From 5e387b4654bf5574da0eec92d86710cc96e23451 Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Wed, 18 Jun 2025 23:34:23 -0500
Subject: [PATCH] s

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+# Topic 4: The quantum version of probabilistic concepts
+
+> In mathematics, on often speaks of non-commutative instead of quantum constructions.
+
+## Section 1: Generalities about classical versus quantum systems
+
+In classical physics, we assume that a systema's properties have well-defined values regardless of how we choose to measure them.
+
+### Basic terminology
+
+#### Observables
+
+#### Set of states
+
+The preparation of a system builds a convex set of states as our initial condition for the system. 
+
+For a collection of $N$ system. Let procedure $N_1=\lambda P_1$ be a preparation procedure for state $P_1$, and $N_2=(1-\lambda) P_2$ be a preparation procedure for state $P_2$. The state of the collection is $N=\lambda N_1+(1-\lambda) N_2$.
+
+#### Set of effects
+
+The set of effects is the set of all possible outcomes of a measurement. $\Omega=\{\omega_1, \omega_2, \ldots, \omega_n\}$. Where each $\omega_i$ is an associated effect, or some query problems regarding the system. (For example, is outcome $\omega_i$ observed?)
+
+#### Registration of outcomes
+
+A pair of state and effect determines a probability $E_i(P)=p(\omega_i|P)$. By the law of large numbers, this probability shall converge to $N(\omega_i)/N$ as $N$ increases.
+
+**Quantum states, observables, and effects can be represented mathematically by linear operators on a Hilbert space.**
+
+## Section 2: Examples of physical experiment in language of mathematics
+
+### Sten-Gernach experiment
+
+**State preparation:** Silver tams are emitted from a thermal source and collimated to form a beam.
+
+**Measurement:** Silver atoms interact with the field produced by the magnet and impinges on the class plate.
+
+**Registration:** The impression left on the glass pace by the condensed silver atoms.
+
+## Finite probability spaces in the language of Hilbert space and operators
+
+> Superposition is a linear combination of two or more states.
+
+A quantum coin can be represented mathematically by linear combination of $|0\rangle$ and $|1\rangle$.$\alpha|0\rangle+\beta|1\rangle\in\mathscr{H}\cong\mathbb{C}^2$.
+
+> For the rest of the material, we shall take the $\mathscr{H}$ to be vector space over $\mathbb{C}$.
+
diff --git a/pages/Math401/_meta.js b/pages/Math401/_meta.js
index 10a5ddc..32a5e15 100644
--- a/pages/Math401/_meta.js
+++ b/pages/Math401/_meta.js
@@ -9,4 +9,5 @@ export default {
     Math401_T1: "Math 401, Topic 1: Probability under language of measure theory",
     Math401_T2: "Math 401, Topic 2: Finite-dimensional Hilbert spaces",
     Math401_T3: "Math 401, Topic 3: Separable Hilbert spaces",
+    Math401_T4: "Math 401, Topic 4: The quantum version of probabilistic concepts",
 }
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