Cite this problem as Problem 14.
An error model is an -dimensional vector space of operators acting on an -dimensional Hilbert space . A quantum code is a subspace , and is said to correct , if the projector onto satisfies for all , and suitable scalars .
*Given and , find the largest such that we can assert the existence of a code of dimension without further information about .
*Find “tough error models” for which this bound is (nearly) tight.
For an introduction to quantum error-correction see, for example, .
See , where a lower bound of is given.
A trivial upper bound on comes from taking orthogonal projections of roughly equal dimension as the error model. Since the channel with these Kraus operators (a Lüders-von Neumann projective measurement) has capacity at most , it is impossible to find larger code spaces. Hence .
 E. Knill, R. Laflamme, A. Ashikhmin, H. Barnum, L. Viola, and W. H. Zurek, Introduction to Quantum Error Correction, quant-ph/0207170 (2002).
 E. Knill, R. Laflamme, and L. Viola, Theory of Quantum Error Correction for General Noise, Phys. Rev. Lett. 84, 2525 (2000) and quant-ph/9908066.