Trance-0/NoteNextra-origin
1
0
Fork 0
Code Issues Pull Requests Actions 1 Packages Projects Releases Wiki Activity

Update Math401_T4.md

Browse Source
This commit is contained in:
Zheyuan Wu
2025-06-25 12:52:56 -05:00
parent eacffecb21
commit 316873f7ba
1 changed files with 5 additions and 3 deletions
Download Patch File Download Diff File Expand all files Collapse all files

8
pages/Math401/Math401_T4.md
View File

@@ -203,7 +203,7 @@ Let $A,B\in\mathscr{F}$ be the set of all events in the classical probability se
> >
> This can be found in linear algebra. [Orthogonal projection](https://notenextra.trance-0.com/Math429/Math429_L28#definition-655) > This can be found in linear algebra. [Orthogonal projection](https://notenextra.trance-0.com/Math429/Math429_L28#definition-655)
Let $P,Q\in\mathscr{P}$ be the event in probability space. $R(\cdot)$ is the range of the operator. $P^\perp$ is the orthogonal complement of $P$. Let $P,Q\in\mathscr{P}$ be the event in non-commutative (_quantum_) probability space. $R(\cdot)$ is the range of the operator. $P^\perp$ is the orthogonal complement of $P$.
| Classical | Classical interpretation | Non-commutative (_Quantum_) | Non-commutative (_Quantum_) interpretation | | Classical | Classical interpretation | Non-commutative (_Quantum_) | Non-commutative (_Quantum_) interpretation |
| --------- | ------- | -------- | -------- | | --------- | ------- | -------- | -------- |
@@ -222,7 +222,7 @@ In quantum probability theory, $P\land(Q\lor R)\neq(P\land Q)\lor(P\land R)$ in
#### Definition of states (non-commutative (_quantum_) probability theory) #### Definition of states (non-commutative (_quantum_) probability theory)
A state on $(\mathscr{H},\mathscr{P})$ is a map $\mu:\mathscr{P}\to[0,1]$ such that: A state on $(\mathscr{B}(\mathscr{H}),\mathscr{P})$ is a map $\mu:\mathscr{P}\to[0,1]$ such that:
1. $\mu(O)=0$, where $O$ is the zero projection. 1. $\mu(O)=0$, where $O$ is the zero projection.
2. If $P_1,P_2,\cdots,P_n$ are pairwise disjoint orthogonal projections, then $\mu(P_1\lor P_2\lor\cdots\lor P_n)=\sum_{i=1}^n\mu(P_i)$. 2. If $P_1,P_2,\cdots,P_n$ are pairwise disjoint orthogonal projections, then $\mu(P_1\lor P_2\lor\cdots\lor P_n)=\sum_{i=1}^n\mu(P_i)$.
@@ -247,6 +247,8 @@ $$
(under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal) (under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal)
Every basis of $\mathscr{H}$ can be decomposed to these forms.
#### Theorem: Born's rule #### Theorem: Born's rule
Let $\rho$ be a density operator on $\mathscr{H}$. then Let $\rho$ be a density operator on $\mathscr{H}$. then
@@ -259,7 +261,7 @@ Defines a probability measure on the space $\mathscr{P}$.
[Proof ignored here] [Proof ignored here]
#### Theorem: Gleason's theorem #### Theorem: Gleason's theorem (very important)
Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that