3.1.1 Measuring both bits

Measuring (or “looking at”) two probabilistic bits works the same way as Eq. 1.32 for a single bit. You get one out of four possible outcomes ($00$, $01$, $10$, or $11$) with the corresponding probability:

(3.4)

The state of both bits after the measurement is no longer a distribution over four possibilities but rather just a single option that corresponds to the measurement outcome you observed. To emphasize this difference, we use light blue for probabilistic bits and gray for deterministic ones after the measurement. How can we measure both bits in Quirky? We simply use the probability display, like so:

Note that, by default, the probability display is connected to both wires, so it shows the probabilities for both bits. The order of probabilities is as in the 4-vector notation in Eq. 3.1. You don’t have to remember the order, though. Simply hover the table with your mouse cursor to remind yourself (thanks, Craig!).

For example, if the two probabilistic bits are in state

then we obtain as measurement outcomes either 00 or 11, each with 50% probability. Notice that this state is special – if we see that the measurement outcome of the first bit is $0$, we immediately learn that the outcome from the second bit also has to be $0$, and similarly for when either of the two outcomes is $1$. Since the measurement outcomes of both bits are always equal, we call two bits in state (3.5) perfectly correlated. We will see below how such states can be created.