Work in collaboration with Ge Bai and Valerio Scarani (NUS, Singapore), Dom Šafránek (IBS, Korea) and Joe Schindler (UAB, Spain), published today on Quantum.

In his 1932 book, von Neumann not only introduced the now familiar von Neumann entropy, but also discussed another entropic quantity that he called “macroscopic”. He argued that this macroscopic entropy, rather than the von Neumann entropy, is the key measure for understanding thermodynamic systems. Here we revisit and extend a concept derived from von Neumann’s macroscopic entropy, observational entropy (OE) — an information-theoretic quantity that measures both the intrinsic uncertainty of a system and the additional uncertainty introduced by the measurement we use to observe it.

Typically, the definition of OE assumes a “uniform prior,” i.e., it starts with an assumption of maximum uncertainty about the state of the system. However, this assumption is not always tenable, especially in more complex systems, such as those influenced by energy constraints or infinite dimensional systems, where instead other priors, such as the Gibbs distribution, would be preferable, both physically and mathematically.

Measurement is our window to the microscopic world, but it’s like a stained glass window: what’s on the other side looks distorted and coarse-grained. That’s where the second law comes in.

To fill this gap and extend OE to arbitrary priors, we first show how OE can be interpreted in two ways: as a measure of how much a measurement scrambles the true state of a system (statistical deficiency), and as the difficulty of inferring the original state from the measurement results (irretrodictability). These two aspects provide complementary insights into how much we lose or gain in our knowledge of the original state of a system when we make observations on it.

This conceptual insight leads us to introduce three generalized versions of OE: two that capture either statistical deficiency or irretrodictability, but are inherently incompatible; and a third, based on Belavkin-Staszewski relative entropy, that instead is able to combine both perspectives and provide a unified view of commuting and non-commuting priors alike. We expect that our results will pave the way for a consistent treatment of the second law of thermodynamics and fluctuation relations in fully quantum scenarios. From the abstract:

Observational entropy captures both the intrinsic uncertainty of a thermodynamic state and the lack of knowledge due to coarse-graining. We demonstrate two interpretations of observational entropy, one as the statistical deficiency resulting from a measurement, the other as the difficulty of inferring the input state from the measurement statistics by quantum Bayesian retrodiction. These interpretations show that the observational entropy implicitly includes a uniform reference prior. Since the uniform prior cannot be used when the system is infinite-dimensional or otherwise energy-constrained, we propose generalizations by replacing the uniform prior with arbitrary quantum states that may not even commute with the state of the system. We propose three candidates for this generalization, discuss their properties, and show that one of them gives a unified expression that relates both interpretations.