Tough error models

Cite this problem as Problem 14.


An error model E is an e-dimensional vector space of operators acting on an n-dimensional Hilbert space H. A quantum code is a subspace C\subset H, and is said to correct E, if the projector P_C onto C satisfies P_C A^{*} B P_C = \lambda(A,B) P_C for all A,B\in E, and suitable scalars \lambda(A,B).

*Given e and n, find the largest c=c(e,n) such that we can assert the existence of a code C of dimension c without further information about E.

*Find “tough error models” for which this bound is (nearly) tight.


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

See [2], where a lower bound of c(e,n) > n/(e^{2} (e^{2}+1)) is given.

A trivial upper bound on c(e,n) comes from taking orthogonal projections of roughly equal dimension n/e as the error model. Since the channel with these Kraus operators (a Lüders-von Neumann projective measurement) has capacity at most n/e, it is impossible to find larger code spaces. Hence c(e,n)\leq \lceil n/e\rceil.


[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.