於1995年時，物理學家Benjamin Schumacher在其著名的論文中，將Shannon著名的Shannon第一編碼定理成功地推廣到量子訊源，自此之後，有越來越多的物理學家，數學家，以及訊息理論學家投入量子訊息理論的研究，使得此研究領域於近十年中蓬勃發展，目前已成為一個重要的研究領域。
雖然 Schumacher 成功的將 Shannon 第一編碼定理推廣到量子訊源，然而在 Schumacher 的論文中所提出的壓縮方式是會失真的量子資料壓縮，也就是說若利用 Schumacher 的方式對一量子訊源做壓縮和解壓縮之後，並無法得到百分之百和原先的量子訊源相同的資料。基於此一觀察之下， Boström 和 Felbinger 提出一新架構並且利用量子可變長度碼來做無失真量子資料壓縮。在其論文中， Boström和 Felbinger定義了一個新的量『基礎長度』來當作度量量子可變長度碼之字碼的一個標準。由於平均基礎長度和 Boström及 Felbinger 所提出的這個無失真量子資料壓縮的壓縮率成一比例關係，而我們也知道壓縮率是評斷一個壓縮方式是好或不好的一個重要指標，因此在本篇論文裡，我們會著重在平均基礎長度這個量，並且推導出一些平均基礎長度的新下界。
在知道一量子訊源所對應的密度算子部分資訊（如：特徵值，密度算子所處在的空間之維度）的情況之下，我們推導出許多平均基礎長度的新下界，並且利用一些電腦計算出來的數據來提供更多的直覺和驗證我們的理論。

In this thesis, we consider the lossless quantum data compression scheme proposed by
Boström and Felbinger. We first obtain a lower bound on the average base length of their
proposed compression scheme in terms of the eigenvalues of the density operator associated
with the quantum source. Then we use this result to derive several new lower bounds on
the average base length. When no partial information is available about the dimension of
the source space and the eigenvalues of the density operator, we obtain a new lower bound
on the average base length in terms of the von Neumann entropy. We next consider the
case that only the largest eigenvalue of the density operator is available as side information,
and obtain several new lower bounds on the average base length by using the method of Lagrange
multiplier. Furthermore, given that the dimension of the source space and/or some
eigenvalues of the density operator are available, we derive new lower bounds in terms of the
von Neumann entropy and the available side information.

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