### Memory Effect in Upper Bound of Heat Flux Induced by Quantum Fluctuations *[Cross-Listing]*

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We develop a model of quantum open systems as a quantum Brownian motion coupled to a classical heat bath by introducing a mathematical definition of operator differentials. We then define a heat operator by extending the stochastic energetics and show that this operator satisfies properties corresponding to the first and second laws in thermodynamics. We further find that the upper bound of the heat flux depends on the memory effect induced by quantum fluctuations and hence the maximum extractable work can be qualitatively modified in quantum thermodynamics.