Measurement, superposition and collapse

“Schrödinger’s cat is both dead and alive” is the most repeated and most misread sentence in the subject.

The misreading is that it sounds like the cat is in some blurred condition and we merely do not know which. Quantum superposition is precisely not that. This module separates the two.

Superposition is relative to a basis

In the $z$ basis this is a superposition: measuring $\sigma_z$ gives $\pm1$ half the time each.

In the $x$ basis it is an eigenstate of $\hat\sigma_x$: measuring $\sigma_x$ gives $+1$ with certainty and no randomness whatsoever.

One state, and changing the basis turns “superposition” into “definite”. So “the system is in a superposition” carries no information unless you say with respect to which basis.

On the Bloch sphere this becomes obvious: the state is a fixed arrow on the sphere, and the measurement basis is an axis you can point wherever you like. When the arrow lies along the axis the outcome is certain; when it is perpendicular it is fifty-fifty. Nothing changed except the question you asked.

Superposition ≠ mixture

Now compare two statements that sound similar:

• A (superposition): the system is in $\ket{+} = \tfrac{1}{\sqrt2}(\ket{0}+\ket{1})$.
• B (mixture): the system is $\ket{0}$ with probability 50% and $\ket{1}$ with probability 50%, and we do not know which.

Measured along $z$, the two give identical statistics: half and half.

Measured along $x$, they could not be more different: A gives $+1$ every time; B is still fifty-fifty.

In density-matrix form:

The whole difference sits in the off-diagonal elements, the coherences. On the Bloch sphere, A is on the surface at the equator ($|\vec r|=1$) while B is at the centre ($|\vec r|=0$).

Measurements in sequence: order has consequences

Chaining Stern–Gerlach analysers is the sharpest way to see what measurement does:

1. Split along $Z$, block the lower beam → everything left is $\ket{0}$;
2. Another $Z$ → 100% upper. Measurement is repeatable.
3. Switch to $X$ → fifty-fifty. $\ket{0}$ is a superposition in the $x$ basis.
4. Add a third analyser along $Z$ → the lower beam is back.

Step 4 is the point. Step 1 removed “lower” cleanly, and the $X$ measurement in step 3 put it back. So the $X$ measurement was not passively reading a property; it changed the state.

Algebraically this is just $[\hat\sigma_x,\hat\sigma_z]\ne0$: non-commuting observables share no eigenstates and cannot both be definite.

An honest word about “collapse”

The projection postulate is a rule for calculating: after obtaining eigenvalue $a_n$, continue with $\hat P_n\ket{\psi}/\|\hat P_n\ket{\psi}\|$. It has survived every experimental test.

Whether collapse is a physical process is another matter, and the interpretations differ sharply:

The good news is that all of them predict the same experimental results so far. So you can learn to use the projection postulate as a tool first and pick a side later.

Think it through

1. In the Bloch scene set the state to $\ket{0}$ and the axis to σx, then measure once. Switch back to σz — what is the probability of $\ket{1}$ now? Map that onto step 4 of the Stern–Gerlach chain.
2. $\rho_A$ and $\rho_B$ are indistinguishable along $z$. Which axes do you need to measure to tell them apart? (Hint: you have to reconstruct the whole Bloch vector.)
3. If “measurement” is just the apparatus becoming entangled with the system, then the apparatus should also end up in a superposition. Why have we never seen a pointer in one? Module 12 gives a partial answer.