-outcome histograms over windows of detected pairs. The protocol uses a fixed two-qubit polarization Hilbert space and only standard coincidence-based polarization measurement at the receiver, with the symbol alphabet enlarged not by increasing the carrier Hilbert-space dimension but by exploiting temporal integration over a window of pairs to resolve distinguishable joint-outcome probability distributions. Under an idealized model in which the source is maximally entangled, the encoding operation is strictly local, and the channel imprints no parameter-dependent signature on the traveling photon, the reduced state of the traveling photon is maximally mixed and independent of the encoded parameter. Consequently, passive interception of the in-flight subsystem alone yields no information about the encoded message. We further develop a quantitative trace-distance framework for bounding any parameter-dependent leakage that may arise from non-ideal channel and device effects in practical implementations. The protocol introduces protocol-level design parameters not available in fixed-alphabet two-way QSDC, specifically a runtime-tunable alphabet size and native support for continuous-time analog modulation, which may be advantageous in operating environments where channel conditions vary on protocol-relevant timescales.