在這篇論文中，我們將會探討一組貝爾不等式以及它們對不同量子態之糾纏深度的驗證能力。我們會將討論分成兩大部份，第一部份主要探討與裝置無關的情形，也就是我們的結論將不需要考量任何對於實驗裝置及設定的假設。接下來我們會運用一些與裝置無關的界限值去驗證不同量子態的糾纏深度。糾纏深度是對於多體糾纏程度的一種度量方式。在第二部份中，我們將會試著用與裝置有關的角度去討論，代表我們會具體的提出測量使用的算符及可觀測量，然後找出對應的量子界限值，接著我們便可以用這組量子界限值來觀測某些量子態的糾纏深度。在這兩種狀態下，我們都計算了他們的雜訊容忍度，我們的不等式之優勢在於他不只可以檢測到糾纏，更可以檢測糾纏的程度。

In this thesis we will discuss a family of Bell-like inequalities and their ability to witness
entanglement depth against different states. The duscussion will be separated into two
major parts. In the first part, we would discuss the inequalities in a device-independent
setting, which means that the conclusions drawn do not rely on any assumption about
the experimental measurements and devices. We then use the device-independent
bound to witness the entanglement depth, which is a quantity that characterizes the
extent of many-body entanglement.
For the other part we would discuss the inequality in the device-dependent setting,
where we consider a certain set of measurements and try to find the corresponding
quantum bound of the inequalities. We then use the quantum bound to witness the
entanglement depth of some quantum state.
For each of these cases we also compute their visibility, a quantity related to noise
tolerance. The advantage of our inequalities is that it not only can certify that the
state is entangled, but also the extent of many-body entanglement.

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