Citation: Bény, Cédric, and Ognyan Oreshkov. "General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels." Physical review letters 104.12 (2010): 120501.
Web: https://arxiv.org/abs/0907.5391
Tags: Error-correcting-codes, Approximate-structure
In this paper, the authors give a borad generalization of the Knill-Laflamme condition. In particular, this condition applies in the setting of approximate error correcting codes. Really, what the authors introduce is a lovely duality theorem for recovery channels which happens to have application to approximate quantum error correction.
Here is the main theorem, in words. Suppose that N and M are two quantum channels. We can consider the fidelity of the best possible recovery map for turning N into M. That is, max_R (F(RN,M)). Similarly, one can consider the fidelity of the best possible recovery map for turning hat(M) into hat(N), max_R(F(hat(N),R hat(M))). The theorem is that these two quantities are equal. Here, "hat" refers to the complimentary channel where we trace out the main system instead of the ancilla in some dilation,
The Knill-Laflamme conditions come from the case where M=id_L is the identity channel. The maximization on one side is equal to 1 iff R correct N, and the maximization on the other side is 1 if and only if hat(N) is constant. By checking coefficients, we can see that hat(N) is constant iff the Knill-Laflamme condition is satisfied.