An accelerated density matrix purification scheme using generalized non-purifying functions

Yale Yauk, Yuhan Liu, Ignacio Cirac

Tensor network methods provide powerful analytical and numerical tools for characterizing quantum phases of matter. While the mathematical structure of matrix product states (MPS) is well understood through the MPS fundamental theorem, an analogous understanding for mixed-state tensor networks remains largely absent: if two purification tensors generate the same density matrix, how are they related? In this work, we initiate the study of a fundamental theorem for matrix product locally purifiable density operators (LPDOs) and focus on sequentially generated LPDOs (sLPDOs), a broad subclass admitting an interpretation in terms of successive applications of quantum channels on an initial state. We prove that, under suitable invertibility or cyclic conditions, two sLPDO representations generate the same density matrix for arbitrary system sizes if and only if they are related by a matrix product isometry acting on the purification bonds. Beyond the sLPDO setting, we provide a counterexample that suggests an obstruction to a general fundamental theorem for LPDOs with periodic boundary conditions. Finally, we discuss implications for mixed-state symmetry-protected topological phases, including the possibility of nontrivial phases protected only by weak symmetry conditions.

Jaehoon Kim, Jae‐Hoon Kim

Matthias Kleinmann, Hermann Kampermann, Theresa Meyer +1 more

We prove that no physical map can provide a perfect purification of an unknown quantum state. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, for which the reduced density matrix is identical to the original state. We investigate the possibility of performing an imperfect purification, by either admitting a non-pure output state, or a non-zero distance of its reduced state to the original. Surprisingly, the restriction of the input set to two states leads to non-trivial purifying maps.

Alan Coleman, Vyacheslav I. Yukalov