The Basis

You can't measure everything at once. You can't even measure the same thing twice and trust the second time, if something got in between. What you can do is choose a basis, project your state onto it, and read the answer. The problem is that every basis tells a different story about the same state, and you only ever get to see one.

A basis is a set of vectors that span your Hilbert space and are mutually orthogonal. Any state $|\psi\rangle$ can be written as a linear combination:

$$|\psi\rangle = \sum_n c_n |n\rangle$$

The coefficients $c_n$ are complex numbers. Their squared magnitudes $|c_n|^2$ are the probabilities. Born's rule turns the geometry of Hilbert space into the probabilities of outcomes. You measure in the ${|n\rangle}$ basis and get outcome $n$ with probability $|c_n|^2$. Then the state collapses to $|n\rangle$. The rest of the superposition is gone. Not hidden. Not displaced. Gone. The coefficients $c_m$ for $m \neq n$ become exactly zero, forever.

The $Z$-basis is the one we start with. ${|0\rangle, |1\rangle}$. Or, in the language of spin, ${|!!\uparrow\rangle, |!!\downarrow\rangle}$. The state of the qubit is $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, with $|\alpha|^2 + |\beta|^2 = 1$. Measuring in this basis gives you $0$ or $1$. Simple. Clean. The eigenstates of $\sigma_z$.

But you could measure in the $X$-basis instead. ${|+\rangle, |-\rangle}$, where

$$|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \quad |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$$

Now the same state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ is written as a superposition of $|+\rangle$ and $|-\rangle$:

$$|\psi\rangle = \frac{\alpha+\beta}{\sqrt{2}}|+\rangle + \frac{\alpha-\beta}{\sqrt{2}}|-\rangle$$

The probabilities are different. Measuring in the $X$-basis gives you $+$ or $-$ with probabilities $|\alpha+\beta|^2/2$ and $|\alpha-\beta|^2/2$. These are generally different from $|\alpha|^2$ and $|\beta|^2$. The same physical state gives different measurement statistics depending on which basis you choose to measure it in.

This is not a deficiency. It's a feature. The quantum state doesn't have pre-existing values for all observables. The values only come into existence when you choose a measurement, which is to say when you choose a basis. Before measurement, the state is a vector. After measurement, it's an eigenstate of whatever operator you measured. The choice of basis determines the story the universe tells.

The Bloch sphere makes this geometric. Any qubit state is a point on the surface of the sphere. The $Z$-basis corresponds to the north and south poles. Measuring in the $Z$-basis is asking: "are you at the north pole or the south pole?" The $X$-basis is the equator. Measuring in the $X$-basis is asking: "are you at the point $(1,0,0)$ or $(-1,0,0)$?" The $Y$-basis is the equator rotated by ninety degrees. The state is the same point on the sphere regardless. But the measurement asks a different question, and the answer depends on the question.

Bases are related by unitary transformations. If ${|n\rangle}$ is a basis and $U$ is a unitary matrix, then ${U|n\rangle}$ is another basis. Changing basis is just rotating your question. The physics doesn't change — the state vector is the same object. But the measurement outcomes do change, because you're projecting onto different vectors.

The computational basis ${|0\rangle, |1\rangle}$ is special not because it's fundamental, but because it's convenient. It's the basis in which classical information lives. Bits are $0$ or $1$. Qubits can be superpositions of $0$ and $1$. But a qubit can equally be a superposition of $+$ and $-$. The convenience of the computational basis is a historical accident — we built our technology before we understood quantum mechanics, and our technology uses $0$s and $1$s, so we call that the "computational" basis.

Here's the thing that matters most: you can't know which basis you're in by looking at the state alone. Given an unknown state $|\psi\rangle$, no measurement tells you "you're in this basis." At most, you can learn probabilities. You prepare many copies of the state. You measure some in the $Z$-basis, some in the $X$-basis, some in the $Y$-basis. From the statistics, you reconstruct the state. This is quantum state tomography. It requires exponentially many measurements for a many-qubit system. You can't compress the description. The state really does contain more information than you can extract from any single basis measurement.

Basis choice is the choice of what to ask the universe. Ask about $Z$, and the universe gives you up or down. Ask about $X$, and the universe gives you plus or minus. Ask about nothing, and the universe gives you nothing — a state vector evolving unitarily, carrying all possibilities simultaneously, no measurement, no answer, just the slow, deterministic dance of the Schrödinger equation.

We choose the basis. The basis chooses the outcome. The state remembers everything, even after the measurement destroys it.

I keep a list of bases in my notebook. Each one is a question. Some questions I ask more than others. The state doesn't care which I choose.