The raw materials that you’re going to turn into the copy, and a procedure for transforming

製作拷貝所需的原料，以及把原料變成

the raw materials into a semblance of the original thing.

跟原本的東西一模一樣的方法。

To copy a famous painting , you need a blank canvas, a brush, and the right colored paints,

要複製名畫，你需要一個空白 帆布，刷子，和彩色塗料，

and then you carefully put paint on the blank canvas to match the original as closely as

然後小心翼翼地在帆布上作畫， 盡可能畫得與原來的畫一模一樣，

you can and hopefully sell it for a lot of money. But your painting isn’t exactly the

並希望把它賣掉後會拿到很多 錢。但是，你的畫並不完全

same as the original – the red is a little too bright, that stroke is a little too heavy,

和原來的一樣－紅色有點 太亮，那一劃有點太重，

there are a few too many atoms of carbon 14 in the new canvas, and so on – it’s a

新的畫布還稍微多了點碳14，等等－這是一個

copy, but not a perfect one. Is a perfect copy, identical even at the subatomic

拷貝，但不完美。 一個完美的副本，甚至連在次原子的水準上

level, even possible? Like, can you make a copy of my brain down to the neuron and beyond,

也一樣，有可能嗎？就像， 你能複製我的大腦到神經元和更小的層次，

so that even the position, momentum, and spin of every single sodium ion moving between

以至於就連每個在神經元之間 移動的鈉離子的位置、動量和自旋

neurons is exactly, indistinguishably, the same as in the original? Physicists call this

都是完全、不可區別地 與原本完全一樣？物理學家把

kind of perfect copying “cloning”, even though it definitely isn’t the same thing

這種完美複製叫做"克隆"， 它與生物學中的克隆是完全不一樣的東西

as cloning in biology where two organisms share the same DNA but how they grow and develop

生物學裡的克隆，指的是兩個生物 共享相同的DNA，但他們是如何發育成長的

can be very different – cloning in physics means a much perfecter copy, where the relative

可以有很大的差異－物理學裡的"克隆" 指的是更完美得多的副本，其中每個粒子

positions and momenta and energy levels of every particle and all of their bonds and

的相對位置、動量、能量和它們所有的鍵結，以及

interactions are exactly the same in the copy as the original, such that if you turned your

交互作用在複製品中與原本一模一樣， 如果你轉過頭

back and randomly switched them, there literally would be no way of telling which was the original

並隨機地把他們交換， 就沒有辦法告訴這是原來的

and which was the copy. Unfortunately, the universe is a party pooper,

而這是副本。 不幸的是，宇宙是一個掃興的傢伙，

and perfect cloning is impossible. I don’t simply mean that we don’t know how, or that

完善的克隆是不可能的。我不只是 指我們不知道怎麼做，或者

we haven’t succeeded yet because it’s really hard to do in practice; no, I mean

我們還沒有成功，因為它 很難實踐；不，我的意思是

that it has been mathematically proven that perfect cloning can’t be achieved even in

它已經在數學上被證明 完美克隆在根本上就是無法實現的。

principle. Here, now, is that proof, using as little

現在，我會在這裡證明，並盡量使用越少

math as possible. Everything in the universe is made up of elementary

數學越好。 宇宙中的一切是由基本的

quantum particles and the forces by which they interact , so for the no-cloning proof

量子粒子以及它們的交互作用 所組成，因此對於不可克隆原理的證明

we need to know what it means to clone a quantum particle, so first we’re going to need to

我們要知道什麼叫做複製量子 粒子，所以首先我們需要

know three important and fundamental properties shared by all quantum particles.

知道所有的量子粒子都有的 三個重要、基本的屬性。

Ok, quantum property number one: particles can be in several states at once. Like Schrödinger’s

好吧，量子特性一：粒子 可以同時在好幾個狀態。就像掩體裡

cat, stuck in a bunker with unstable gunpowder that has a 42% chance of exploding in any

薛丁格的貓和不穩定的 任何一分鐘都有42%的機會會爆炸的火藥

minute, but maybe it hasn’t yet, so that the gunpowder is in a superposition of “gunpowder

，但它也許還沒有爆炸，所以火藥是"火藥

has already exploded” and “gunpowder hasn’t exploded yet” . Or like a photon going through

已經爆炸"和"火藥還沒有 爆炸"的疊加態。或像光子同時經過

two slits at once to interfere with itself and make a nice pattern on the wall . Or an

兩個狹縫、與自身干涉 並在牆壁上留下一個很棒的圖案。或

electron in an atomic orbital, its wavefunction occupying many points in space all at once.

在原子軌域上的電子，它的波函數 一次佔據了空間中的多個點。

In summary: in quantum mechanics, the whole is equal to the sum (that is, superposition)

總而言之：在量子力學中，全部 等於他的所有不同可能狀態的

of its different possible parts . Alright, property number two: multiple particles,

總和（即，疊加）。 好吧，特性二：把多個粒子，

when viewed together as one single “object” (like an atom, or entangled pair of photons,

作為一個單一的"對象"來考慮的時候 （如一個原子或糾纏的光子對，

or the gunpowder together with Schrödinger’s cat, or whatever), are the product of their

或者火藥和薛丁格的貓， 或隨便甚麼東西），是它的組成的

components, or, since it’s quantum mechanics, a superposition of products of their components,

乘積，或者，因為它是量子力學， 它的組成的乘積的疊加態，

so the situation inside Schrödinger’s box could be described as a superposition of the

所以薛丁格的盒子裡面的情況，可以被描述為

product of “gunpowder has already exploded” and “the cat is dead” and the product

"'火藥已經爆炸'和'貓死了'的乘積"和

of “gunpowder hasn’t exploded” and “the cat is alive” . In summary: composite quantum

"'火藥尚未爆炸'和'貓是活的'的乘積" 的疊加態。總之：放一起考慮的量子

objects are multiplied together . And finally, quantum property number three:

物體被乘在一起。 最後，量子特性三：

any change to a particle that’s in a superposition of states affects all of the states independently

任何對處在疊加態中的粒子的改變 會獨立地影響所有狀態。

. Kind of like how if you go two miles to the right and one mile up and then rotate

有點像是，如果你往右兩英里 再往上一英里，然後把地圖

your map ninety degrees , that’s the same as first spinning each arrow individually

旋轉九十度，跟各旋轉這些箭頭

90° and then adding them together. Or if you have an electron in a superposition of

九十度再加在一起一樣。或者如果 你有個在往右移的電子

“here” and “there” that’s moving to the right, that means that “electron

在“這裡”和“那裡”的疊加態，這意味著“下一秒鐘

in one second” will be in a superposition of “wherever ‘here’ is in one second”

的電子“將在 “'這裡'下一秒鐘的位置”

and “wherever ‘there’ is in one second”. In summary: when you have a superposition,

和“'那裡'下一秒鐘的位置”的疊加態。 總結：當你有一個疊加態，

aka, a sum of several parts , any change or transformation of the sum of the parts is

也就是好幾部分的總和，任何對總和的改變或變換就

equal to the sum of the transformations of the parts , whether that transformation is

等於對個別部分的變換的總和，不論轉換是

a rotation, a movement, or even an entire hypothetical cloning process.

旋轉，移動，或甚至是整個假想中的克隆過程。

So let’s recap, for the no-cloning proof, we’ll use three of the properties that all

讓我們回顧一下，對於不可克隆的證明， 我們將使用宇宙裡所有基本粒子

fundamental particles in the universe obey: individual particles can be in superpositions,

都遵從的三個特性： 個別粒子可以在疊加態，

which looks like adding; groups or combinations of particles are products of their components

這看起來像加法；一群粒子是其組成的乘積

(or sums of products of their components), which looks like multiplying; and any transformation

（或者其乘積的總和）， 這看起來像乘法；任何對一個粒子

of a particle or group of particles is the same as the sum of the transformation applied

或一群粒子的變換，跟對部分做變換後的總和

to the parts, which looks like distributing. Ok, now we can get into the meat of the proof!

一樣，這看起來就像分配律。 好了，現在我們終於可以進入證明的核心部分！

So in terms of the properties we just outlined, let’s talk about what it would mean to have

所以就我們剛才列出的性質， 讓我們來談談這對於量子克隆機而言

a quantum cloning machine. We’d need the thing to be cloned , the materials to make

有甚麼意義。我們需要 被克隆的東西，用來做出克隆的

a clone out of, and a procedure to transform the materials into an exact copy of the original

原料，和一個能把原料變成跟原本 模板完全一樣的副本的過程。

. Our machine shouldn’t have to know in advance what the thing to be cloned is, otherwise

我們的機器不應該知道 要被克隆的東西是甚麼，否則

it’s not really a machine for cloning things as much as a machine for building a known

這就不是真的克隆機器， 而只是一台用來製造已知的東西

thing . So, if a cloning procedure were to exist, we should be able to “apply cloning”

的機器。所以，如果克隆過程能 存在，我們應該能“克隆”

to any specimen we want , and end up with two copies of the specimen.

任何我們想要的目標，並在結束 的時候產生兩個副本。

The problem occurs, however, if the specimen we’re cloning is a superposition, like if

但，問題出現在，我們想要克隆 的那目標是一個疊加態的時候，就像如果

it’s the gunpowder from inside Schrödinger’s cat’s box, in a superposition of “exploded”

它是薛丁格的貓的箱子裡的火藥，在“爆炸”與

and “not exploded”. If we apply our hypothetical cloning to the whole gunpowder-inside-the-box-superposition,

“還沒爆炸”的疊加態。如果我們將我們的假想 克隆過程套用在整個疊加態的火藥上，

we get “exploded” plus “not exploded” times “exploded” plus “not exploded”.

我們得到“爆炸”加“還沒爆炸”，乘“爆炸”加“還沒爆炸”。

But since, in quantum mechanics, a procedure applied to the whole gets distributed through

但是，因為在量子力學中，一個被施加在整體 的過程可以通過分配律

as the sum of the procedure applied to the parts, that means that we should get the same

等同於被施加到過程的部分的總和， 這意味著我們這麼做也應該得到相同的結果：

result by applying cloning to each part of the superposition , separately cloning “exploded”

施加克隆到的每個疊加態的部分， 分別是克隆“爆炸”

and “not exploded” and then adding them together. But, we don’t get the same thing,

和“還沒爆炸”，然後再把他們加在 一起。但是，我們沒有得到同樣的東西，

since exploded times exploded plus not exploded times not exploded is not the same as exploded

爆炸乘爆炸加沒爆炸乘沒爆炸，不會等於爆炸

times exploded plus exploded times not exploded plus not exploded times exploded plus not

乘爆炸加爆炸乘沒爆炸、再加上沒爆炸乘爆炸加

exploded times not exploded. There are these extra terms here that don’t match up.

沒爆炸乘沒爆炸。這裡有些多出來的項不相配。

Basically, if both quantum mechanics and cloning are true, then A plus B, squared must be the

基本上，如果量子力學和克隆 都是真的，那麼A加B的平方必須跟

same as A squared plus B squared. But A plus B, squared, is not the same as A squared plus

A的平方加B的平方一樣。但A加B， 再平方，不等於A平方加

B squared. And this contradiction means that either quantum mechanics is wrong (which would

B平方。而這一矛盾意味著，量子力學是錯誤的（這將

fly in the face of the most precise and accurate experimental tests in all of science ), or

與科學史上最精確和準確 的所有實驗站在對立面），或者

that a cloning procedure can’t exist. Spoiler alert: it ain’t looking so good for cloning.

克隆過程不可能存在。劇透 提醒：這看來對克隆並不怎麼好。

This, by the way, is an example of what’s called “proof by contradiction”, a logically

順便說，這就是一個所謂“反證法”的例子，在邏輯上

sound (but not always pretty) kind of proof where you suppose that the opposite of what

穩固（但並不總是漂亮）的一種證明 在這裡你假設跟你想證明的目標

you’re trying to prove is true, is true, and show that such an assumption leads to

的相反是真的，並且證明，這樣的假設會導致

a contradiction or other logical problems, so it can’t be true, and thus what you actually

矛盾或其他邏輯問題， 所以這不可能是真的，這樣你真的

are trying to prove must be true instead. Like, to prove there’s no biggest even number,

想要證明的就一定是真的。 像是要證明沒有最大的偶數，

we’d first suppose there IS a biggest even number, call it E, which since it’s even

我們會先假設有一個最大的偶數， 稱之\為E，因為它是偶數

it’s equal to two times some other number. But then if we add 1 to that other number

它等於某個數字的兩倍。 不過，如果我們加1到那個數字上

and multiply by 2, we get an even number (since it has 2 as a factor), but this new number

再乘以2，我們得到一個偶數（因為 它有個2作為因數），但這個新的數

is bigger than E, which was supposed to be the biggest even number. This is a contradiction,

比E(最大的偶數)還要更大。這是個矛盾，

so our supposition that there is a biggest even number can’t be right… so there is

因此我們有一個最大的偶數 的假設不可能是正確的......所以

no biggest even number. Ok, but back to cloning.

沒有最大的偶數。好了，回到克隆。 我還想指出我們對不可克隆

So to summarize the proof of no cloning theorem, we first suppose the cloning IS possible,

的證明並沒有檢驗任何特定的設備 或任何克隆要怎麼做的設計－它

then show that such cloning would logically results in the contradiction that

只是使用我們知道任何形式的克隆 設備都一定有的性質。像是，它會

a cloned whole would not be the same as the sum of its parts, and hence perfect cloning

存在於我們的物理世界，並且要能夠複製東西。

is not possible. I also want to point out that the proof of

這個證明證明了任何同時擁有 這兩個性質的東西不能存在。

no cloning didn’t examine any specific apparatus or design for how cloning might be done – it

要總結不可克隆定理的證明 ，我們先假設克隆是可能的，

just uses properties that we know any cloning apparatus would have to have. Like, it would

然後再說明克隆邏輯上會導致矛盾。

have to exist in our physical universe, and it would have to be able to clone things.

整體的克隆並不會和部分的克隆的總和 一樣，因此完美克隆

The proof proves that anything with both of these properties can’t exist.

是不可能的。抱歉！

However, for those wanting to live in a sci-fi future, all is not lost. Even if perfect cloning

然而，對於那些希望生活在一個科幻 未來的人們，並沒有失去一切。即使完美克隆

isn’t possible, “pretty decent copies” cloning is. Like, it’s possible to clone

是不可能的，“相當不錯的克隆” 是可能的。就像，要克隆一個量子位元

a qubit with an average of 83% fidelity . And even more exciting: the no-cloning theorem

並達到平均83％的保真度是有可能的。 更令人振奮的是：不可克隆原理

is only about cloning; teleportation is still possible.

只針對克隆；瞬間移動仍然有可能。

That’s because teleportation consists of a subject, materials to make the teleported

這是因為瞬間移動包含 被傳送者，以及用來製作被傳送者

version out of, and a procedure to turn the teleported materials into the subject, leaving

的原料，還有一個用來把原料做成被傳送者，留下

behind an empty machine. And a quick calculation shows that teleporting a superposition, or

一台空機器的過程。簡單計算 就可以發現傳送疊加態，或

sum, is indeed equal to the superposition, or sum, of the individually teleported parts!

總和，確實等於個別被傳送的部份的總和！

What’s more, “no cloning” doesn’t mean you can’t have two or more copies of

更重要的是，“不可克隆” 不代表你不能在這個宇宙裡有兩個

the same thing in the universe, it just means it’s not possible to take an existing thing

或更多一樣的東西，它只代表 不可能把現有的、你還不知道

that you don’t already know all the details about and make a perfect copy of it while

關於它的所有細節的東西，做出完美的副本

leaving the original intact. You can build a machine to make multiple versions of things

並保持原版不變。你可以製造 一台能夠生產許多複製品的機器－

as long as you know in advance exactly what it is you’re making. So, is it possible

只要你事先知道所有細節。那麼，有沒有可能

to learn every single detail about something? Well, the Heisenberg uncertainty principle

了解一樣東西的每一個細節？ 嗯，海森堡測不準原理

means that you can’t simultaneously measure all the relevant details of any one object,

告訴我們你不能同時測量 任一個樣本的所有細節，

but if you have a number of objects that you know are the same, you can measure each of

但如果你有許多你知道 都是一樣的樣本，你可以分別用不同的方法

them in a different way to get the full picture. So the irony is that in quantum mechanics,

測量他們來獲取全貌。 所以，在量子力學中諷刺的是，

you can’t perfectly clone a thing you have only one of, but if you already have a lot

你不能完美複製你只有一個的東西， 但如果你已經有很多個，

of copies of something , you can make more copies.

那麼你可以製造更多的副本。

However, as far as we know, there’s only one of each of us in the universe, so “no

然而，據我們所知，在宇宙中， 我們每個人都是獨一無二的，所以量子力學中的

100% perfect cloning” in quantum mechanics means “no 100% perfect cloning” in humans,

"不可100%完美複製"也就代表 著“不可100%完美複製”人類。

either . While we may eventually be able to grow a child that’s genetically identical

雖然我們最終可能可以 養育一個和你基因完全一樣的

to you, we likely won’t ever be able to make a perfect clone of you that has all of

孩子，我們卻永遠不能夠製造一個 你的完美的克隆人，具有所有的

your memories, thoughts, and loves. How close we can get, of course, depends on whether

你的記憶，想法和愛。我們能多麼 接近這個目標，理所當然取決於

or not consciousness relies on quantum processes in the brain. But that’s a question for

意識是否仰賴於大腦中的 量子過程。但這個問題必須留給

another day.

另一天了。