Towards an information theory of scrambling

          @misc{ pirsa_PIRSA:25060025,
            doi = {10.48660/25060025},
            url = {https://pirsa.org/25060025},
            author = {Wang, Jinzhao},
            keywords = {Quantum Gravity, Quantum Information},
            language = {en},
            title = {Towards an information theory of scrambling},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {jun},
            note = {PIRSA:25060025 see, \url{https://pirsa.org}}
          }
          

Abstract

A scrambling unitary never destroys information according to quantum information/Shannon theory. However, this framework alone doesn’t capture the fact that scrambled information can be effectively inaccessible. This limitation points to the need for a new kind of information theory—one that quantifies how much information is scrambled, rather than how much is lost to noise. To address this, we propose introducing a new family of entropies into physics: free entropy. Unlike conventional quantum entropies, which are extensive under tensor independence, free entropy has the defining feature of extensivity under freeness—the appropriate notion of independence pertaining to quantum scrambling. I will present a preliminary result showing how free entropy naturally arises in a variant of Schumacher compression, providing it with an operational interpretation as the quantum minimum description length of quantum states. I will sketch how this interpretation extends to observables and unitaries, allowing free entropy to capture an operational aspect of quantum scrambling. Finally, I will highlight striking parallels between free entropy and von Neumann entropy, suggesting that free entropy may form the foundation of a new, complementary information theory.