RECOVERY AND RECONSTRUCTION IN QUANTUM SYSTEMS

Abstract

Quantum systems are prone to noises. Accordingly, many techniques are developed tocancel the action of a quantum operation, or to protect the quantum information against the noises. In this dissertation, I discuss two such schemes, namely the recovery channel and the quantum error correction, and various scenarios in which they are applied.

The first scenario is the perfect recovery in the Gaussian fermionic systems. When therelative entropy between two states remains unchanged under a channel, the perfect recovery can be achieved. It is realized by the Petz recovery map. We study the Petz recovery map in the case where the quantum channel and input states are fermionic and Gaussian. Gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so-called Gaussian channels. Using a Grassmann representation of fermionic Gaussian maps, we show that the Petz recovery map is also Gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel. As a by-product, we obtain a formula for the fidelity between two fermionic Gaussian states. This scenario is based on the work [1].

The second scenario is the approximate recovery in the context of quantum field theory.When perfect recovery is not achievable, the existence of a universal approximate recovery channel is proven. The approximation is in the sense that the fidelity between the recovered state and the original state is lower bounded by the change of the relative entropy under the quantum channel. This result is a generalization of previous results that applied to type-I von Neumann algebras in [2]. To deal with quantum field theory, the type of the von Neumann algebras is not restrained here. This induces qualitatively new features and requires extra proving techniques. This results hinge on the construction of certain analytic vectors and computations/estimations of their Araki-Masuda Lp norms. This part is based on the work [3].

The third scenario is applying quantum error correction codes on tensor networks on hyperbolicplanes. This kind of model is proposed to be toy models of the AdS/CFT duality, thus also dubbed holographic tensor networks. In the case when the network consists of a single type of tensor that also acts as an erasure correction code, we show that it cannot be both locally contractible and sustain power-law correlation functions. Motivated by this no-go theorem, and the desirability of local contractibility, we provide guidelines for constructing networks consisting of multiple types of tensors which are efficiently contractible variational ansatze, manifestly (approximate) quantum error correction codes, and can support power-law correlation functions. An explicit construction of such networks is also provided. It approximates the holographic HaPPY pentagon code when variational parameters are taken to be small. This part is based on the work [4].

Supplementary materials and technical details are collected in the appendices.