Protected Subspaces in Quantum Information

TL;DR

This paper explores protected subspaces in quantum information, focusing on error correction and dark subspaces, using recent mathematical results to construct examples and demonstrate their properties in various quantum channels.

Contribution

It introduces new constructions of dark and error correction subspaces for specific quantum channels using higher rank numerical ranges.

Findings

Dark spaces constructed for rank 2 maps and specific channels

Error correction subspaces identified for certain unitary noise models

Examples demonstrate the feasibility of protected subspaces in complex quantum channels

Abstract

In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by 'coding' the state space of the system before transmission into a 'protected' part of a larger state space, and by applying a proper 'decoding' map afterwards. We investigate two kinds of protected subspaces: the error correction subspaces and the dark subspaces. In neither case information leaks out of the system, but in the second it is more difficult to restore the state. Recent results on higher rank numerical ranges of operators are used to construct examples. In particular, dark spaces are constructed for any map of rank 2, for a biased permutations channel and for certain separable maps acting on multipartite systems. Furthermore, error correction subspaces are provided for a class of three unitary noise models.