Permutations and Transformations – Qubit Operations

In preparation for the application quantum cryptography in the next station, here, we first examine basis transformations for a single spin. This qubit describes spin “up” – but is it an eigenstate? Yes and no – it depends on the rotation axis chosen! For rotations around the y-axis, spin “up” is not an eigenstate, since the direction of the spin is changed upon rotation. Up becomes down – then up again … a permutation!

However, spin up is an eigenstate for a rotation around the z-axis: Spin “up” stays where it is – the direction of spin does not change. This is true in general: For a given qubit, the rotation around an arbitrary axis results in a cone. Only if the direction of spin and the rotation axis match, the state is an eigenstate! If the direction of spin is orthogonal to the rotation axis, the cone degenerates into a plane.

In U2-7 Slide 3, we discussed the Stern-Gerlach experiment. The direction of the inhomogeneous magnetic field defines the z-axis. Due to the interaction of the arbitrary initial spin with the magnetic field, the spin direction changes to become an eigenstate.

The initial state must, so to speak, align with the direction of the slot … Why does the magnetic field have to be inhomogeneous? If the magnetic field were homogeneous, the spin would just rotate around the cone.

The inhomogeneous magnetic field causes the arrow to spiral towards the z-axis and thereby to become shorter. The initial state is projected onto the z-axis. If the initial state is orthogonal to the z-axis, the length of the arrow becomes zero, leading to 50% probability for spin up or down – a completely mixed state.

An eigenstate does not change at all upon projection. Here, the probability for spin “down” is 100%. Now we rotate the direction of the inhomogeneous magnetic field by 90 °.

Then, the same initial state is no longer an eigenstate. The projection results in a completely mixed state with a 50% probability for parallel or anti-parallel alignment of the spin with the magnetic field. Thus, depending on the measuring axis, the same initial state either leads to a predictable or to a random result!

The state corresponds to certain coordinates – which are interpreted in one basis or another. This basis transformation is the essential step for quantum cryptography, where this idea is realized with single photons, as we will see in the following chapter.