Why can’t quantum states be cloned?

You may have heard of the “no clone” theorem in quantum computing, why is it a thing?
One of the basic processes in computation is copying data: take a bit $b_1$ and copy it to a new bit $b_2$ . Think of this in terms of functions

$f: B\times B\rightarrow B, \quad f(b_1,\&b_2)=b_1$

in which we are using C pointer notation (ampersand) for the memory location of the blank bit $b_2$ that will be overwritten with the copy of $b_1$ . Now let’s look at some quantum computing functions (gates).

Classical bits are $b=0$ , $b=1$ , quantum bit is a superposition, which is at the core of the question

$|q\rangle=a|0\rangle+b|1\rangle$

only measurable things are $|a|$ and $|b|$ since $|q\rangle$ and $e^{i\phi}|q\rangle$ represent the same quantum state.
A quantum system with $n$ qubits contains the information of $2^n$ complex numbers, so modeling such a system with a classical computer will require storage of a vast amount of floating point data.

Hadamard gate

In other words

$|x,\, y\rangle \rightarrow |x, \, XOR(x,y)\rangle$

$\begin{array}{ccc}\hline x & y & XOR(x,y)\\ \hline 0 & 0 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0\end{array}$

If $\hat{U}$ is a single-qubit unitary operator, a control-U gate is

$|0,0\rangle\rightarrow |0,0\rangle,\quad |0,1\rangle\rightarrow |0,1\rangle$

but

$|1,0\rangle\rightarrow |1\rangle\otimes \hat{U}|0\rangle,\quad |1,1\rangle\rightarrow |1\rangle\otimes \hat{U}|1\rangle$

Fool around with some gate operations:
Start with

Apply control NOT with control on the second cubit

$\rightarrow a|00\rangle+b|11\rangle$

Apply control NOT with control on the first cubit

$\rightarrow a|00\rangle+b|10\rangle$

Apply control NOT with control on the second cubit

Start with

Apply control NOT with control on the second cubit

$\rightarrow a|10\rangle+b|01\rangle$

Apply control NOT with control on the first cubit

$\rightarrow a|11\rangle+b|01\rangle$

Apply control NOT with control on the second cubit

this sequence of gates will swap states

$CN_2\cdot CN_1\cdot CN_2|\phi\rangle\otimes|\psi\rangle=|\psi\rangle\otimes|\phi\rangle$

In terms of matrices

$CN_1\left(\begin{array}{c}00\\ 01\\ 10\\ 11\end{array}\right)=\left(\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0\end{array}\right)\left(\begin{array}{c}00\\ 01\\ 10\\ 11\end{array}\right), \qquad CN_2\left(\begin{array}{c}00\\ 01\\ 10\\ 11\end{array}\right)$

$=\left(\begin{array}{cccc}1&0&0&0\\ 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\end{array}\right)\left(\begin{array}{c}00\\ 01\\ 10\\ 11\end{array}\right)$

and the swapping operation is

$CN_2\cdot CN_1\cdot CN_2\left(\begin{array}{c}00\\ 01\\ 10\\ 11\end{array}\right)=\left(\begin{array}{cccc}1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\end{array}\right)\left(\begin{array}{c}00\\ 01\\ 10\\ 11\end{array}\right)$

$=CN_1\cdot CN_2\cdot CN_1\left(\begin{array}{c}00\\ 01\\ 10\\ 11\end{array}\right)$

Cloning
In the Hilbert space $\mathcal{H}$ , make a copy of arbitrary, unknown basis state $|e_k\rangle$ , perhaps “overwriting” or transforming some template or “blank” state $|e_1\rangle$ into a copy of $|e_k\rangle$ using a function $\hat{U}$ ( aquantum operator)

$|e_k\rangle\otimes|e_1\rangle\rightarrow |e_k\rangle\otimes|e_k\rangle, \qquad \hat{U}(|e_k\rangle\otimes|e_1\rangle)= |e_k\rangle\otimes|e_k\rangle$

Experimentor \#1 has a state $|e_k\rangle$ that she can measure, and access (at least for now) to a template state $|e_1\rangle$ , and experimentor \#1 wishes to provide experimentor \#2 with a copy of their state.
Try to clone a superposition with a linear operator;

$\hat{U}\Big(a|e_k\rangle+b|e_j\rangle\Big)\otimes |e_1\rangle=a\hat{U}\Big(|e_k\rangle\otimes |e_1\rangle\Big)+b\hat{U}\Big(|e_j\rangle\otimes |e_1\rangle\Big)$

$=a\Big(|e_k\rangle\otimes |e_k\rangle\Big)+b\Big(|e_j\rangle\otimes |e_j\rangle\Big)$

which is not what we want:

$\ne \Big(a|e_k\rangle+b|e_j\rangle\Big)\otimes\Big(a|e_k\rangle+b|e_j\rangle\Big)$

there is nonlinearity inherent in the cloning process, and quantum transformations are linear unitary transformations. A universal quantum state copier would need to violate linearity. It all boils down to the fact that qubits are superpositions and quantum operations are linear operators as you saw in our gates examples.

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Why can’t quantum states be cloned?

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Why can’t quantum states be cloned?

Published by Jeff on October 3, 2021October 3, 2021

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