Chapter 8 - Decoherence-Free/Noiseless Subsystems

Introduction

In the last chapter we saw that it is possible, at least in principle, to detect and correct errors in quantum systems. Here a different method for protecting against errors is explored. This method encodes information into quantum states such that the information avoids errors. The information is encoded in such a way that it is invariant under the errors produced by the system-bath Hamiltonian. The initial work involved what are called decoherence-free subspaces and was later generalized to subsystems. (These terms are defined below. Reviews may be found in Whaley/Lidar and Byrd/Wu/Lidar.)

In what follows, a general quantum system will be assumed to be coupled non-trivially to a bath such that the entire system-bath Hamiltonian is given by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = H_S\otimes I_B + I_S\otimes H_B + H_I, \,\!}	(8.1)

where $H_{S}\,\!$ acts only on the system, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_B\,\!} acts only on the bath, and

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_I = \sum_\alpha S_\alpha\otimes B_\alpha, \,\!}	(8.2)

is the interaction Hamiltonian with the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_\alpha\,\!} acting only on the system and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_\alpha\,\!} acting only on the bath.