# Can Eve perform this operation?

I am a beginner in quantum computing. Please consider the following scenario:

Suppose Alice wants to send $$\frac{1}{\sqrt{N}}\sum_{j=0,1,2,..N-1} |j\rangle$$ to Bob.

Eve has intercepted the state and want to do the following operation - add his $$|0_{\text{Eve}}\rangle$$ and get $$\frac{1}{\sqrt{N}}\sum_{j=0,1,2,..N-1} |j\rangle |0_{\text{Eve}}\rangle$$

Next he wants to apply apply Unitary Transformation $$U|j\rangle|0_{\text{Eve}}\rangle = |j\rangle|j\rangle$$ and convert the state to $$|0_{\text{Eve}}\rangle$$ and get $$\frac{1}{\sqrt{N}}\sum_{j=0,1,2,..N-1} |j\rangle |j\rangle$$

Does such unitary operation/transformation exist? if no is it because conflicting no-cloning theorem that you cannot copy a state as it is?

It does exist. It's basically the controlled-not gate generalised to higher dimensional systems.

The important thing to realise is that means that the bit that Bob ends up with will be highly entangled with what Eve has, and that will have significant observable consequences. As part of a cryptographic protocol, for exampe, Bob could detect that Eve is doing this before sending any sensitive informaiton.

Please be careful with your notation; don't get confused with the number of qubits that are in $$|0_{Eve}\rangle$$ (I'm not necessarily saying that you are - just mentioning this as a precaution). Since the state $$|\psi\rangle = \frac{1}{\sqrt{N}}\sum_{j \in 0,1,2....N-1}|j\rangle$$ that Alice wants to send is a $$n$$-qubit state with $$N=2^{n}$$, Alice needs to send $$n$$ qubits to Bob - possibly via Eve. Therefore, the state $$|0_{Eve}\rangle$$ that you describe is a $$n$$-qubit state as well.

A unitary $$U$$ that copies the state $$|\psi\rangle$$:

The $$2n$$-qubit unitary that you describe is possible, but on the presumption that Eve knows exactly what Alice will send. Then, a series of $$n$$ $$CX$$ gates from the $$i$$th qubit to the $$(i+n)$$th qubit will suffice - it will copy any state $$|j\rangle$$ in the first $$n$$ qubit to the second $$n$$ qubits, thereby obtaining the total state $$|j\rangle \otimes |j\rangle$$. By linearity, it also copies a superposition of different $$|j\rangle$$-states. So we have: $$$$U_{copy} = \sum_{i\in \{1,2...n\}}CX^{(i,i+n)},$$$$

where $$CX^{(i,i+n)}$$ is the CNOT operation with qubit $$i$$ as the control qubit and qubit $$(i+n)$$ as the target qubit.

A unitary that copies any state $$|\phi\rangle$$:

However, if Eve does not know what state Alice sends, there is no unitary that will perform the task (of copying a unknown $$n$$-qubit state $$|\phi\rangle$$ into the second $$n$$ qubits). The proof of impossibility is straightforward via the condition that the operation needs to be linear; for a proof see a proof of the no cloning theorem, which applies here.