43 lines
1.2 KiB
Markdown
43 lines
1.2 KiB
Markdown
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Vectors to describe quantum states can get tedious really quick, so we introduce a new notation.
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We can rewrite a quantum state in the following way:
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![[Pasted image 20260730101012.png]]
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To represent the vector $e_i$ as its classical possibility we use $\ket{}$ .
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For a classical possibility x, which is the i-th possibility represented by $e_i$ we use:
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![[Pasted image 20260730101435.png]]
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Example:
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![[Pasted image 20260730101723.png]]
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Also we have 2 special cases:
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![[Pasted image 20260730102024.png]]
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$\ket{+} = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}$
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$\ket{-} = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix}$
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The Teleportation example from [[Quantum Circuits]] with ket notation:
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![[Pasted image 20260730102305.png]]
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![[Pasted image 20260730102312.png]]
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![[Pasted image 20260730102318.png]]
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## Bra and Braket notation
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Additionally to ket notation, bra notation, $\bra{}$ denotes the conjugate transpose of $\ket{}$:
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$\bra{\psi} := \ket{\psi}^\dagger$
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Example:
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![[Pasted image 20260730103526.png]]
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Combined, we get the Braket notation $\braket{\psi||\phi}$
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Example:
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![[Pasted image 20260730103836.png]]
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