Quantum Computing Explained

13

TOOLS OF QUANTUMINFORMATION THEORY

In this chapter we consider several important aspects of quantum information theory that are important for a thorough understanding of quantum computers. First we discuss the no-cloning theorem, which shows that you cannot make copies of an unknown quantum state. After recognizing this fact, we see how to measure the closeness of two states to each other. We will do this by looking at trace distance and fidelity. We can characterize the amount of entanglement in a state by looking at the concurrence, and we can determine the resources needed to create a given entangled state by calculating the entanglement of formation.

We conclude the chapter by considering how to characterize the information content in a state. This is done by calculating entropy.

THE NO-CLONING THEOREM

A routine task performed in information processing is making copies of data. We take it for granted that we can make as many copies as we like of something—whether it’s a word processing file or a bit of music. As we have seen, the remarkable power of a quantum computer comes from the fact a qubit can exist in a superposition |ψ〉 = α|0〉 + β|1〉. Given this fact, can we make an exact copy of an arbitrary qubit?

It turns out the answer is no. This result, which we state below, is known as the no-cloning theorem, and it was derived by Wooters and Zurek in 1982.

Consider two pure states |ψ〉 and |ϕ〉, and suppose that there exists a unitary operator U such that

for some target ...