Quantumania 14 - Schrödinger's Umlaut

Before we get back to that exciting word “gate”, let’s consider some actual physics. Yes, weird, I know. Also I’m going to immediately break the rule-slash-guideline I set for myself last time, where $\hat{H}$ was forever going to stand for Hadamard. Well, this time it’s going to stand for Hamiltonian. This is an operator that characterises the energy of a system, and is something that QM inherits wholesale from classical physics.

Suppose you could measure the energy of a system. For now we’ll gloss over what energy actually is (it’s a deep, circular topic.) Instead, just as we thought of spin orientation along a chosen axis as something we could measure about a system, now we’re saying there’s a numerical quantity called the energy of the system that we can also measure. The state vector encodes everything there is to know about the state of the system, so it has all the information there is about the energy too.

So as always in a two dimensional QM system, anything about it we can measure can only have two possible values. For energy we’ll call them $E_1$ and $E_2$. These are just real numbers. QM tells us to throw these values into a diagonal matrix, $\operatorname{diag}(E_1, E_2)$:

\[\operatorname{diag}(E_1, E_2) = \begin{bmatrix} E_1 & 0 \\ 0 & E_2 \end{bmatrix}\]

There is something particularly important about energy: it is connected with how the state vector of the system changes as time passes. I teased this when complaining about the so-called measurement problem.

QM gives us probability distributions over measurement outcomes that occur when we “make an observation”, a kind of interaction that discontinuously snaps the state vector into alignment with an eigenvector of the observable’s hermitian operator, but the dynamics of the theory contain nothing to suggest this actually ever happens, which is, on the face of it, ridiculous. Now we’re going to examine what the theory does allow to happen to a state vector as time passes.

We’re in effect saying that our state vector is a function of time: $\vert \psi(t) \rangle$. But one claim we’ve made about QM is that the state vector contains every scintilla of physical information about the system. This means it knows the future! As usual the easiest first step is to align our state vector with a basis vector, that is, an eigenvector of the observable in question. Let’s label the eigenvectors for the two energy levels $\vert E_1 \rangle$ and $\vert E_2 \rangle$. At “the beginning” when $t = 0$:

\[\vert \psi(0) \rangle = \vert E_1 \rangle\]

In that case, what happens as time passes is:

\[\vert \psi(t) \rangle = \vert E_1 \rangle e^{-i E_1 t / \hbar}\]

That is, it’s just a pure unit phase factor, and $\hbar$ is defined as $h/2\pi$, where $h$ is Planck’s constant, which is ~$6.6 \times 10^{-34}$. This means the phase factor is oscillating with frequency $\nu = E_1 / h$ (one of the earliest formulae discovered in QM was $E = h \nu$, the Planck relation.) Actual physical implementations of qubits have energies of the order of $10^{-24}$ joules, so the frequency is in the GHz range. This thing is, notionally at least, “spinning” pretty fast.

But because it’s just an overall unit phase factor, this means the alignment of the state vector is not changing. In terms of the physical difference this makes to the state of an isolated system, it’s like multiplying by 1. Physically it means nothing’s happening. The energy, if it was to be measured, would be $E_1$, forever.

But $\vert \psi(0) \rangle$ doesn’t have to be one of the eigenstates. It could be any unit vector in the state space, which, if we choose to write it in terms of the energy eigenstates, would be some combination of them that comes out as a unit vector.

\[\vert \psi(0) \rangle = \alpha \vert E_1 \rangle + \beta \vert E_2 \rangle\]

But the complex phase factor spinning as time passes applies to both components:

\[\vert \psi(t) \rangle = \alpha \vert E_1 \rangle e^{-i E_1 t / \hbar} + \beta \vert E_2 \rangle e^{-i E_2 t / \hbar}\]

Each has its own frequency given by its energy level. $\vert E_1 \rangle$ and $\vert E_2 \rangle$ are the unit vectors of the basis we’re working in:

\[\vert E_1 \rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad\quad \vert E_2 \rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\]

So scaling each and summing them is a trivial thing:

\[\vert \psi(t) \rangle = \begin{bmatrix} \alpha e^{-i E_1 t / \hbar} \\ \beta e^{-i E_2 t / \hbar} \end{bmatrix}\]

The spinning phase factors, scaled by the initial coordinates, just become the time-dependent coordinates of the vector in this basis. Although nothing is changing in the $\alpha$ and $\beta$ factors, the relative phase of the two coordinates is going absolutely bonkers.

What does this mean? Refer back to the Bloch sphere, which began as a way to define the state of a qubit in terms of a pair of orthogonal basis states and two real parameters: the inter-coordinate mixing factor, $\theta$, and the phase difference, $\phi$:

\[\vert \psi \rangle = \cos \left( \frac{\theta}{2} \right) \vert u \rangle + e^{i\phi} \sin \left( \frac{\theta}{2} \right) \vert d \rangle\]

and ended up with the realisation that it actually directly describes physical directions in space as points on a sphere, given by the angles $(\theta, \phi)$. Changing the relative phase changes the direction in physical space represented by the state vector, moving around the Bloch sphere in a circle of constant latitude.

Can we say the qubit is continuously changing the direction it is pointing in, in physical space? This is considered a step too far. The least controversial thing we can say is that, if you were to choose an alignment in physical space to measure its orientation along, at any given time there is theoretically one specific alignment along which you’d get a definite answer, but you have no idea what that is, because the exact phase difference at any moment is unknown to you, being impossible to discover, and is oscillating rather quickly.

But doesn’t this suggest there something special about the energy eigenvectors and the directions in space they relate to? We’ve always been clear that the observable spin orientation is something that can have any alignment in physical space. Space has rotational symmetry. There are no “special” directions.

Well, if your qubit is the spin state of an electron, that changes the moment you apply a uniform magnetic field across it. The magnetic field has an orientation, and so in effect picks out a special alignment across the electron. If the electron’s spin at the moment you switch the field on happened to be perfectly aligned with the magnetic field, then it will stay aligned with it. The state vector’s corresponding point on the Bloch sphere will stay put. But if it was misaligned, the spin will precess - or I mean to say, the point on the Bloch sphere will trace out a circle, at a fixed latitude, around the pole.

This is why the Hamiltonian, the operator representing the energy observable, is built directly from those same reflection operators we saw last time, most conveniently the diagonal Z matrix form $\operatorname{diag}(1, -1)$.