Bloch sphere

In plain terms

Here is the useful part. The states of the simplest quantum system -- a qubit -- map perfectly onto an ordinary sphere. Measurement settings are directions; perfectly distinguishable states are antipodal points; angles set the statistics; mixed states fill the interior with the coin-flip state at dead centre. Every quantum intuition you can have in three dimensions, you can have here.

Here is the trap. All of that is an accident of two levels. Go to three-level systems and the state space is not a sphere, is not any familiar shape, and supports none of the same tricks. The picture that teaches everyone the subject also quietly teaches them falsehoods about its general case.

Both halves matter: use the sphere, and know when to put it down.

In CSD

The corpus computes on the sphere wherever qubits appear -- Bell settings, the singlet correlations, the qubit Born rule -- and flags the accident explicitly rather than trading on it. At two levels the projective space IS a sphere, so structure that in general is placed in a fibre above the state space can sit on the space itself, and nothing warns you this is happening.

The programme's fibre analysis makes the warning concrete: contextual content can live on the base at N equals two purely because of this accident, and from three levels up the natural ways of keeping it there stop working. That finding narrowed the programme's own claims. The sphere is where everyone's intuition is formed; the corpus treats it as a crime scene to be handled with gloves.

Mathematically

CP^1 is diffeomorphic to the two-sphere; antipodal points are orthogonal states; the Fubini-Study measure is the normalised round area; trace distance is half the Euclidean chord. Mixed states fill the closed ball.

For N above two, CP^(N-1) is not a sphere, has nontrivial topology, and admits no comparable elementary picture -- which is precisely why qubit-calibrated intuition transfers badly and why the corpus's N-of-three-and-up results needed different methods.

The name

Felix Bloch (1905-1983) was born in Zurich, studied at the ETH, and took his doctorate under Heisenberg at Leipzig -- the thesis founded the band theory of solids, no small warm-up act. Jewish, he left Germany in 1933 and settled at Stanford for the rest of his career, with war work at Los Alamos and on radar at Harvard.

The sphere comes from his work on spins precessing in magnetic fields -- nuclear magnetic resonance, for which he shared the 1952 Nobel Prize with Purcell, and which became MRI. He also served as CERN's first Director-General in 1954, lasting about a year before deciding administration was not physics. Quantum information adopted his sphere decades later, mostly without asking.