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Zheyuan Wu
2025-07-10 14:55:56 -05:00
parent d64d10ef6d
commit da283e28cc
5 changed files with 24 additions and 14 deletions
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22
content/Math401/Math401_T6.md
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@@ -49,12 +49,26 @@ A state $|\psi\rangle$ is entangled if it cannot be expressed as a product state
Example: the Bell state $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ is entangled. Example: the Bell state $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ is entangled.
Assume it can be written as $|\psi\rangle=|\psi_1\rangle\otimes|\psi_2\rangle$ where $|\psi_1\rangle=a|0\rangle+b|1\rangle$ and $|\psi_2\rangle=c|0\rangle+d|1\rangle$. Then: Assume it can be written as $|\psi\rangle=|\psi_1\rangle\otimes|\psi_2\rangle$ where $|\psi_1\rangle=a|0\rangle+b|1\rangle$ and $|\psi_2\rangle=c|0\rangle+d|1\rangle$. Then:
$$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$$
$$
|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle
$$
Setting this equal to $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ gives: Setting this equal to $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ gives:
$$ac|00\rangle+ad|01\rangle+bc|10\rangle+bd|11\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$$
$$
ac|00\rangle+ad|01\rangle+bc|10\rangle+bd|11\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)
$$
This requires: This requires:
$$ac=bd=\frac{1}{2}$$
$$ad=bc=0$$ $$
ac=bd=\frac{1}{2}
$$
$$
ad=bc=0
$$
This is a contradiction, so $|\psi^+\rangle$ is entangled. This is a contradiction, so $|\psi^+\rangle$ is entangled.

2
content/Math401/_meta.js
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@@ -21,4 +21,6 @@ export default {
}, },
Math401_P1: "Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)", Math401_P1: "Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)",
Math401_P1_1: "Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration", Math401_P1_1: "Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration",
Math401_P1_2: "Math 401, Paper 1, Side note 2: Page's lemma",
Math401_P1_3: "Math 401, Paper 1, Side note 3: Levy's concentration theorem",
} }

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8
next.config.mjs
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@@ -2,13 +2,7 @@ import nextra from 'nextra'
import withBundleAnalyzer from '@next/bundle-analyzer' import withBundleAnalyzer from '@next/bundle-analyzer'
const withNextra = nextra({ const withNextra = nextra({
latex: { latex: true,
renderer: 'katex',
options: {
// suppress warnings from katex for `\\`
strict: false,
}
},
mdxOptions: { mdxOptions: {
format: 'detect' format: 'detect'
}, },

6
package.json
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@@ -1,10 +1,10 @@
{ {
"scripts": { "scripts": {
"dev": "next dev", "dev": "next --turbopack",
"build": "cross-env NODE_OPTIONS='--max-old-space-size=16384' next build && npm run postbuild", "build": "cross-env VERCEL_FORCE_NO_BUILD_CACHE=1 NODE_OPTIONS='--max-old-space-size=16384' next build && npm run postbuild",
"build:test": "cross-env ANALYZE=true NODE_OPTIONS='--inspect --max-old-space-size=4096' next build", "build:test": "cross-env ANALYZE=true NODE_OPTIONS='--inspect --max-old-space-size=4096' next build",
"build:analyze": "cross-env ANALYZE=true NODE_OPTIONS='--max-old-space-size=16384' next build", "build:analyze": "cross-env ANALYZE=true NODE_OPTIONS='--max-old-space-size=16384' next build",
"postbuild": "pagefind --site .next/server/app --output-path public/_pagefind", "postbuild": "next-sitemap && pagefind --site .next/server/app --output-path public/_pagefind && cp -r ./public/_pagefind ./out",
"start": "next start" "start": "next start"
}, },
"dependencies": { "dependencies": {