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.)

General Considerations

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


	$H=H_{S}\otimes I_{B}+I_{S}\otimes H_{B}+H_{I},\,\!$	(8.1)

where $H_{S}\,\!$ acts only on the system, $H_{B}\,\!$ acts only on the bath, and


	$H_{I}=\sum _{\alpha }S_{\alpha }\otimes B_{\alpha },\,\!$	(8.2)

is the interaction Hamiltonian with the $S_{\alpha }\,\!$ acting only on the system and the $B_{\alpha }\,\!$ acting only on the bath. The "error algebra" is denoted ${\mathcal {A}}\,\!$ and is the algebra generated by the set $\{H_{S},S_{\alpha }\}\,\!$ . The $S_{\alpha }\,\!$ obviously cause errors because they describe the interaction between the system and the bath. The reason the error algebra contains other terms is that when the system and bath together evolve unitarily, the exponential of the Hamiltonian $H\,\!$ means that products of the $S_{\alpha }\,\!$ with each other and also with $H_{S}\,\!$ will be present in the unitary evolution. In the case that $H_{S}\,\!$ is identically zero, or we can remove it by changing basis (to a rotating frame), the problem simplifies to the consideration only of the algebra of the $S_{\alpha }\,\!$ or the modified $S_{\alpha }\,\!$ ( $S_{\alpha }\,\!$ in the rotating frame) need be considered, respectively.

At this point, the objective is to find a set of states which will be immune to the errors which are present. Such states are identified using the error algebra. The way this is done is to put the algebra in a form which is block-diagonal. This type of algebra is said to be "reducible" which means that one may always block-diagonalize it. (See Appendix D.) Suppose that each element of the algebra can be put into the same block-diagonal form using a particular unitary transformation $U_{dns}\,\!$ . Then for any element of the error algebra, $A_{\alpha }\,\!$ , $U_{dns}A_{\alpha }U_{dns}^{\dagger }\,\!$ is block-diagonal. If the information is stored in states which are acted upon by these blocks, and only these blocks, then the information is protected because the information stays in the states which are defined by these blocks. If the blocks are $1\times 1\,\!$ blocks, i.e., $1\times 1\,\!$ submatrices, which are just numbers, then the states which make up such a system are called decoherence-free "subspaces". If the blocks are larger, then they are called decoherence-free subsystems, or noiseless subsystems.