Seminars

We introduce a family of positive linear maps in the algebra of 3 x3 complex matrices, which generalizes the seminal positive non-decomposable map originally proposed by Choi. Necessary and sufficient conditions for decomposability are derived and demonstrated. The proposed maps offer a new method for the analysis of bound entangled states of two qutrits.

Based on: https://arxiv.org/abs/2212.03807

In this work, we report a deterministic and exact algorithm to reverse any unknown qubit-encoding isometry operation. We present the semidefinite programming (SDP) to search the Choi matrix representing a quantum circuit reversing any unitary operation. We derive a quantum circuit transforming four calls of any qubit-unitary operation into its inverse operation by imposing the SU(2)×SU(2) symmetry on the Choi matrix. This algorithm only applies only for qubit-unitary operations, but we extend this algorithm to any qubit-encoding isometry operations. For that, we derive a subroutine to transform a unitary inversion algorithm to an isometry inversion algorithm by constructing a quantum circuit transforming finite sequential calls of any isometry operation into random unitary operations.

Recently it was shown, in the three-dimensional Anderson model [1] and the avalanche model of ergodicity breaking transitions [2], that the spectral form factor and the Thouless time extracted from the spectral form factor are scale invariant quantities at eigenstate transition. Thus they represent useful measures for characterisation of eigenstate transition. In the literature, an alternative definition of the Thouless time was given in terms of survival probability [3,4] which measure the stability of an initial state. Motivated by this fact, we investigate the survival probability measure and possible connections to the spectral form factor measure.

We focus on differences in behavior of the survival probability across the eigenstate transitions. Remarkably, we observe scaling invariant power-law decay of the survival probability at the transition in three physically relevant models: the three-dimensional Anderson model, one-dimensional Aubry-Andre model, and the avalanche model of ergodicity breaking transitions. We discuss connections of this universality to the universality of the spectral form factor measure. Our study [5] demonstrate that both quantities, the survival probability and the spectral form factor, are useful tool for detection of the eigenstate transitions.

[1] J. Šuntajs, T. Prosen and L. Vidmar, Annals of Physics 435, 168469 (2021)

[2] J. Šuntajs and L. Vidmar, Phys. Rev. Lett. 129, 060602 (2022)

[3] M. Schiulaz, E. J. Torres-Herrera, and L. F. Santos, Phys. Rev. B 99, 174313 (2019)

[4] T. L. M. Lezama, E. J. Torres-Herrera, F. Pérez-Bernal, Y. Bar Lev, and L. F. Santos, Phys. Rev. B 104, 085117 (2021)

[5] M. Hopjan and L. Vidmar, ArXiv:2212.13888