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We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signaling function of quantum order–disorder transitions. We discuss the possibility for such transitions to characterize interesting physical phenomena, as quantum phase transitions, or abrupt variations in correlation distributions. We apply our measure on two exactly solvable Hamiltonian models: the 1D-Quantum Ising Model (in the single-particle reduced state), and on Heisenberg XXZ spin-1/2 chain (in the two-particle reduced state). We analyze its behavior across quantum phase transitions for finite system sizes, as well as in the thermodynamic limit by using Bethe Ansatz technique.

We introduce a new kind of Entanglement Witness which is appropriate for studying genuine multipartite entanglement in large systems. e witness operator has a form that ts naturally to quantum states represented by tensor networks. It opens up novel routes for investigating properties of condensed ma er systems, and gives a new perspective to understand many-body quantum correlations. We illustrate the potential of this new operator with a GHZ state, and apply the method to a transverse Ising model with the well-known approach of Matrix Product States and Matrix Product Operators for a hundred spins.