Achieving the volume-law entropy regime with random-sign Dicke states

Manipulating entanglement, which reflects non-local correlations in a quantum system and defines the complexity of describing its wave function, represents the extremely tough challenge in the fields of quantum computing, quantum information, and condensed matter physics. In this work, by the example of the well-structured Dicke states we demonstrate that the complexity of these real-valued wave functions can be accurately tuned by introducing a random-sign structure, which allows us to explore the regime of the volume-law entanglement. Importantly, setting nontrivial sign structure one can increase the entanglement entropy of the Dicke state to the values that are close to Page's estimates for Haar-random states. The practical realization of these random-sign Dicke states is possible on different physical platforms with shallow quantum circuits. On the level of the measurements the change in the quantum state complexity due to sign structure can be traced out with the dissimilarity measure that estimates multi-scale variety of patterns in bit-string arrays.