Generalized $\alpha$-Observational Entropy

Traditional measures of entropy, like the Von Neumann entropy, while fundamental in quantum information theory, are insufficient when interpreted as thermodynamic entropy due to their invariance under unitary transformations, which contradicts observed entropy increases in isolated systems. Recognizing this limitations of existing measures for thermodynamic entropy, recent research has focused on observational entropy (OE) as a promising alternative, offering practical applicability and theoretical insights. In this work, we extend the scope of observational entropy by generalizing it to a parameterized version called $\alpha$-Observational entropy ($\alpha$-OE). $\alpha$-OE is expressed in terms of the Petz-R\'{e}nyi relative entropy between the states on which a quantum-to-classical channel is applied. The $\alpha$-OE reduces to OE under $\alpha\rightarrow 1$. We prove various properties of the $\alpha$-OE, which are the generalization of the properties of OE, including the monotonically increasing of $\alpha$-OE as a function of refinement of coarse-graining. We further explore the role of $\alpha$-OE in thermodynamic contexts, particularly for the entropy production in open and closed quantum systems and its relation with the Helmholtz free energy.