From 89b3a26aa731c64a66cc907e8367dac050d65ec3 Mon Sep 17 00:00:00 2001 From: Stefano Pigozzi Date: Tue, 4 Jun 2024 08:16:05 +0200 Subject: [PATCH] Progressi in IQIP --- 0 - Generale/formula di Eulero.md | 8 + 0 - Generale/operatore aggiunto.md | 22 + .../1 - Concetti base/bra.md | 9 + .../1 - Concetti base/proiezione inversa.md | 15 + .../1 - Concetti base/proiezione.md | 14 + .../1 - Concetti base/qbit.md | 1 + .../regola di Born generalizzata.md | 8 + .../stato base di un qbit.md | 21 + .../stato base di un qubit.md | 16 - .../1 - Concetti base/★ concetti base.canvas | 25 +- .../circuito quantistico a caso.png | Bin 0 -> 6151 bytes .../circuito quantistico.md | 12 + .../gate quantistico universale.md | 30 ++ .../2 - Gates semplici/gate quantistico.md | 4 + .../★ gates semplici.canvas | 10 +- .../3 - Gates complessi/Swap gate.md | 13 + .../controlled Pauli X gate.md | 18 + ...gate quantistico controllato universale.md | 17 + ...zione uguale.md => equal superposition.md} | 6 - .../modello computazionale generale.md | 2 +- .../★ modello computazionale.canvas | 2 +- .../5 - Cose strane/Hardy state.md | 15 + .../costruire un Hardy state, old.md | 509 ++++++++++++++++++ .../costruire un Hardy state.md | 77 +++ .../no-approximation theorem.md | 0 .../no-cloning theorem.md | 50 +- .../spooky action at a distance.md | 23 + .../5 - Cose strane/★ cose strane.canvas | 14 + .../6 - Teoremi/★ teoremi.canvas | 4 + 29 files changed, 908 insertions(+), 37 deletions(-) create mode 100644 0 - Generale/formula di Eulero.md create mode 100644 0 - Generale/operatore aggiunto.md create mode 100644 7 - Introduction to quantum information processing/1 - Concetti base/proiezione inversa.md create mode 100644 7 - Introduction to quantum information processing/1 - Concetti base/proiezione.md create mode 100644 7 - Introduction to quantum information processing/1 - Concetti base/regola di Born generalizzata.md create mode 100644 7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qbit.md delete mode 100644 7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qubit.md create mode 100644 7 - Introduction to quantum information processing/2 - Gates semplici/circuito quantistico a caso.png create mode 100644 7 - Introduction to quantum information processing/2 - Gates semplici/circuito quantistico.md create mode 100644 7 - Introduction to quantum information processing/2 - Gates semplici/gate quantistico universale.md create mode 100644 7 - Introduction to quantum information processing/3 - Gates complessi/Swap gate.md create mode 100644 7 - Introduction to quantum information processing/3 - Gates complessi/controlled Pauli X gate.md create mode 100644 7 - Introduction to quantum information processing/3 - Gates complessi/gate quantistico controllato universale.md rename 7 - Introduction to quantum information processing/4 - Modello computazionale/{superposizione uguale.md => equal superposition.md} (86%) create mode 100644 7 - Introduction to quantum information processing/5 - Cose strane/Hardy state.md create mode 100644 7 - Introduction to quantum information processing/5 - Cose strane/costruire un Hardy state, old.md create mode 100644 7 - Introduction to quantum information processing/5 - Cose strane/costruire un Hardy state.md rename 7 - Introduction to quantum information processing/{5 - Teoremi => 5 - Cose strane}/no-approximation theorem.md (100%) rename 7 - Introduction to quantum information processing/{5 - Teoremi => 5 - Cose strane}/no-cloning theorem.md (59%) create mode 100644 7 - Introduction to quantum information processing/5 - Cose strane/spooky action at a distance.md create mode 100644 7 - Introduction to quantum information processing/5 - Cose strane/★ cose strane.canvas create mode 100644 7 - Introduction to quantum information processing/6 - Teoremi/★ teoremi.canvas diff --git a/0 - Generale/formula di Eulero.md b/0 - Generale/formula di Eulero.md new file mode 100644 index 0000000..832e701 --- /dev/null +++ b/0 - Generale/formula di Eulero.md @@ -0,0 +1,8 @@ +[[Formula]] che mette in relazione [[funzione trigonometrica|funzioni trigonometriche]] alla [[funzione esponenziale complessa]]. + +$$ +\def \varX {{\color{yellow} x}} +\def \varI {{\color{hotpink} i}} +\Huge +e^{\varI \varX} = \cos \varX + \varI \sin \varX +$$ diff --git a/0 - Generale/operatore aggiunto.md b/0 - Generale/operatore aggiunto.md new file mode 100644 index 0000000..d03e8c7 --- /dev/null +++ b/0 - Generale/operatore aggiunto.md @@ -0,0 +1,22 @@ +[[Operazione]] [[numero complesso|complessa]] [[matrice|matriciale]] che consiste nella combinazione dell'[[operatore coniugato]] e dell'[[operatore trasposto]]. + +$$ +\huge \mathbf{M}^\dagger = \mathbf{M}'^{*} +$$ + +$$ +\begin{bmatrix} +1 + {\color{hotpink}1i} & 1 + {\color{hotpink}2i} \\ +2 + {\color{hotpink}1i} & 2 + {\color{hotpink}2i} +\end{bmatrix}^\dagger += +\begin{bmatrix} +(1 + {\color{hotpink}1i})^* & (2 + {\color{hotpink}1i})^* \\ +(1 + {\color{hotpink}2i})^* & (2 + {\color{hotpink}2i})^* +\end{bmatrix} += +\begin{bmatrix} +1 - {\color{hotpink}1i} & 2 - {\color{hotpink}1i} \\ +1 - {\color{hotpink}2i} & 2 - {\color{hotpink}2i} +\end{bmatrix} +$$ diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/bra.md b/7 - Introduction to quantum information processing/1 - Concetti base/bra.md index 58187c3..8758414 100644 --- a/7 - Introduction to quantum information processing/1 - Concetti base/bra.md +++ b/7 - Introduction to quantum information processing/1 - Concetti base/bra.md @@ -91,3 +91,12 @@ $$ \bra{11111111} = \bra{255}_8 } $$ +Corrispondono all'[[operatore aggiunto]] applicato ad un [[ket]]: +$$ +\Huge +\bra{0} = \ket{0}^\dagger +$$ +$$ +\Huge +\ket{0} = \bra{0}^\dagger +$$ \ No newline at end of file diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/proiezione inversa.md b/7 - Introduction to quantum information processing/1 - Concetti base/proiezione inversa.md new file mode 100644 index 0000000..9108ce5 --- /dev/null +++ b/7 - Introduction to quantum information processing/1 - Concetti base/proiezione inversa.md @@ -0,0 +1,15 @@ +[[Operazione]] che restitusce il coefficiente dello stato $\ket{1}$ di un dato [[qbit]]. + +$$ +\Huge +\mathbf{\dot{n}}_0 += +\begin{bmatrix} +0 & 0 \\ +0 & 1 +\end{bmatrix} += +1 - +\ket{0} +\bra{0} +$$ diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/proiezione.md b/7 - Introduction to quantum information processing/1 - Concetti base/proiezione.md new file mode 100644 index 0000000..559efec --- /dev/null +++ b/7 - Introduction to quantum information processing/1 - Concetti base/proiezione.md @@ -0,0 +1,14 @@ +[[Operazione]] che restitusce il coefficiente dello stato $\ket{0}$ di un dato [[qbit]]. + +$$ +\Huge +\mathbf{n}_0 += +\begin{bmatrix} +1 & 0 \\ +0 & 0 +\end{bmatrix} += +\ket{0} +\bra{0} +$$ diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/qbit.md b/7 - Introduction to quantum information processing/1 - Concetti base/qbit.md index 707d862..052b7a5 100644 --- a/7 - Introduction to quantum information processing/1 - Concetti base/qbit.md +++ b/7 - Introduction to quantum information processing/1 - Concetti base/qbit.md @@ -4,6 +4,7 @@ aliases: - qubit - stato di un qbit - fase di un qbit + - stato --- [[Valore]] di un sistema quantistico che può trovarsi contemporaneamente in due [[stato di un qbit|stati]] con intensità complementari tra loro. diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/regola di Born generalizzata.md b/7 - Introduction to quantum information processing/1 - Concetti base/regola di Born generalizzata.md new file mode 100644 index 0000000..33b0ebe --- /dev/null +++ b/7 - Introduction to quantum information processing/1 - Concetti base/regola di Born generalizzata.md @@ -0,0 +1,8 @@ +Generalizzazione della [[regola di Born]] a più [[qbit]]. + +$$ +\mathrm{measure}\ \ket{x} \rightarrow +\begin{cases} + \Psi & il\braket{x | \Psi}\ delle\ volte +\end{cases} +$$ diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qbit.md b/7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qbit.md new file mode 100644 index 0000000..b3ff37f --- /dev/null +++ b/7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qbit.md @@ -0,0 +1,21 @@ +Quando un [[qbit]] si trova in uno stato con massima intensità, si dice che esso si trova in uno stato base: + +$$ +\Large +\begin{bmatrix} + 1\\ + 0 +\end{bmatrix} +\qquad +\begin{bmatrix} + 0\\ + 1 +\end{bmatrix} +$$ + +Gli stati base solitamente vengono rappresentati con un [[ket]] dedicato: + +$$ +\LARGE +\ket{0} \qquad \ket{1} +$$ \ No newline at end of file diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qubit.md b/7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qubit.md deleted file mode 100644 index c3594f0..0000000 --- a/7 - Introduction to quantum information processing/1 - Concetti base/stato base di un qubit.md +++ /dev/null @@ -1,16 +0,0 @@ -Quando un [[qbit]] si trova in uno [[stato di un qbit|stato]] con massima intensità, si dice che esso si trova in uno stato base: - -$$ -\Huge -\begin{bmatrix} - 1\\ - 0 -\end{bmatrix} -\qquad -\begin{bmatrix} - 0\\ - 1 -\end{bmatrix} -$$ - -Si trovano rappresentati spesso in notazione [[ket]]. diff --git a/7 - Introduction to quantum information processing/1 - Concetti base/★ concetti base.canvas b/7 - Introduction to quantum information processing/1 - Concetti base/★ concetti base.canvas index c0bd93e..46f78c2 100644 --- a/7 - Introduction to quantum information processing/1 - Concetti base/★ concetti base.canvas +++ b/7 - Introduction to quantum information processing/1 - Concetti base/★ concetti base.canvas @@ 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\left(\frac{\theta}{2}\right) = \frac{{\color{Orange} \sqrt{10}}}{\sqrt{12}}\\\ \\ + +{\color{Green} \ket{\_1}}: & e^{i \phi} \sin \left(\frac{\theta}{2}\right) = \frac{{\color{Lime} \sqrt{2}}}{\sqrt{12}}\\ + +\end{cases} + +$$ + + + +Impostando le due dimensioni non-vincolate $\phi$ e $\lambda$ a $0$, abbiamo che: + + + +$$ + +\begin{cases} + +\cos \left(\frac{\theta}{2}\right) = \frac{{\color{Orange} \sqrt{10}}}{\sqrt{12}}\\ + +e^i \sin \left(\frac{\theta}{2}\right) = \frac{{\color{Lime} \sqrt{2}}}{\sqrt{12}}\\\phi = 0\\ + +\lambda = 0\\ + +\end{cases} + +$$ + + + +Ci basta risolvere una delle due equazioni per trovare $\theta$, quindi decidiamo di risolvere quella sopra: + + + +$$ + +\begin{cases} + +\theta = 2 \cdot \cos^{-1} \left(\frac{{\color{Orange} \sqrt{10}}}{\sqrt{12}}\right) \approx 0.841\\ + +\phi = 0\\ + +\lambda = 0\\ + +\end{cases} + +$$ + + + +Per ottenere il bit di destra dello stato $\ket{\Psi}$ dobbiamo usare il gate $\mathbf{U_0}(0.841,\ 0,\ 0)$, che chiameremo $\mathbf{U_A}$! + + + +### Costruzione del gate $\mathbf{U_B}$ da applicare al qbit 1 + + + +Ora dobbiamo configurare il qbit di sinistra, entangleandolo correttamente con il bit di destra. + + + + + + + +Lo stato $\ket{\_0}$ che avevamo raccolto prima era composto da: + + + +$$ + +\frac{1}{\sqrt{10}}\ ({\color{Orange} 3}\ {\color{Red} \ket{0\_}} + {\color{lime} 1}\ {\color{Green} \ket{1\_}}) + +$$ + + + +Allora, possiamo costruire il gate $\mathbf{U_B}$ usando un gate $\mathbf{U_1}(\theta, \phi, \lambda)$ tale che: + + + +$$ + +\mathbf{U_B} \ket{0\_} = \frac{1}{\sqrt{10}}\ ({\color{Orange} 3}\ {\color{Red} \ket{0\_}} + {\color{lime} 1}\ {\color{Green} \ket{1\_}}) + +$$ + + + +Usiamo lo stesso processo che abbiamo usato prima (saltando alcuni passaggi): + + + +$$ + +\begin{cases} + +{\color{Red} \ket{0\_}}: & \cos \left(\frac{\theta}{2}\right) = \frac{{\color{Orange} 3}}{\sqrt{10}}\\\ \\ + +{\color{Green} \ket{1\_}}: & e^{i \phi} \sin \left(\frac{\theta}{2}\right) = \frac{{\color{lime} 1}}{\sqrt{10}}\\ + +\end{cases} + +$$ + + + +Semplificandoci ancora la vita: + + + +$$ + +\begin{cases} + +\cos \left(\frac{\theta}{2}\right) = \frac{{\color{Orange} 3}}{\sqrt{10}}\\ + +e^{i \phi} \sin \left(\frac{\theta}{2}\right) = \frac{{\color{lime} 1}}{\sqrt{10}}\\ + +\phi = 0\\ + +\lambda = 0\\ + +\end{cases} + +$$ + + + +E risolvendo ancora solo l’equazione sopra, abbiamo che: + + + +$$ + +\begin{cases} + +\theta = 2 \cdot \cos^{-1} \left(\frac{{\color{orange} 3}}{\sqrt{10}}\right) \approx 0.643\\\phi = 0\\ + +\lambda = 0\\ + +\end{cases} + +$$ + + + +Per ottenere il bit di sinistra dello stato $\ket{\Psi}$ quando il bit di destra è impostato a zero dobbiamo usare il gate $\mathbf{U_1}(0.643,\ 0,\ 0)$, che chiameremo $\mathbf{U_B}$! + + + +### Costruzione del gate controllato $\mathbf{U_{C}}$ + + + +Infine, dobbiamo costruire il gate che configura il bit di sinistra dello stato $\ket{\_1}$. + + + +Esso deve costruire il seguente stato: + + + +$$ + +\frac{1}{\sqrt{2}}\ ({\color{DodgerBlue} 1} \ket{0\_} \ {\color{Turquoise} -\ 1}\ \ket{1\_}) + +$$ + + + +Inoltre, esso deve annullare le modifiche apportate da $\mathbf{U_B}$, ovvero: + + + +$$ + +\frac{1}{\sqrt{10}}\ ({\color{OrangeRed} 3} \ket{0\_} + {\color{Goldenrod} 1} \ket{1\_}) + +$$ + + + +Usiamo un gate $\mathbf{U_{01}}(\theta, \phi, \lambda)$, che chiamiamo $\mathbf{U_C}$. + + + +Dobbiamo trovare i suoi parametri, in modo tale che: + + + +$$ + +\mathbf{U_C}(\theta, \phi, \lambda) \otimes \frac{1}{\sqrt{10}}\ ({\color{OrangeRed} 3} \ket{0\_} + {\color{Goldenrod} 1} \ket{1\_}) = \frac{1}{\sqrt{2}}\ ({\color{DodgerBlue} 1} \ket{0\_} \ {\color{Turquoise} -\ 1}\ \ket{1\_}) + +$$ + + + +Abbiamo quindi che: + + + +$$ + +\mathbf{U_C} (\theta, \phi, \lambda)\ \frac{\sqrt{2}}{\sqrt{12}} \left[ \begin{matrix} + +{\color{Gray} Out} & {\color{Gray} In_{\ket{\_0}}} & {\color{Gray} In_{\ket{\_1}}} \\\ \\ + +{\color{Gray} \ket{\_0}} & {\color{OrangeRed} 3}\ \cdot\ {\color{DodgerBlue} 1} & {\color{Goldenrod} 1}\ \cdot\ {\color{DodgerBlue} 1} + +\\\ \\ + +{\color{Gray} \ket{\_1}} &{\color{OrangeRed} 3}\ \cdot\ {\color{Turquoise} -1} & {\color{Goldenrod} 1}\ \cdot\ {\color{Turquoise} -1} + +\end{matrix} \right] = 1 + +$$ + + + +E allora, che: [qual è il passaggio matematico qui?] + + + +$$ + +\begin{cases} + +\frac{\sqrt{2}}{\sqrt{12}} & ({\color{OrangeRed} 3}\ \cdot\ {\color{DodgerBlue} 1}) \cos \frac{\theta}{2} & - &({\color{Goldenrod} 1}\ \cdot\ {\color{DodgerBlue} 1})\ e^{i\lambda} \sin \frac{\theta}{2} & = 1\\ + +\\ + +\frac{\sqrt{2}}{\sqrt{12}} & \left({\color{OrangeRed} 3}\ \cdot\ {\color{Turquoise} -1} \right) e^{i \phi} \sin \frac{\theta}{2} & + & ({\color{Goldenrod} 1}\ \cdot\ {\color{Turquoise} -1})\ e^{i\phi + i\lambda} \cos \frac{\theta}{2} & = 1 + +\end{cases} + +$$ + + + +Semplificandoci ancora una volta la vita: + + + +$$ + +\begin{cases} + +\frac{\sqrt{2}}{\sqrt{12}} & ({\color{OrangeRed} 3}\ \cdot\ {\color{DodgerBlue} 1}) \cos \frac{\theta}{2} & - &({\color{Goldenrod} 1}\ \cdot\ {\color{DodgerBlue} 1}) \sin \frac{\theta}{2} & = 1\\ + +\\ + +\frac{\sqrt{2}}{\sqrt{12}} & ({\color{OrangeRed} 3}\ \cdot\ {\color{Turquoise} -1}) \sin \frac{\theta}{2} & + & ({\color{Goldenrod} 1}\ \cdot\ {\color{Turquoise} -1}) \cos \frac{\theta}{2} & = 1\\ + +\\ + +\phi = 0\\ + +\\ + +\lambda = 0 + +\end{cases} + +$$ + + + +E allora: + + + +$$ + +\begin{cases} + +({\color{OrangeRed} 3}\ \cdot\ {\color{DodgerBlue} 1}) \cos \frac{\theta}{2} & - &({\color{Goldenrod} 1}\ \cdot\ {\color{DodgerBlue} 1}) \sin \frac{\theta}{2} & = &\left({\color{OrangeRed} 3}\ \cdot\ {\color{Turquoise} -1} \right) \sin \frac{\theta}{2} & + & ({\color{Goldenrod} 1}\ \cdot\ {\color{Turquoise} -1}) \cos \frac{\theta}{2}\\ + +\\ + +\phi = 0\\ + +\\ + +\lambda = 0 + +\end{cases} + +$$ + + + +Che risulta in: + + + +$$ + +\begin{cases} + +4 \cos \frac{\theta}{2} + 2 \sin \frac{\theta}{2} = 0\\ + +\phi = 0\\ + +\lambda = 0 + +\end{cases} + +$$ + + + +Quindi: + + + +$$ + +\begin{cases} + +\theta \approx -2.214\\ + +\phi = 0\\ + +\lambda = 0 + +\end{cases} + +$$ + + + +Per ottenere il bit di sinistra dello stato $\ket{\Psi}$ quando il bit di destra è impostato a uno dobbiamo usare il gate $\mathbf{U_{01}}(-1.571,\ 0,\ 0)$, che chiameremo $\mathbf{U_C}$! + + + +### Risultato + + + +![Untitled](Glossario%206f22ab79f2da4bd4a0fcd670c58cde62/Untitled.png) + + + + \ No newline at end of file diff --git a/7 - Introduction to quantum information processing/5 - Cose strane/costruire un Hardy state.md b/7 - Introduction to quantum information processing/5 - Cose strane/costruire un Hardy state.md new file mode 100644 index 0000000..ed1e401 --- /dev/null +++ b/7 - Introduction to quantum information processing/5 - Cose strane/costruire un Hardy state.md @@ -0,0 +1,77 @@ +Per creare un [[Hardy state]] partendo da $\ket{00}$, è necessario: + +==TODO: Formattare con sintassi matematica decente.== + +1. Separare i [[qbit]] nell'equazione dello stato: + $$ + \def \noteA {{\color{grey} a}} + \def \noteB {{\color{grey} b}} + + \displaylines{ + \ket{00} = \ket{0}_\noteA \otimes \ket{0}_\noteB \\ + \ket{01} = \ket{0}_\noteA \otimes \ket{1}_\noteB \\ + \ket{10} = \ket{1}_\noteA \otimes \ket{0}_\noteB \\ + \ket{11} = \ket{1}_\noteA \otimes \ket{1}_\noteB + } + $$ +2. Raccogliere i bit dello stato: + $$ + \frac{1}{\sqrt{12}} + \cdot + {\LARGE(\ } + (\ 3 \ket{0}_\noteA + 1 \ket{1}_\noteA\ ) \otimes \ket{0}_\noteB + + + (\ 1 \ket{0}_\noteA - 1 \ket{1}_\noteA\ ) \otimes \ket{1}_\noteB + {\ \LARGE)} + $$ +3. Determinare la somma dei quadrati dei coefficienti: + $$ + \large + \begin{matrix} + \ket{0}_\noteB & : & \frac{\sqrt{3^2 + 1^2}}{\sqrt{12}} &=& \frac{\sqrt{10}}{\sqrt{12}} \\ + \ket{1}_\noteB & : & \frac{\sqrt{1^2 + 1^2}}{\sqrt{12}} &=& \frac{\sqrt{2}}{\sqrt{12}} + \end{matrix} + $$ +4. Determinare i parametri del [[gate quantistico universale]] per il secondo qbit $\mathbf{U}_\noteB (\theta, \phi, \lambda)$: + $$ + \large + \displaylines{ + \begin{cases} + \cos \frac{\phi}{2} &=& \frac{\sqrt{10}}{\sqrt{12}} \\ + e^{i \theta} \sin \frac{\phi}{2} &=& \frac{\sqrt{2}}{\sqrt{12}} \\ + \end{cases} + \\\\\updownarrow\\\\ + \begin{cases} + \phi &=& 2 \arccos \frac{\sqrt{10}}{\sqrt{12}} \\ + \theta &=& 0 \\ + \lambda &=& 0 + \end{cases} + } + $$ +5. Determinare la somma dei quadrati dei coefficienti quando il bit $\noteB$ è $\ket{0}$: + + $$ + \large + \begin{matrix} + \ket{0}_\noteA \otimes \ket{0}_\noteB & : & \frac{3}{\sqrt{12}} \\ + \ket{1}_\noteA \otimes \ket{0}_\noteB & : & \frac{1}{\sqrt{12}} + \end{matrix} + $$ +6. Determinare i parametri del [[gate quantistico universale]] per il primo qbit $\mathbf{U}_\noteA$: + $$ + \large + \displaylines{ + \begin{cases} + \cos \frac{\phi}{2} &=& \frac{3}{\sqrt{12}} \\ + e^{i \theta} \sin \frac{\phi}{2} &=& \frac{1}{\sqrt{12}} \\ + \end{cases} + \\\\\updownarrow\\\\ + \begin{cases} + \phi &=& 2 \arccos \frac{3}{\sqrt{10}} \\ + \theta &=& 0 \\ + \lambda &=& 0 + \end{cases} + } + $$ + +==TODO: Non lo so, mi sono perso.== diff --git a/7 - Introduction to quantum information processing/5 - Teoremi/no-approximation theorem.md b/7 - Introduction to quantum information processing/5 - Cose strane/no-approximation theorem.md similarity index 100% rename from 7 - Introduction to quantum information processing/5 - Teoremi/no-approximation theorem.md rename to 7 - Introduction to quantum information processing/5 - Cose strane/no-approximation theorem.md diff --git a/7 - Introduction to quantum information processing/5 - Teoremi/no-cloning theorem.md b/7 - Introduction to quantum information processing/5 - Cose strane/no-cloning theorem.md similarity index 59% rename from 7 - Introduction to quantum information processing/5 - Teoremi/no-cloning theorem.md rename to 7 - Introduction to quantum information processing/5 - Cose strane/no-cloning theorem.md index 20f47d4..ef64590 100644 --- a/7 - Introduction to quantum information processing/5 - Teoremi/no-cloning theorem.md +++ b/7 - Introduction to quantum information processing/5 - Cose strane/no-cloning theorem.md @@ -20,7 +20,7 @@ $$ \mathbf{U}_f \left( \varB \otimes \ket{0} \right) = \varB \otimes \varB $$ -Creando una [[superposizione]] generica, e usando la [[linearità]] per risolverla: +Creando una [[superposizione]] generica, e usando [[proprietà distributiva]] data dalla [[linearità]] per risolverla: $$ \displaylines{ \mathbf{U}_f \left( @@ -68,8 +68,52 @@ $$ ab \cdot ( \varA \otimes \varB ) + ab \cdot ( \varB \otimes \varA ) - } $$ -I risultati sono diversi, il che è impossibile! +I risultati possono essere uguali solo se: +$$ +\small +a \cdot \left( + \varA \otimes \varA +\right) ++ +b \cdot \left( + \varB \otimes \varB +\right) += +a^2 \cdot ( \varA \otimes \varA ) ++ +b^2 \cdot ( \varB \otimes \varB ) ++ +ab \cdot ( \varA \otimes \varB ) ++ +ab \cdot ( \varB \otimes \varA ) +$$ +Ovvero, quando: +$$ +\begin{cases} +a &=& a^2 \\ +b &=& b^2 +\end{cases} +$$ +Cioè: +$$ +\begin{cases} +a \cdot b = 0 \\\\ +a = 0 \\ +b = 1 +\end{cases} +\quad +\bigcup +\quad +\begin{cases} +a \cdot b = 0 \\\\ +a = 1 \\ +b = 0 +\end{cases} +$$ +Il gate $\mathbf{U}_f$ esiste quindi solo per gli stati [[ortogonale|ortogonali]]. + +> [!Note] +> Per gli stati $\ket{0}$ e $\ket{1}$, il gate $\mathbf{U}_f$ è il [[controlled Pauli X gate]] $\mathbf{X}_n$! diff --git a/7 - Introduction to quantum information processing/5 - Cose strane/spooky action at a distance.md b/7 - Introduction to quantum information processing/5 - Cose strane/spooky action at a distance.md new file mode 100644 index 0000000..3a31c2e --- /dev/null +++ b/7 - Introduction to quantum information processing/5 - Cose strane/spooky action at a distance.md @@ -0,0 +1,23 @@ +Fenomeno particolare che si manifesta con due [[qbit]] in [[entanglement]]. + +Anche se questi due [[qbit]] vengano fisicamente separati e messi a grande distanza tra loro, le [[gate quantistico|operazioni]] che vengono effettuate su di essi li portano a influenzare immediatamente i risultati delle [[misura|misure]] effettuate su di essi. + +> [!Quote] Albert Einstein +> _“I cannot seriously believe in it because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky action at a distance.”_ + +## Con l'[[Hardy state]] + +Sono dati due [[qbit]] a grande distanza nell'[[Hardy state]] e due [[entità]] che li controllano, dette *Alice* e *Bob*, incaricati di [[misura|misurarli]]. + +Si lascia loro la possibilità di scegliere se prima della misura vada applicato un [[Hadamard gate]] al proprio [[qbit]]. + +I risultati possibili sono i seguenti: + +| Alice | Bob | Stato (non normalizzato) | Note | +| ------------------------ | ------------------------ | --------------------------------------------------- | -------------------------------------- | +| $\color{Grey}\mathbf{1}$ | $\color{Grey}\mathbf{1}$ | $3 \ket{00} + 1 \ket{01} + 1 \ket{10} - 1 \ket{11}$ | Tutti i risultati sono possibili. | +| $\color{Grey}\mathbf{1}$ | $\color{Lime}\mathbf{H}$ | $3 \ket{00} + 1 \ket{01} + \ket{11}$ | È impossibile il risultato $\ket{10}$. | +| $\color{Lime}\mathbf{H}$ | $\color{Grey}\mathbf{1}$ | $3 \ket{00} + 1 \ket{10} + \ket{11}$ | È impossibile il risultato $\ket{01}$. | +| $\color{Lime}\mathbf{H}$ | $\color{Lime}\mathbf{H}$ | $\ket{00} + \ket{01} + \ket{10}$ | È impossibile il risultato $\ket{11}$. | + +In qualche modo, se si considerano i due [[qbit]] come oggetti fisici separati, essi devono avere trasmesso informazioni in qualche momento per raggiungere a quello stato, o alternativamente deve significare che l'universo non è [[principio di località|locale]]. diff --git a/7 - Introduction to quantum information processing/5 - Cose strane/★ cose strane.canvas b/7 - Introduction to quantum information processing/5 - Cose strane/★ cose strane.canvas new file mode 100644 index 0000000..162a2e8 --- /dev/null +++ b/7 - Introduction to quantum information processing/5 - 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