Superdense coding: Difference between revisions

Revision as of 15:31, 2 September 2021

Revision as of 15:40, 2 September 2021

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1. If Alice wants to send the classical two-bit string 00 to Bob, then she applies the identity quantum gate, <math>\mathbb{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}</math>, to her qubit, so that it remains unchanged. The resultant entangled state is then

: <math>|\Phi ^{+}\rangle := |B_{00}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle)</math>

In other words, the entangled state shared between Alice and Bob has not changed, i.e. it is still <math>|\Phi ^{+}\rangle</math>. The notation <math>|B_{00}\rangle</math> is also used to remind us of the fact that Alice wants to send the two-bit string 00.

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2. If Alice wants to send the classical two-bit string 01 to Bob, then she applies the [[Quantum logic gate|quantum ''NOT'' (or ''bit-flip'') gate]], <math>X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} </math>, to her qubit, so that the resultant entangled quantum state becomes

:<math> |\Psi ^{+}\rangle := |B_{10}\rangle = \frac{1}{\sqrt{2}}(|1_A0_B\rangle + |0_A1_B\rangle)</math>

3. If Alice wants to send the classical two-bit string 10 to Bob, then she applies the [[Quantum logic gate|quantum ''phase-flip'' gate]] <math>Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}</math> to her qubit, so the resultant entangled state becomes

:<math>|\Phi ^{-}\rangle := |B_{01}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle - |1_A1_B\rangle)</math>

4. If, instead, Alice wants to send the classical two-bit string 11 to Bob, then she applies the quantum gate <math>Z*X</math> to her qubit, so that the resultant entangled state becomes

:<math>|\Psi ^{-}\rangle := |B_{11}\rangle = \frac{1}{\sqrt{2}}(|0_A1_B\rangle -|1_A0_B\rangle ) </math>

The matrices <math>X</math> and <math>Z</math> are two of the [[Pauli matrices]]. The quantum states <math>|\Phi ^{+}\rangle</math>, <math>|\Psi ^{+}\rangle</math>, <math>|\Phi ^{-}\rangle</math> and <math>|\Psi ^{-}\rangle</math> (or, respectively, <math>B_{00}, B_{01}, B_{10}</math> and <math>B_{11}</math>) are the [[Bell states]].