Tough error models

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Cite as: http://qig.itp.uni-hannover.de/qiproblems/14

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Problem

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

  • Given [math]e[/math] and [math]n[/math], find the largest [math]c=c(e,n)[/math] such that we can assert the existence of a code [math]C[/math] of dimension [math]c[/math] without further information about [math]E[/math].
  • 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 [math]c(e,n) \gt n/(e^{2} (e^{2}+1))[/math] is given.

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

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.