From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/superdense-coding/index.md | 94 +++++++++++++++++--------- 1 file changed, 62 insertions(+), 32 deletions(-) (limited to 'source/know/concept/superdense-coding') 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: - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + +
$(a_1, a_2)$OperatorResult
$00$$\hat{I}$$\ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$
$01$$\hat{\sigma}_z$$\ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$
$10$$\hat{\sigma}_x$$\ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$
$11$$\hat{\sigma}_x \hat{\sigma}_z$$\ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$
+ $$(a_1, a_2)$$ + + Operator + + Result +
+ $$00$$ + + $$\hat{I}$$ + + $$\displaystyle \ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$$ +
+ $$01$$ + + $$\hat{\sigma}_z$$ + + $$\displaystyle \ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$$ +
+ $$10$$ + + $$\hat{\sigma}_x$$ + + $$\displaystyle \ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$$ +
+ $$11$$ + + $$\hat{\sigma}_x \hat{\sigma}_z$$ + + $$\displaystyle \ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$$ +
-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. -- cgit v1.2.3