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[td]As a thought experiment in quantum physics, a large number of KCBS pentagrams are imagined, each with colorings hidden. On each pentagram, theoretical observer uncovers 2 vertices that share a common edge. <!-- I cut out a chunk of text as follows:[/td]Revision as of 11:24, 31 August 2025
[/td][td]As a thought experiment in quantum physics, a large number of KCBS pentagrams are imagined, each with colorings hidden. On each pentagram, theoretical observer uncovers 2 vertices that share a common edge. <!-- I cut out a chunk of text as follows:[/td] [td]"This random choice is necessary because if the pentagram producers had been able to guess your choice for each pentagram in advance, he could have "conspired" to fool you."[/td]
[td]"This random choice is necessary because if the pentagram producers had been able to guess your choice for each pentagram in advance, he could have "conspired" to fool you."[/td] [td]I don't know much about this topic, so if this is important, please help reword it and put it back in, otherwise just delete this comment. -->Doing this shows that no matter which edge you choose, it always ends with finding blue-blue with a probability of <math>1-\frac{2}{\sqrt 5}</math>, red-blue with <math>\frac{1}{\sqrt 5}</math>, and blue-red with <math>\frac{1}{\sqrt 5}</math>. So, the expectation value of the sum of mismatches is <math>2\sqrt 5 \approx 4.47 > 4</math>.[/td]
[td]I don't know much about this topic, so if this is important, please help reword it and put it back in, otherwise just delete this comment. -->Doing this shows that no matter which edge is chosen, it always ends with finding blue-blue with a probability of <math>1-\frac{2}{\sqrt 5}</math>, red-blue with <math>\frac{1}{\sqrt 5}</math>, and blue-red with <math>\frac{1}{\sqrt 5}</math>. So, the expectation value of the sum of mismatches is <math>2\sqrt 5 \approx 4.47 > 4</math>.[/td] [td][/td]
[td][/td] [td]To explain, each pentagram is a 3D quantum system with an orthonormal basis <math>\left\{ |A\rangle, |B \rangle, |C\rangle \right\}</math>, and is initialized to <math>|C\rangle</math>. Each vertex is assigned a 1D projector projecting to <math>\frac{1}{\sqrt{\sqrt{5}}}|C\rangle + \sqrt{1-\frac{1}{\sqrt{5}}} \left[ \cos\left( \frac{4\pi n}{5}\right)|A\rangle + \sin\left( \frac{4\pi n}{5} \right)|B\rangle \right]</math>, ''n'' = 0, ..., 4 .[/td]
[td]To explain, each pentagram is a 3D quantum system with an orthonormal basis <math>\left\{ |A\rangle, |B \rangle, |C\rangle \right\}</math>, and is initialized to <math>|C\rangle</math>. Each vertex is assigned a 1D projector projecting to <math>\frac{1}{\sqrt{\sqrt{5}}}|C\rangle + \sqrt{1-\frac{1}{\sqrt{5}}} \left[ \cos\left( \frac{4\pi n}{5}\right)|A\rangle + \sin\left( \frac{4\pi n}{5} \right)|B\rangle \right]</math>, ''n'' = 0, ..., 4 .[/td]
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