Chapter 8 - Decoherence-Free/Noiseless Subsystems

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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 acts only on the system, acts only on the bath, and


(8.2)

is the interaction Hamiltonian with the acting only on the system and the acting only on the bath.