3.1 Two probabilistic bits ‣ Quest 3: Wizard of entanglement ‣ The Quantum Quest‣ Quest 3: Wizard of entanglement ‣ The Quantum Quest

When you have two coins placed on a table, they can be in one out of four configurations:

These correspond to the four possible bit strings¹²¹² 12 A ‘string’ means a sequence of symbols (in this case: a sequence of bit values). It will be our favourite notation when we are dealing with multiple coins or bits.: 00, 01, 10, 11.

If you have a pair of probabilistic bits, their state is described by a probability distribution over these four possible deterministic states. In other words, their state is specified by four numbers $p_{00},p_{01},p_{10},p_{11}\geq 0$ such that $p_{00}+p_{01}+p_{10}+p_{11}=1$ . Just like in the case of a single bit, we could write this down as a vector

\begin{pmatrix}p_{00}\\ p_{01}\\ p_{10}\\ p_{11}\end{pmatrix}.

(3.1)

While accurate, this representation is quite clumsy because keeping track of which probability is assigned to which bit configuration can be hard (does $p_{10}$ or $p_{01}$ go first?!), and it will become only harder once we have more than two bits. It is much more convenient to use a notation that lets us directly keep track of the probabilities assigned to particular bit strings. We therefore extend the notation we introduced in Eq. 1.7 for a single probabilistic bit and write the above two-bit state as follows:

p_{00}[00]+p_{01}[01]+p_{10}[10]+p_{11}[11].

(3.2)

If you wish, you can always relate this notation back to the 4-vector notation in Eq. 3.1 using the following dictionary that generalizes Eq. 1.6:

[00]=\begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix},\qquad[01]=\begin{pmatrix}0\\ 1\\ 0\\ 0\end{pmatrix},\qquad[10]=\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix},\qquad[11]=\begin{pmatrix}0\\ 0\\ 0\\ 1\end{pmatrix}.

(3.3)

One advantage of this notation in the case of multiple bits is that we can simply omit entries that are zero. For example, we can simply write

\displaystyle\frac{1}{2}[00]+\frac{1}{2}[11]

instead of the more clumsy

\displaystyle\frac{1}{2}[00]+0\,[01]+0\,[10]+\frac{1}{2}[11].

With this notation it is also much easier to describe measurements and operations on several probabilistic bits. We can explore two probabilistic bits using Quirky, which has again gained new powers since last week. To begin, go to:

https://www.quantum-quest.org/quirky

and click on “Quest 3” and select “Two Bits”. Your web browser will look similarly to Fig. 3.1. Note that we now have two wires, corresponding to two bits, which are initialized in state $[00]$ . Perhaps surprisingly, the first bit corresponds to the bottom wire and the second bit to the top wire. In addition, there is one new box: $\bullet$ (but the mystery box is gone). We will discuss the meaning of this box later in this chapter.