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"# Correlation in Entangled States Lab\n",
"\n",
"## Objectives:\n",
"* Experimentally determine the difference between two particles in a product state vs an entangled state using the entanglement simulator https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/entanglement/entanglement.html\n",
"* Apply the idea of basis changing to explain the correlation that is observed"
]
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"## Questions\n",
"Alice and Bob each measure one of two qubits with a Stern-Gerlach apparatus. Start with **both SGAs along the z-axis.** \n",
"\n",
"1. Try sending pairs of particles in a product state$|↑A〉|↓B〉$. **What do Alice and Bob measure individually?**"
]
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"2. Try sending pairs of particles in an entangled state:$1√2(|↑A〉|↓B〉− |↓A〉|↑B〉)$. **What do Alice and Bob measure individually?**"
]
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"3. If Alice measures her spin, **would you be able to predict Bob’s result:\n",
"\n",
" (a) In the product state?\n",
" \n",
" (b) In the entangled state?**"
]
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"Now rotate **both SGAs along the x-axis**\n",
"\n",
"4. Try sending pairs of particles in a product state $|↑A〉|↓B〉$. **What do Alice and Bob measure individually?**"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": []
},
{
"cell_type": "markdown",
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"source": [
"5. Try sending pairs of particles in an entangled state $1√2(|↑A〉|↓B〉− |↓A〉|↑B〉)$. **What do Alice and Bob measure individually?**"
]
},
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"cell_type": "markdown",
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{
"cell_type": "markdown",
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"source": [
"6. If Alice measures her spin, **would you be able to predict Bob’s result:\n",
"\n",
" (a) In the product state?\n",
"\n",
" (b) In the entangled state?**"
]
},
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"7. Convert the product state $|↑A〉|↓B〉$ into the x-basis and use it to **explain the observations in the x-basis.** Recall that $|↑〉= 1√2(|+〉+ |−〉)$ and $|↓〉=1√2(|+〉− |−〉)$."
]
},
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"8. Convert the entangled state $1√2(|↑A〉|↓B〉−|↓A〉|↑B〉)$ into the x-basis and use it to **explain the measurements in the x-basis.**"
]
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"9. Suppose that there are two possible sources of particles. Source #1 randomly emits two particles in either the state $|↑A〉|↓B〉$ or $|↓A〉|↑B〉$ with equal probability. Source #2 emits two particles in the entangled state $1√2(|↑A〉|↓B〉−|↓A〉|↑B〉)$. **How can Alice and Bob tell whether the source is #1 or #2?**"
]
},
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{
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"source": [
"# Superposition vs. Mixed States Lab\n",
"\n",
"## Objectives\n",
"\n",
"* Experimentally determine the difference between particles in a superposition state and amixed state using the superposition states and mixed states simulator https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/superposition/superposition-mixed-states.html\n",
"* Apply the idea of basis changing to explain the experimental results.\n",
"* Compute the probability amplitudes given measurement results.\n",
"\n",
"## Questions\n",
"\n",
"1. We send 100 electrons of unknown spin into a Stern-Gerlach apparatus. We measure that 50 are spin up and 50 are spin down. **We can conclude that:**\n",
"\n",
" (a) 100 electrons were in a 50/50 superposition state of up and down (superposition state).\n",
" \n",
" (b) The electrons were a classical mixture of 50 electrons spin up and 50 spindown (mixed state).\n",
" \n",
" (c) Not enough information"
]
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"2. Use the simulator to compare the measurement outcomes of the mixed particles vs. the superposition particles. **What are the similarities and differences?**"
]
},
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"3. By making a basis change with $|0〉=1√2|+〉+1√2|−〉$ and $|1〉=1√2|+〉− 1√2|−〉$ , **can you explain the similarities and differences mathematically?**"
]
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"4. Which of the two inputs labelled “Superposition or mixture?” and “Superposition or mixture??” is a random mixture and which is a superposition?"
]
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"5. The mixture consists of a fraction $A$ of spin up particles and a fraction $B$ of spin down particles. **Find these fractions, $A$ and $B$.**"
]
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"6. The superposition state can be written as $α|0〉+β|1〉$. **Find the amplitudes $α$ and $β$ assuming they are real and positive.**"
]
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"7. Use a basis change to **show that the amplitudes α and β give the correct probabilities** in both the x- and z-basis"
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