Notes - Quantum Information HT24, Tensor products
Flashcards
Suppose
\[|\alpha\rangle = \begin{pmatrix}\alpha_0 \\\\ \vdots \\\\ \alpha_{d_A -1}\end{pmatrix}, \quad |\beta\rangle = \begin{pmatrix}\beta_0 \\\\ \vdots \\\\ \beta_{d_B -1}\end{pmatrix}\]
Then what is $ \vert \alpha\rangle \otimes \vert \beta\rangle$?
\[|\alpha\rangle \otimes |\beta\rangle :=
\left(
\begin{array}{c}
\alpha_0 \beta_0 \\\\
\vdots \\\\
\alpha_0 \beta_{d_B-1} \\\\
\vdots \\\\
\alpha_{d_A-1} \beta_0 \\\\
\vdots \\\\
\alpha_{d_A-1} \beta_{d_B-1}
\end{array}
\right)\]
Suppose $ \vert \Psi\rangle \in \mathcal H _ A \otimes \mathcal H _ B$ and that we are considering the computational basis
\[\\{ |m\rangle \otimes |n\rangle \mid m = 0, \ldots, d_A - 1, \, n = 0, \ldots, d_B - 1 \\}\]
How can you expand $ \vert \Psi\rangle$?
\[|\Psi\rangle = \sum_{m=0}^{d_A-1} \sum_{n=0}^{d_B-1} c_{mn} |m\rangle \otimes |n\rangle\]
What is the product bra
\[\langle \alpha | \otimes \langle \beta |\]
equal to in terms of product kets?
\[(| \alpha \rangle \otimes |\beta \rangle)^\dagger\]
What is
\[(\langle \alpha | \otimes \langle \beta |) (| \alpha' \rangle \otimes |\beta' \rangle)\]
?
\[\langle \alpha | \alpha'\rangle \langle \beta | \beta'\rangle\]
What is
\[(A \otimes B)(|\alpha \rangle \otimes |\beta \rangle)\]
?
\[A|\alpha\rangle \otimes B|\beta\rangle\]
What is
\[(A \otimes B)(A' \otimes B')\]
?
\[AA' \otimes BB'\]
What is
\[\left( \langle\alpha | \otimes \langle\beta | \right) (A \otimes B) \left( |\alpha'\rangle \otimes |\beta'\rangle \right)\]
?
\[\langle\alpha | A |\alpha'\rangle \langle\beta | B |\beta'\rangle\]
What is
\[(A \otimes B)^\dagger\]
?
\[A^\dagger \otimes B^\dagger\]