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We present a new inequality which holds in the thermodynamical processes withmeasurement and feedback controls with using only the Helmholtz free energy andthe entanglement of formation: $W_{\mathrm{ext}}\le-\Delta F-k_{B}T\DeltaE_{F}$. The quantity $-\Delta E_{F}$, which is positive, expresses the amountof entanglement transfer from the system $S$ to the probe $P$ through theinteraction $\hat{U}_{SP}$ during the measurement. It is easier to achieve theupper bound in the new inequality than in the Sagawa-Ueda inequality [Phys.Rev. Lett. 100 080403 (2008)]. The new inequality has clear physical meaning:in the above thermodynamical processes, the work which we can extract from thethermodynamic system is greater than the upper bound in the conventionalthermodynamics by the amount of the entanglement extracted by the measurement.

Second law of the information thermodynamics with entanglement transfer