One-to-one correspondence between entanglement mechanics and black hole thermodynamics

Abstract

We establish a one-to-one mapping between entanglement entropy, energy, and temperature (quantum entanglement mechanics) with black hole entropy, Komar energy, and Hawking temperature, respectively. We show this explicitly for 4-D spherically symmetric asymptotically flat and non-flat space-times with single and multiple horizons. We exploit an inherent scaling symmetry of entanglement entropy and identify scaling transformations that generate an infinite number of systems with the same entanglement entropy, distinguished only by their respective energies and temperatures. We show that this scaling symmetry is present in most well-known systems starting from the two-coupled harmonic oscillator to quantum scalar fields in spherically symmetric space-time. The scaling symmetry allows us to identify the cause of divergence of entanglement entropy to the generation of (near) zero-modes in the systems. We systematically isolate the zero-mode contributions using suitable boundary conditions. We show that the entanglement entropy and energy of quantum scalar field scale differently in space-times with horizons and flat space-time. The relation $E=2TS$, in analogy with the horizon's thermodynamic structure, is also found to be universally satisfied in the entanglement picture. We then show that there exists a one-to-one correspondence leading to the Smarr-formula of black hole thermodynamics for asymptotically flat and non-flat space-times.