Quantum Information HT24, Tensor products
Flashcards
Suppose
\[\vert \alpha\rangle = \begin{pmatrix}\alpha _ 0 \\\\ \vdots \\\\ \alpha _ {d _ A -1}\end{pmatrix}, \quad \vert \beta\rangle = \begin{pmatrix}\beta _ 0 \\\\ \vdots \\\\ \beta _ {d _ B -1}\end{pmatrix}\]
Then what is $ \vert \alpha\rangle \otimes \vert \beta\rangle$?
\[\vert \alpha\rangle \otimes \vert \beta\rangle :=
\left(
\begin{array}{c}
\alpha _ 0 \beta _ 0 \\\\
\vdots \\\\
\alpha _ 0 \beta _ {d _ B-1} \\\\
\vdots \\\\
\alpha _ {d _ A-1} \beta _ 0 \\\\
\vdots \\\\
\alpha _ {d _ A-1} \beta _ {d _ B-1}
\end{array}
\right)\]
Suppose $ \vert \Psi\rangle \in \mathcal H _ A \otimes \mathcal H _ B$ and that we are considering the computational basis
\[\\{ \vert m\rangle \otimes \vert n\rangle \mid m = 0, \ldots, d _ A - 1, \, n = 0, \ldots, d _ B - 1 \\}\]
How can you expand $ \vert \Psi\rangle$?
\[\vert \Psi\rangle = \sum _ {m=0}^{d _ A-1} \sum _ {n=0}^{d _ B-1} c _ {mn} \vert m\rangle \otimes \vert n\rangle\]
What is the product bra
\[\langle \alpha \vert \otimes \langle \beta \vert\]
equal to in terms of product kets?
\[( \vert \alpha \rangle \otimes \vert \beta \rangle)^\dagger\]
What is
\[(\langle \alpha \vert \otimes \langle \beta \vert ) ( \vert \alpha' \rangle \otimes \vert \beta' \rangle)\]
?
\[\langle \alpha \vert \alpha'\rangle \langle \beta \vert \beta'\rangle\]
What is
\[(A \otimes B)( \vert \alpha \rangle \otimes \vert \beta \rangle)\]
?
\[A \vert \alpha\rangle \otimes B \vert \beta\rangle\]
What is
\[(A \otimes B)(A' \otimes B')\]
?
\[AA' \otimes BB'\]
What is
\[\left( \langle\alpha \vert \otimes \langle\beta \vert \right) (A \otimes B) \left( \vert \alpha'\rangle \otimes \vert \beta'\rangle \right)\]
?
\[\langle\alpha \vert A \vert \alpha'\rangle \langle\beta \vert B \vert \beta'\rangle\]
What is
\[(A \otimes B)^\dagger\]
?
\[A^\dagger \otimes B^\dagger\]