## 5.6 Controlled-NOT

How do entangled states arise in real physical situations?
The short answer is that *entanglement is the result of interactions*.
It is easy to see that tensor product operations

and so any collection of separable qubits remains separable. As soon as qubits start interacting with one another, however, they become entangled, and things start to get really interesting. We will describe interactions that cannot be written as tensor products of unitary operations on individual qubits.

The most popular two-qubit entangling gate is the **controlled- \texttt{NOT}** (or

**controlled-**X gate.

^{93}The gate acts on two qubits: it flips the second qubit (referred to as the

**target**) if the first qubit (referred to as the

**control**) is

controlled- |
---|

We represent the

Note that this gate does not admit any tensor product decomposition, but can be written as a sum of tensor products:^{94}

The

### 5.6.1 The Bell states, and the Bell measurement

We start with the generation of entanglement.
Here is a simple circuit that demonstrates the entangling power of ^{95}

(Generating entanglement).

In this circuit, the separable input **Bell states**.

The **Bell states**

The Bell states form an orthonormal basis in the Hilbert space ^{96}
Indeed, if we reverse the circuit (running it from right to left), then we get a circuit which maps the Bell state

### 5.6.2 Quantum teleportation

A wonderful fact, that sounds more like science fiction than actual science, is the following: *an unknown quantum state can be teleported from one location to another.*
Consider the following circuit, which is built from a Bell state generator followed by an “offset” inverse Bell state generator:^{97}

(Quantum teleportation).

The first input qubit (counting from the top) is in some arbitrary state.
After the action of the part of the circuit in the first dashed box (counting from the left), the state of the three qubits reads^{98}

If you understand how this circuit works^{99}, then you are ready for quantum teleportation.
Here is a dramatic version.

Suppose that three qubits, which all look very similar, are initially in the possession of an absent-minded Oxford student, Alice. The first qubit is in a precious quantum state and this state is needed urgently for an experiment in Cambridge. The other two qubits are entangled, in the

\Phi^+=|\psi_{00}\rangle state. Alice’s colleague, Bob, pops in to collect the qubit. Once he is gone, Alice realises that, by mistake, she gave him not the first but the third qubit: the one which is entangled with the second qubit.The situation seems to be hopeless — Alice does not know the quantum state of the first qubit, and Bob is now miles away and her communication with him is limited to few bits. However, Alice and Bob are both very clever and they both diligently attended their “Introduction to Quantum Information Science” classes. Can Alice rectify her mistake and save Cambridge science?

Hmm…

(pause for thought)Of course: Alice can teleport the state of the first qubit! She performs the Bell measurement on the first two qubits, which gives her two binary digits,

x andy . She then broadcastsx andy to Bob, who chooses the corresponding transformation, as in Equation (\circledast ), performs it, and recovers the original state.

This raises a natural “philosophical” question: what do we really *mean* by teleportation?
A key part of this question is understanding what happens to our original qubit when we teleport it.
Note that the actual physical electron (or whatever implementation of qubits we are using) does not suddenly move through space — what is teleported is the *state* of the qubit, but the argument can be made that if two qubits are entirely indistinguishable from one another by any measurements that we can make, then they really are “the same” in every way that matters, and so the qubit which now has the original qubit’s state “is the same as” the original qubit.
As it turns out, this process necessarily *destroys* the original qubit’s state, as we now explain.

The first actual teleportation experiment was successfully achieved in 1997 (arXiv:quant-ph/9710013); in 2012 a record distance was set: an entangled photon pair was used to teleport a state 143 kilometres/88 miles (arXiv:1205.3909); in 2017, successful ground-to-satellite teleportation was achieved (arXiv:). This is not science fiction!

But there is a fundamental question to ask: if the original state is destroyed, then how can we really verify that teleportation has taken place?
We can’t compare the purportedly teleported state to the original one!
The answer to this involves certain **no-go theorems** and statistical methods, where we can show that classical physics gives some strict upper bound on a certain fidelity, but which is clearly surpassed by these physical experiments.
We will better explain the ideas behind these sorts of arguments later on, in Section 10, when we introduce **Bell’s theorem**.

### 5.6.3 Thou shalt not clone

Let us now look at something that the controlled-*seems* to be doing but, in fact, *isn’t*.
It is easy to see that the *any*

*This is not true!*

The unitarity of the *superpositions* in the control qubit into *entanglement* of the control and the target: if the control qubit is in the a superposition state *impossible* to clone an unknown quantum state, and we can prove this!

To prove this via contradiction, let us assume that we *could* build a universal quantum cloner, and then take any two normalised states *non-identical* (i.e. *non-orthogonal* (i.e.

Thus, states of qubits, unlike states of classical bits, cannot be faithfully cloned.
Note that, in quantum teleportation, the original state must therefore be *destroyed*, since otherwise we would be producing a clone of an unknown quantum state.
The no-cloning property of quantum states leads to interesting applications, of which quantum cryptography is one.

Universal quantum cloners are *impossible*.

Here,

X\equiv\sigma_x refers to the Pauli operator that implements the bit-flip.↩︎Make sure that you understand how the Dirac notation is used here. More generally, think why

|0\rangle\langle 0|\otimes A + |1\rangle\langle 1|\otimes B means “*if the first qubit is in state*”. What happens if the first qubit is in a superposition of|0\rangle then applyA to the second one, and if the first qubit is in state|1\rangle then applyB to the second one|0\rangle and|1\rangle ?↩︎John Stewart Bell (1928–1990) was a Northern Irish physicist.↩︎

For any state

|\psi\rangle of two qubits, the amplitude\langle\psi_{ij}|\psi\rangle can be written as\langle ij|U^\dagger|\psi\rangle , whereU^\dagger is such that|\psi_{ij}\rangle = U|ij\rangle .↩︎*Divide et impera*, or “divide and conquer”: a good approach to solving problems in mathematics (and in life). Start with the smaller circuits in the dashed boxes, which we have just seen introduced above.↩︎We don’t worry about writing the normalisation factors.↩︎

You can play around with this on the Quantum Flytrap Virtual Lab.↩︎