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In this work, we study the fluctuation relation and the second law ofthermodynamics within a quantum linear oscillator externally driven over theperiod of time t = tau. To go beyond the standard approach (the two-pointprojective measurement one) to this subject and also render it discussed inboth quantum and classical domains on the single footing, we recast thisstandard approach in terms of the Wigner function and its propagator in thephase space (x,p). With the help of the canonical transformation from (x,p) tothe angle-action coordinates (\phi,I), we can then derive a measurement-free(classical-like) form of the Crooks fluctuation relation in the Wignerrepresentation. This enables us to introduce the work W_{I_0,I_{tau}}associated with a single run from (I_0) to (I_{tau}) over the period tau, whichis a quantum generalization of the thermodynamic work with its roots in theclassical thermodynamics. This quantum work differs from the energy differencee_{I_0,I_{tau}} = e(I_{tau}) - e(I_0) unless beta, hbar --> 0. Consequently, wewill obtain the quantum second-law inequality Delta F_{beta} \leq <W>_{P} \leq<e>_{P} = Delta U, where P, Delta F_{beta}, and <W>_P denote the work(quasi)-probability distribution, the free energy difference, and the averagework distinguished from the internal energy difference Delta U, respectively,while <W>_P --> Delta U in the limit of beta, hbar --> 0 only. Therefore, wecan also introduce the quantum heat Q_q = Delta U - W even for a thermallyisolated system, resulting from the quantum fluctuation therein. This is a morefine-grained result than <W>_P = Delta U obtained from the standard approach.Owing to the measurement-free nature of the thermodynamic work W_{I_0,I_{tau}},our result can also apply to the (non-thermal) initial states rho_0 = (1-gamma)rho_{beta} + gamma sigma with sigma \ne rho_{beta}.

Quantal-classical fluctuation relation and the second law of thermodynamics: The quantum linear oscillator