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Diffstat (limited to 'source/know/concept/superdense-coding/index.md')
-rw-r--r-- | source/know/concept/superdense-coding/index.md | 94 |
1 files changed, 62 insertions, 32 deletions
diff --git a/source/know/concept/superdense-coding/index.md b/source/know/concept/superdense-coding/index.md index 5c1e4ca..ba6e898 100644 --- a/source/know/concept/superdense-coding/index.md +++ b/source/know/concept/superdense-coding/index.md @@ -20,48 +20,78 @@ She could send a qubit, which has a larger state space than a classical bit, but it can only be measured once, thereby yielding only one bit of data. However, they are already sharing an entangled pair of qubits -in the [Bell state](/know/concept/bell-state/) $\ket{\Phi^{+}}_{AB}$, -where $A$ and $B$ are qubits belonging to Alice and Bob, respectively. +in the [Bell state](/know/concept/bell-state/) $$\ket{\Phi^{+}}_{AB}$$, +where $$A$$ and $$B$$ are qubits belonging to Alice and Bob, respectively. -Based on the values of the two classical bits $(a_1, a_2)$, -Alice performs the following operations on her side $A$ +Based on the values of the two classical bits $$(a_1, a_2)$$, +Alice performs the following operations on her side $$A$$ of the Bell state: <table style="width:70%;margin:auto;text-align:center;"> - <tr> - <th>$(a_1, a_2)$</th> - <th>Operator</th> - <th>Result</th> - </tr> - <tr> - <td>$00$</td> - <td>$\hat{I}$</td> - <td>$\ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$</td> - </tr> - <tr> - <td>$01$</td> - <td>$\hat{\sigma}_z$</td> - <td>$\ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$</td> - </tr> - <tr> - <td>$10$</td> - <td>$\hat{\sigma}_x$</td> - <td>$\ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$</td> - </tr> - <tr> - <td>$11$</td> - <td>$\hat{\sigma}_x \hat{\sigma}_z$</td> - <td>$\ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$</td> - </tr> +<tr> + <th markdown="1"> + $$(a_1, a_2)$$ + </th> + <th> + Operator + </th> + <th> + Result + </th> +</tr> +<tr> + <td markdown="1"> + $$00$$ + </td> + <td markdown="1"> + $$\hat{I}$$ + </td> + <td markdown="1"> + $$\displaystyle \ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$$ + </td> +</tr> +<tr> + <td markdown="1"> + $$01$$ + </td> + <td markdown="1"> + $$\hat{\sigma}_z$$ + </td> + <td markdown="1"> + $$\displaystyle \ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$$ + </td> +</tr> +<tr> + <td markdown="1"> + $$10$$ + </td> + <td markdown="1"> + $$\hat{\sigma}_x$$ + </td> + <td markdown="1"> + $$\displaystyle \ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$$ + </td> +</tr> +<tr> + <td markdown="1"> + $$11$$ + </td> + <td markdown="1"> + $$\hat{\sigma}_x \hat{\sigma}_z$$ + </td> + <td markdown="1"> + $$\displaystyle \ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$$ + </td> +</tr> </table> -Her actions affect the state on Bob's side $B$ due to entanglement. -Alice then sends her qubit $A$ to Bob over the quantum channel, +Her actions affect the state on Bob's side $$B$$ due to entanglement. +Alice then sends her qubit $$A$$ to Bob over the quantum channel, so he has both sides of the entangled pair. Finally, Bob performs a measurement of his pair in the Bell basis, which will yield a Bell state that he can then look up in the table above -to recover the values of the bits $(a_1, a_2)$. +to recover the values of the bits $$(a_1, a_2)$$. In the end, Alice only sent a single qubit, and the rest of the information transfer was via entanglement. |