SciPost Phys. Core 6, 048 (2023) ·
published 10 July 2023

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We analyze the effect of a simple coin operator, built out of Bell pairs, in a 2d Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the Entangling Power of the coin operator. Secondly, we compute the Generalized Relative Rényi Entropy between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the Entangling Power and Generalized Relative Rényi Entropy behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the 2d DQRW into two 1d massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.
SciPost Phys. 11, 002 (2021) ·
published 8 July 2021

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An intriguing correspondence between ingredients in geometric function theory related to the famous Bieberbach conjecture (de Branges' theorem) and the nonperturbative crossing symmetric representation of 22 scattering amplitudes of identical scalars is pointed out. Using the dispersion relation and unitarity, we are able to derive several inequalities, analogous to those which arise in the discussions of the Bieberbach conjecture. We derive new and strong bounds on the ratio of certain Wilson coefficients and demonstrate that these are obeyed in oneloop $\phi^4$ theory, tree level string theory as well as in the Smatrix bootstrap. Further, we find two sided bounds on the magnitude of the scattering amplitude, which are shown to be respected in all the contexts mentioned above. Translated to the usual Mandelstam variables, for large $s$, fixed $t$, the upper bound reads $\mathcal{M}(s,t)\lesssim s^2$. We discuss how Szegö's theorem corresponds to a check of univalence in an EFT expansion, while how the Grunsky inequalities translate into nontrivial, nonlinear inequalities on the Wilson coefficients.