Study On The Dimensions Of Spontaneous Emission Error Designs

Quantum information science is a newly emerging interdisciplinary subject,which mainly uses microscopic particles as carriers and completes some tasks that classical communication,computation and cryptography cannot achieve by virtue of the unique properties of quantum mechanics.Quantum error correction code is a reliable guarantee for quantum computation and quantum communication.It can be used to solve problems such as reliability,error correction,error avoidance and error prevention.Therefore,quantum coding has naturally become a research hotspot in information science,physics and mathematics.In quantum optical systems,spontaneous emission of photons can cause quantum errors.In order to correct the quantum errors generated in the spontaneous decay process,Beth and Charnes studied quantum jump code and described its combinational properties.Then,they introduced a combination configuration called a spontaneous emission error design(SEED)contacting with quantum jump code.Let v,k,t,m be integers with 0<t<k<v and V be a set of v elements.A t-(v,k;m)SEED is a system B of some k-subsets of V with a partition B1,B2,…,Bm of B satisfying that | {B ∈ Bi:E(?)B} |/| Bi |=λE,for any 1≤i≤m and E(?)V,|E|≤t,where λE is a constant depending only on E.In addition,they also proved that a spontaneous emission error design is a quantum jump code with the same parameters.Therefore,the parameter m is called the dimension of a t-(v,k;m)SEED.This article mainly studies the upper bound and lower bound of a t-SEED.Chapter 1 is the introduction.Section 1.1 briefly introduces the research background and significance of a t-SEED;Section 1.2 analyzes the research history and the latest research status in related fields;Section 1.3 gives the main conclusions of this article.Chapter 2 utilizes the action of twisted affine groups on finite fields and obtain new lower bounds on the dimensions of 2-(q2,k;m)SEED through the Cauchy-Frobeniusburnside lemma,where q is an odd prime power,some of which outperform the known ones.Chapter 3 presents general upper bounds on the dimensions of 2-(v,5;m)SEED and describe the concrete leave graphs of the 2-SEED attaining the stated upper bounds.Chapter 4 improves the lower bounds of 3-(v,4;m)SEED by studying mutually disjoint 2-chromatic SQS(v)s when v≡2,4(mod 6).Following Phelps and Rosa’s work,we show maximum sets of mutually disjoint 2-chromatic SQS(v)s by applying direct constructions for suitable collections of triple representatives of(Fq,+)by utilizing automorphism groups in the finite fields Fq for prime powers q.Chapter 5 is the conclusion and prospect,including the main content of this article and further research issues.