Cite this problem as **Problem 14**.

**Problem**

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.

**Background**

For an introduction to quantum error-correction see, for example, [1].

See [2], 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 .

**References**

[1] E. Knill, R. Laflamme, A. Ashikhmin, H. Barnum, L. Viola, and W. H. Zurek, *Introduction to Quantum Error Correction*, quant-ph/0207170 (2002).

[2] 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.