Entanglement and Approximate Quantum Error Correction

The possibility of performing quantum error correction obviously lies behind and justifies the vast efforts made up to now in order to develop quantum computation techniques, since it allows fault-tolerant computationeven when quantum systems—in fact extremely sensitive to noise—are considered as the basic carriers of information. Besides well-known algebraic conditions for exact quantum error correction, which directly lead to algebraic quantum error correcting codes (for a thorough presentation of quantum error correction theory and a detailed account about the enormous literature about it, see e. g. [2, 3]), an informationtheoretical approach to quantum error correction [4, 5, 6] can shed some light on the dynamical processes which underlie quantum noise, offering at the same time the opportunity to better understand the conditions under which approximate quantum error correction is feasible [7]. In the present paper, we will be working within the latter scenario. Approximate quantum error correction is not just a theoretical issue: in fact, in all practical implementations the experimenter can only rely upon some confidence level—exact processes exist as abstract mathematical concepts only. Then, conditions for approximate quantum error correction can provide useful ways to test the reliability of a real apparatus. In Ref. [7], Schumacher and Westmoreland proved that an adequate information-theoretical quantity to consider is the coherent information: the loss of coherent information along a quantum noisy channel is small if and only if the quantum noisy channel can be approximately corrected. In a subsequent paper [8], the same Authors provided another criterion,