大多數物理學家或許都對古典參考系轉換中的參考系概念十分熟悉且瞭若指掌。如我
們所知,這些轉換是由伽利略變換所推得,也就是牛頓力學框架下的轉換。新的座標
系是舊座標系的函數,這些轉換包含了空間中的旋轉以及時空中的平移。顯而易見,這些轉換會影響物理量的數值。事實上,所有的觀測與測量都依賴於參考系。對於量
子態也不存在特殊的描述地位,同樣不免對於參考系的依賴。這裡比較有趣的情況,
是考慮由量子物件(例如量子粒子或量子位元)定義的參考系。這便引出了「量子參考
系」的概念,並因此也引入了「量子參考系轉換」的概念。
量子參考系轉換是由希爾伯特空間上的酉變換所主導,其中酉算符將希爾伯特空間
中的一個態向量映射到另一個態向量。然而,在轉換的過程中,量子態通常不會保留
其原本的形式。這些變化同時取決於轉換的性質,以及被選作新參考系的物體所處的
量子態的性質。當糾纏牽涉其中時,後者必須被視為該物體與系統中其他部分所構成
的複合態。這類糾纏在轉換之後將無法被觀察到。原本由子系統與新參考系物體構成
的直積態,也可能在轉換後變為一個在子系統與原參考系物體之間存在糾纏的狀態。
正是這一特性凸顯了量子參考系轉換的重要性,並驅使我們將其作為量子理論中一個
不可或缺的研究領域。
在本論文中,我們將採用 Giacomini 等人以及 Kong 在分析量子參考系轉換時
所使用的方法,並運用 Kong 在其論文《On locality of quantum information in the
Heisenberg picture for arbitrary states》中新引入的「Deutsch-Hayden 矩陣值(Deutsch-
Hayden Matrix Values,簡稱 DHMV)」的框架,來分析在一個由量子位元構成的系統
中,當量子態經歷參考系轉換時,DHMV 是如何變化的。DHMV 是我們稱之為「非對
易值(noncommutative values)」形式主義中的一部分。非對易值是非對易代數中的一
個元素,亦是由量子態所定義的可觀測量代數的一個同構影像。與我們先前提出的非
對易值概念不同的地方是 DHMV 為一個矩陣。我們將其與參考系轉換結合使用,以分
析其在參考系改變過程中的變化。
我們的討論將從古典正則參考系轉換的概念開始,並探討這一概念如何延伸至量子
參考系轉換。接著,我們將考察針對粒子系統的量子轉換,這本身即為一種正則轉換,
並討論參考系平移所帶來的意涵。利用量子參考系的平移,我們將為一個三量子位元
系統構造出一個量子參考系轉換的類比模型,其中一個量子位元將作為參考系。換句
話說,我們將研究當中兩個量子位元的量子參考系轉換,其中一個將被選定為新的參
考系。完成轉換構造之後,我們將分析在參考系轉換前的可觀測量的 DHMV,並與轉
換後的結果進行比較。在此分析中,我們發現系統的量子態與基本可觀測量(如包立矩
陣)在轉換後皆會改變其值,並進一步明確驗證 DHMV 的變化與此一致;即其轉換後
與原始值之間的關係仍然保持有與可觀測量間相同的關聯性。

Most physicists are well-antiquated and well-versed in the concept of reference frames in
the context of classical reference frame transformations. As we know, these are given by
Galilean transformations which are transformations in the context of Newtonian physics.
The new coordinate system is given as a function of the old coordinates, and the trans-
formation includes spatial rotation and translations in both space and time. It becomes
apparent that these transformations will affect the values of the physical quantities. In-
deed, every observation and measurement is reference frame dependent. Descriptions of
quantum states do not possess a special status are not exempt from being reference frame
dependent. The more interesting case here is to consider reference frames defined by quan-
tum objects, say a quantum particle or a qubit. This introduces the notion of a quantum
reference frames, which consequently introduces the notion of a quantum reference frame
transformation.
Quantum reference frame transformations are governed by unitary transformations on
the Hilbert space, wherein the unitary operator takes a state vector in the Hilbert space
to another state vector in the Hilbert space. However, a state generally does not retain its
original form during a transformation. These changes depend both on the nature of the
transformation and on the nature of the state of the object chosen as the new reference
frame. When entanglement is involved, the latter has to be taken as a composite state of
the object together with other parts of the system. Such entanglement can not be seen
after the transformation. A product state of a subsystem and the new frame object may
also be transformed into one with entanglement between the subsystem and the object
as the original frame. It is this aspect that highlights the importance and motivates the
studying of quantum reference frame transformations as an essential part of quantum theory.
In this dissertation, we will use the methods employed by Giacomini et. al. and Kong
in their analysis of quantum reference frame transformations and use the newly introduced
framework of Deutsch-Hayden Matrix Values (DHMVs) of quantum observables by Kong
in the paper “On locality of quantum information in the Heisenberg picture for arbitrary
states” to analyze how a DHMV changes as states undergo a change of reference frame
within a system of qubits. DHMVs are part of a formalism we call the “noncommutative
values (of observables)”. A noncommutative value is an element of an noncommutative
algebra. It is an isomorphic image of the observable algebra defined by the state. A
DHMV, unlike our earlier concept of a noncommutative value, is explicitly a matrix. We
use it in conjunction with a reference frame transformation to analyze how it changes
with a change in reference frame.
We begin our discussion with the concept of classical canonical reference frame trans-
formations and how that extend to quantum reference frame transformations. We then
look at quantum transformations for particle systems, which in and of itself is a canonical
transformation, and discuss the implications of a translation in reference frame. Using the
quantum translation of a reference frame, we will construct an analogue for a quantum
reference frame transformation for a system of three qubits, one of which will act as the
reference frame. That is, we will look at the quantum reference frame transformation
of two qubits; one among which we take will take as the new reference frame. After
constructing the transformation, we will analyze the DHMV of the observable before
the reference frame transformation and compare it to after the reference frame transfor-
mation. In this analysis, we found both the state and the basic observables (the Pauli
matrices) change values after undergoing the transformation, and confirm explicitly that
the DHMVs changes accordingly; maintaining the same relations between the transformed
and the original values as the relations among the observables.

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