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The energy cost of measurement is an interesting fundamental question, and may have profound implications for quantum technologies.
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Since E = F + T S E=F+TS , where F F is the Helmholtz free-energy, the change in energy due to a change in temperature (at constant volume) is d E = d F + S d T + T d S dE=dF+SdT+TdS . By using the explicit expressions for F F and S S in terms of the partition function, one obtains d F + S d T = 0 dF+SdT=0 and thus d E = T d S dE=TdS
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