What Are Probability Amplitudes?
Suppose you toss a coin in the air.
There are two possible outcomes: Heads (H) or Tails (T).
Assume the coin is fair.
Using ket notation, the state of the coin while in the air is given by:
|C〉 = ½ |H〉 + ½ |T〉 .
The coefficients in front of |H〉 and |T〉 indicate the probabilities of the coin landing on head or tail once it reaches the table.
If the coin is biased, its state in the air is:
|C〉 = p_H |H〉 + p_T |T〉 ,
where p_H and p_T represent the probabilities of getting head or tail, respectively.
Since these are the only two possible outcomes, we have:
p_H+p+T=1.
Now, let’s take this analogy a step further to bring it even closer to quantum mechanics.
We define the state of the coin in the air as:
|C〉 = √p_H |H〉 + √p_T |T〉 .
This expression looks very similar to the mathematical description of a single qubit.
It seems that the only difference between the two models is that the coefficients are real in one case and complex in the other.
The similarities are so striking that some computer scientists introduce the mathematics of quantum mechanics using the probabilistic model.
However, despite the resemblance, there is a fundamental—philosophical, if you will—difference between the two:
• In the probabilistic model, uncertainty arises due to our limited knowledge of the system.
• In the quantum model, uncertainty is intrinsic to nature.
In principle, we could build a probabilistic model of computation using complex coefficients—there’s nothing wrong with that.
But as long as it’s based on a classical physical system, superposition will not carry the same meaning as it does in quantum mechanics.
In a classical system, whether deterministic or probabilistic, a measurement "reveals" the preexisting state of the system.
In contrast, in quantum mechanics, a measurement "fixes" the state of the system.
Once again, just like in the case of "Qubits are not 0 and 1 at the same time," it is not that we do not know the state of the quantum coin while it is in the air.
According to quantum mechanics, asking this question simply does not make sense!
You cannot ask about it—because there is no way for us to know!
Quantum mechanics is rather blunt about this!
Certain classical concepts, like preexisting states before measurement, do not apply in the quantum world.
Want to dive deeper? My eBook is a great place to start → https://www.ozatp.com/qaf
