Bloch sphere

In plain terms

A qubit is the simplest quantum system, one with just two perfectly distinguishable states, and its possible states map exactly onto an ordinary sphere. The map is useful. Measurement settings are directions in space, two perfectly distinguishable states sit at opposite poles, and the angle between two points sets the statistics of measuring one when the other was prepared. States of incomplete knowledge, called mixed states, fill the interior, with the state of complete ignorance, the coin flip, at dead centre, so every quantum intuition you can have in three dimensions you can have here.

Then comes the trap, as all of that is an accident of having two levels. Go to a system with three distinguishable states and its state space isn't a sphere, isn't any familiar shape, and supports none of the same tricks, so the picture that teaches everyone the subject also quietly teaches them falsehoods about the 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, in the Bell settings, the singlet correlations and the qubit Born rule, and it flags the accident explicitly rather than trading on it. For two levels the projective space, the space of physically distinct pure states, is itself a sphere, so structure that in general has to be placed in a fibre above the state space, extra data attached over each state, can sit on the space itself, and nothing warns you that this is happening.

The programme's fibre analysis makes the warning concrete. Contextual content, the part of the theory that makes an outcome depend on which other measurements accompany it, can live on the base for two levels 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, and the corpus treats it with corresponding care.

Mathematically

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

For N above two, CP^(N-1) isn't 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 results for N at least three 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 with a thesis that founded the band theory of solids. Being 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, the basis of 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 that administration wasn't physics. Quantum information adopted his sphere decades later.

Related entries

• Pauli matricesThree small matrices carry the entire two-level world, and their refusal to commute is the uncertainty principle in its smallest packaging.
• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• Hilbert spaceHilbert space is the stage on which quantum mechanics is usually performed, and CSD's claim is that it's scenery, the shadow cast by the physical object and not the object itself.
• Stern-Gerlach experimentAtoms sent through an uneven magnetic field split into two clean beams instead of a smear, and sorting them twice along different axes undoes the first sorting, so the whole strangeness of quantum measurement sits in one 1922 apparatus.
• Ramsey interferometryTwo brief nudges separated by a long stretch of doing nothing measure a frequency more sharply than any amount of continuous pushing, and every atomic clock runs on this trick.