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From Quantum Economics and Finance: An Applied Mathematics Introduction by David Orrell

Consider $$H=H_C \otimes H_P$$ where $H=H_C$ is the coin space spanned by $\{|\uparrow\rangle,|\downarrow\rangle\}$ and position space $H_P$ is spanned by a discrete set of position basis states $\{|x_j\rangle:j \in Z \}$

Consider the translation operator of the quantum walk,

$$ T = |\uparrow\rangle \langle\uparrow|\otimes \sum_j|x_{j+1}\rangle\langle x_j|+|\downarrow\rangle \langle \downarrow | \otimes \sum_j |x_{j-1}\rangle\langle x_j| $$

which has the effect of shifting the position $x_j$ up or down depending on whether the coin state is up or down.

If |$\uparrow$> means coin toss represents by vector (1 0), what does <$\uparrow$ | mean and why we need it?

In addition, how should we view |xj+1⟩ w.r.t ⟨xj|?

Thank you :)

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  • $\begingroup$ Welcome to physcs.SE Jessica – you can typeset math with MathJax here. This is preferred to posting images of equations. See the help on notation. $\endgroup$
    – Sebastian Riese
    Commented Mar 17, 2022 at 18:41
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    $\begingroup$ Does "| up arc >" mean $|\uparrow\rangle$ i.e. "up arrow" rather than "up arc"? $\endgroup$
    – John Rennie
    Commented Mar 17, 2022 at 18:46
  • $\begingroup$ One further question: I don't understand what the sentence with the part with the toin coss shall convey. In general $\langle \psi \rvert \colon H \to \mathbb{R}$ is the dual of the vector $\lvert \psi \rangle$, its a linear operator that maps elements from the Hilbert space to the inner product of its argument and $\lvert \psi \rangle$. (So in matrix language one may interpret it as the transpose of the vector). $\endgroup$
    – Sebastian Riese
    Commented Mar 17, 2022 at 18:50
  • $\begingroup$ Thank you Sebastian and John, I add more information now. If ⟨ψ| is the dual of the vector |ψ⟩, how should we understand |$x_j+1$⟩ and ⟨$x_j$|? Because in the setting, $x_j+1$ is the next position of $x_j$? Did I understand something wrongly? $\endgroup$
    – Jessica
    Commented Mar 17, 2022 at 22:47
  • $\begingroup$ Hello! For an introduction to Dirac's bra-ket notation ($|\rangle $, $\langle |$) you can check chapter 1 from Sakurai's Modern Quantum Mechanics. The idea of translation operator for single particle quantum system has been introduced $\endgroup$
    – KP99
    Commented Mar 18, 2022 at 4:44

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If |$\uparrow$> means coin toss represents by vector (1 0), what does <$\uparrow$ | mean and why we need it?

If $|\uparrow\rangle$ is the "ket" represented by the column vector $$ |\uparrow\rangle \to \left( \begin{matrix} 1\\0 \end{matrix} \right) $$ then $\langle \uparrow|$ is the "bra" represented by the row vector $$ \langle\uparrow| \to \left( \begin{matrix} 1 &0 \end{matrix} \right) $$

In addition, how should we view |xj+1⟩ w.r.t ⟨xj|?

Same deal. $\langle x_j|$ is the "bra" corresponding to the "ket" $|x_j\rangle$.

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