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The two-time measurement scheme is well studied in the context of quantum fluctuation theorem. However, it becomes infeasible when the random variable determined by a single measurement trajectory is associated with the von Neumann entropy of the quantum states. We employ the one-time measurement scheme to derive a Jarzynski-like equality of nonequilibrium information production by proposing an information production distribution based on the quantum cross-entropy. The derived equality further enables one to explore the roles of the quantum cross-entropy in quantum communications, quantum machine learning, and quantum thermodynamics.