1.4.3 A mysterious operation

We still have not discussed the mysterious orange box. Let us call this operation $M$ . How can we figure what is going on inside the box? As a first step, let us consider the problem of determining $M\,[0]$ , that is, the result of applying the mysterious operation $M$ to a bit in state $[0]$ . In Quirky, this corresponds to the following setup:

How can we read off $M\,[0]$ ? At this point it is good to remind ourselves that when random bits appear in nature we cannot simply look at them and read off their probabilities. Instead, as explained in Section 1.3, we have to perform many measurements (e.g., toss a coin many times) and estimate the probabilities from the outcomes. The advantage of using a simulator like Quirky is that we do not have to play by these rules – we can use the probability display to determine the state:

Thus, we find that

\displaystyle M\,[0]=0.2\,[0]+0.8\,[1].

Now it is your turn!

Homework 1.6 (Mystery time).

1.

Determine the state $M\,[1]$ .
2.

Do $M\,[0]$ and $M\,[1]$ specify the random operation $M$ completely?
If yes, write down a formula for $M\,\bigl{(}\begin{smallmatrix}1/2\\ 1/2\end{smallmatrix}\bigr{)}$ and verify it in Quirky. If not, explain why.

Hack.

1.

Following Quirky,

we find that

$\displaystyle M[1]=0.7[0]+0.3[1].$

Yes, since any random operation satisfies linearity (Eq. 1.25). Thus we have:

	$\displaystyle M\bigl{(}\begin{smallmatrix}1/2\\ 1/2\end{smallmatrix}\bigr{)}$	$\displaystyle=\frac{1}{2}M[0]+\frac{1}{2}M[1]$
		$\displaystyle=\frac{1}{2}\frac{2}{10}[0]+\frac{1}{2}\frac{8}{10}[1]+\frac{1}{2% }\frac{7}{10}[0]+\frac{1}{2}\frac{3}{10}[1]$
		$\displaystyle=\frac{9}{20}[0]+\frac{11}{20}[1]$
		$\displaystyle=45\%[0]+55\%[1].$

We can also verify this using Quirky:

In the coming weeks we will make the jump from ordinary bits to quantum bits, and learn how to compute with them in increasingly sophisticated ways. Quirky will serve as our trusty tool, gaining new capabilities as we move along. You are warmly encouraged to use it to investigate the theory that you will learn, as well as to help you solve your homework problems.