Unit quaternions and the Bloch sphere

Abstract

The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere rotations to be represented as quaternionic multiplications, simplifying the form of the dynamical equations. Left-multiplications generally correspond to non-unitary transformations, providing a simpler (essentially classical) analysis of time-reversal. But the quaternion viewpoint also reveals a surprisingly large broken symmetry, as well as a potential way to restore it, via a natural expansion of the state space that has parallels to second order fermions. This expansion to "second order qubits" would imply either a larger gauge freedom or a natural space of hidden variables. ‡

Disentangling the Quantum World

“…are available to supply the necessary "fine-tuning" for partially entangled states. Resolution of this question must await a hidden variable model rich enough to encode all such states, although one promising framework can be found in [9]. 6 5 What about Bob?…”

“…This is not the asymmetry-of-signalling that was removed by Erutan, as discussed in the previous section; with Erutan present, neither Ecila nor Bob can signal to each other. 9 But even with Erutan, a further time-asymmetry remains if there's only one fixed value of τ . The standard assumption in this case is that Ecila still has control over τ in this case (up to an additive factor of 90 • ), while Bob does not.…”

Simulating Time Evolution with Fully Optimized Single-Qubit Gates on Parameterized Quantum Circuits

“…In the following, for simplicity we refer to the algorithm described in this section as FQS(1q, 3p), where the 1q denotes its targeting parameterized single-qubit gates, and the 3p denotes the full parameterization of each gate: the (unit) quaternion which can be identified with a set of rotational angle and axis or direct parameters of a single-qubit gate [38]. We emphasize the fact that all parameters of a single-qubit gate are simultaneously optimized in FQS(1q, 3p).…”

Natural Parameterization of Two-Qubit States

“…The first qubit's Bloch ball vector n (1) can then be determined from ρ (1) = (I+n (1) •σ)/2, where σ is the usual vector of Pauli matrices and I is the identity. The components of this vector are given by…”

“…Typically, one ignores a global phase, leaving six (real) angles that parameterize the space of all pure twoqubit states. (Although ignoring a single phase angle cannot be trivially accomplished without introducing topological phase-jumps and also breaking the S 7 symmetry; for a discussion of the analogous situation in H 2 , see [1]. )…”