Prelude to Postulate 2

Introduction

Consider the quantum state

$$\ket{\psi} = \alpha\ket{0} + \beta\ket{1}.$$

Suppose Alice prepares this quantum state in her lab, and shares it with her colleague Bob.

Is it possible for Bob to figure out the coefficients $\alpha$ and $\beta$?

From our real world intuition, it feels as though this should certainly be possible!

For example, when one looks at a moving baseball, one expects to be able to diagnose the physical state of that baseball: its speed, its velocity, its shape, etc, to nearly arbitrary precision given current technology.

Or, if one brings their smartphone into a repair shop, one would expect the technician to be able to look inside and diagnose the physical state of the various parts, such as voltages, resistances, size, shape, to come to an overall diagnosis of the problem with the smartphone.

However, this is not so in quantum mechanics.

Bob, nor Alice, can ever know what the coefficients $\alpha$ and $\beta$ are by "taking a look" (performing a measurement).

However, not all hope is lost.

It is possible to glean at least SOME information from the quantum state, in this case by performing something that is one of the absolute cornerstones of quantum computing. That is, performing a measurement in the computational basis.

Performing a measurement in the computational basis

We have some (potentially complicated) physical device in the lab, with a readout screen, which can output the numbers $0$ or $1$. 

As an example, perhaps 1 corresponds to having measured a photon and receiving the outcome that it was vertically polarized, and 0 corresponds to having measured a photon and receiving the outcome that it was horizontally polarized.

We can also call this readout a classical bit, $0$ or $1$. Some of the time when we perform a measurement (this might look like shining a laser on an atom as another example), the measurement device will output a $0$, other times it will output a $1$.

In fact, as we'll see later when we get to the second postulate of quantum mechanics, for $\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$, with probability $|\alpha|^{2}$ our measurement device outputs a $0$, and with probability $|\beta|^{2}$ it outputs a $1$.

When that output is received, the quantum state is now in the state $\ket{0}$ or $\ket{1}$, depending on whether the measurement readout was a $0$ or a $1$. That's just how it works. $\alpha$ and $\beta$ are gone.  

We won't go into the multitude of different explanations for why this happens, whether it's a "collapse" of the wavefunction, interference from other universes, or other explanations. That would be a full course in and of itself.

By the way, it actually makes sense that it is not possible to learn what $\alpha$ and $\beta$ are, because they could theoretically be used to store an infinite amount of "classical information", for example in the real part of $\alpha$. 

That is, we won't say how, but you could encode an infinite amount of information as a neverending bit string "011010001...

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