# Tough error models

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# Problem

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.

# Background

For an introduction to quantum error-correction see, for example, \cite{KL02}[1].

# Partial Results

See [2], where a lower bound of $c(e,n) \gt n/(e^{2} (e^{2}+1))$ 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 $c(e,n)\leq \lceil n/e\rceil$.

# Literature

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