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Diffstat (limited to 'source/know/concept/no-cloning-theorem')
-rw-r--r-- | source/know/concept/no-cloning-theorem/index.md | 12 |
1 files changed, 6 insertions, 6 deletions
diff --git a/source/know/concept/no-cloning-theorem/index.md b/source/know/concept/no-cloning-theorem/index.md index d4ca0d4..a91ae6f 100644 --- a/source/know/concept/no-cloning-theorem/index.md +++ b/source/know/concept/no-cloning-theorem/index.md @@ -10,13 +10,13 @@ layout: "concept" --- In quantum mechanics, the **no-cloning theorem** states -there is no general way to make copies of an arbitrary quantum state $\ket{\psi}$. +there is no general way to make copies of an arbitrary quantum state $$\ket{\psi}$$. This has profound implications for quantum information. To prove this theorem, let us pretend that a machine exists that can do just that: copy arbitrary quantum states. -Given an input $\ket{\psi}$ and a blank $\ket{?}$, -this machines turns $\ket{?}$ into $\ket{\psi}$: +Given an input $$\ket{\psi}$$ and a blank $$\ket{?}$$, +this machines turns $$\ket{?}$$ into $$\ket{\psi}$$: $$\begin{aligned} \ket{\psi} \ket{?} @@ -24,7 +24,7 @@ $$\begin{aligned} \ket{\psi} \ket{\psi} \end{aligned}$$ -We can use this device to make copies of the basis vectors $\ket{0}$ and $\ket{1}$: +We can use this device to make copies of the basis vectors $$\ket{0}$$ and $$\ket{1}$$: $$\begin{aligned} \ket{0} \ket{?} @@ -36,7 +36,7 @@ $$\begin{aligned} \ket{1} \ket{1} \end{aligned}$$ -If we feed this machine a superposition $\ket{\psi} = \alpha \ket{0} + \beta \ket{1}$, +If we feed this machine a superposition $$\ket{\psi} = \alpha \ket{0} + \beta \ket{1}$$, we *want* the following behaviour: $$\begin{aligned} @@ -47,7 +47,7 @@ $$\begin{aligned} &= \Big( \alpha^2 \ket{0} \ket{0} + \alpha \beta \ket{0} \ket{1} + \alpha \beta \ket{1} \ket{0} + \beta^2 \ket{1} \ket{1} \Big) \end{aligned}$$ -Note the appearance of the cross terms with a factor of $\alpha \beta$. +Note the appearance of the cross terms with a factor of $$\alpha \beta$$. The problem is that the fundamental linearity of quantum mechanics dictates different behaviour: |