A Warped Born Rule Turns an Observer into a Maxwell's Demon
Quantum mechanics seems to be an incredibly delicate theory. Slight alterations to nearly any part of its structure can lead to huge qualitative differences and wildly unphysical predictions. Here, we explore the many-body consequences of replacing the Born rule $p(m)\propto\|P_m|ψ\rangle|^2$ by a warped rule $p_b(m)\propto\|P_m|ψ\rangle\|^{2b}$. Specifically, in systems with conserved particle number, evolution punctuated by repeated local particle observations acquires a probabilistic bias over measurement histories with an effect controlled by the sign of $b-1$. For $b>1$, this bias generically drives the gas to collect into nearly frozen configurations. Similar to the action of Maxwell's demon, the simple act of observation seems to extract entropy from a closed system. For $b<1$, the bias is reversed, generically scouring out long-wavelength fluctuations to produce a hyperuniform gas. We study this effect in a one-dimensional quantum gas, where these two phases are derived analytically for a particular measurement scheme and corroborated numerically for more general procedures with discrete-time projective-measurement circuit models using matrix product state simulations. We additionally discuss how this bias manifests in a common technique that often employs this warped rule, the ``replica trick,'' implying qualitative differences between finite-replica and replica-limit calculations.