Fig. 3: Illustrative diagram of the emergence of the magnetic flux term in the free energy equality.

a Schematic diagram of the forced protocol in Eq. (6). The blue (red) lines denote the force along with the forward (conjugate) propagators, and the dashed lines correspond to the measurement operator in the two-point work measurement scheme²⁴. The symbol ℏ is the Planck constant, ν is the parameter in the characteristic function χ(ν), τ is the time interval of the driving process. b A schematic diagram of the propagators in the work characteristic function. The blue (red) lines are the forward (conjugate) propagators. The two solid-line propagators are from the unitary operators. The dashed red and blue (green) lines are propagators from work measurements (initial distribution³⁶). All propagators contain multiple quantum paths and are idealized as a line. The magnetic flux through the closed-loop leads to a new term in the free energy equality Eq. (4). The symbols and expressions for the propagators K are listed in the legend, where the tilde variables are those for the conjugate process and B is the magnetic field. The expressions for the propagators K are in the Methods section. The x is the position coordinate and f is the forced parameter, with their subscripts denoting time.