These notes are an exposition of the basic facts about the Pauli matrices and the Bloch Sphere. The goal is to give a completely mathematically rigourous exposition of the core facts about the action of the Pauli matrices as rotations on the Bloch Sphere, and to do so in a way where the reasons for this strange correspondence between vectors in and become clearer. To this end, I've tried to avoid matrix multiplications and complicated trig formulas almost completely, and to let the underlying algebraic patterns shine through.
This page is not a good introduction to the properties of the Bloch Sphere. For that I recommend Nielsen and Chuang or Rieffel and Polak. Hopefully it may be of interest to someone who has read those books and would like to see proofs of all the gory details and, perhaps, gain some insight into why this correspondence works so well.
I am also indebted to Ian Glendinning's lovely slide deck on the same topics.
The Pauli Matrices are defined to be:
Routine calculation shows that . All other multiplicative identities involving , , and can be deduced from these. For example, (Of course these identities could also be obtained by direct matrix multiplication.) Note in particular that the distinct anticomute, that is, for . Also, as observed, for . We can summarize the rules for as where are distinct, and is the sign of the permutation .
If we set then . These are the defining relations of the quaternions, and so the span of provides a representation of the algebra of quaternions.
Note that are self-adjoint matrices. If then is also a self-adjoint matrix and it is clear that every self-adjoint matrix can be expressed in this form. Thus span the real vector space of self-adjoint matrices, and since this space is 4-dimensional, they are a basis. Note in particular that and so span the space of trace-zero self-adjoint matrices. Also, by direct computation,
Now we write which is called the Pauli vector and, by slight abuse of notation, for we write Using the anticommutation relations we calculate In particular and if is a unit vector in then .
Now let be a unit vector in and . Clearly is Hermitian, and by the anticommutation relations. Thus the spectrum of is and its spectral projections are and . Thus the (not necessarily normalized) eigenvectors for and respectively are (Note these are orthogonal since .)
If is any matrix with then (In fact this is true in any unital Banach algebra.) Thus if is a unit vector in and then which is a generalized Euler's Identity.
Note that by the anticommutation relations, if then and so Thus So when are distinct For example In the coordinate system of this acts as a rotation of about the axis. In other words where
The analogous computations with and show that these also give rotations of about the and axes respectively.
Thus we write for these operators of rotation about the coordinate axes.
Now for a unit vector we consider the operator and show that this corresponds to a rotation of about the axis . Write in spherical coordinates as for suitable and . Now, a rotation of about can be done by the following steps:
This sequence of rotations is implemented by the following matrix A routine calculation shows that this is indeed :
Given a unit vector , consider the pure state . This is a projection and so can be written as where is a self-adjoint unitary, also known as a symmetry. Since is rank-1, and so . Thus for some and since , so is a point on the unit sphere of .
Thus the mapping gives a correspondence of unit vectors in with unit vectors in . Since is determined up to a unimodular multiple by the range of , it can also be recovered from , at least up to a physically irrelevant multiple of .
The unitary acts on and induces an action on as: We know that conjugation by acts on as rotation of about the axis, and so the action of on corresponds to a rotation of about by an angle of to .
If and are unit vectors in then they are orthogonal if and only if . This happens if and only if (where correspond to respectively). Because the () are linearly independant, if the product is zero then and by the extremal case of the Cauchy-Schartz Inequality, . Conversely if then the product is zero. Thus and are orthogonal if and only if they correspond to antipodal points on the Bloch Sphere.
Let be a unitary complex matrix and consider the map . Let be a unit vector in which corresponds to in the Bloch Spehere. Then is also a unit vector in , which corresponds, say, to in the Bloch Sphere. Thus Thus corresponds to in the Bloch Sphere, where the coordinates of are given by .
Now is a linear map which takes the set of trace-0 self-adjoint matrices to itself. Since is a basis for this set of matrices, induces a linear map of to itself. Moreover, since it follows that if and then Thus induces a real unitary (or orthogonal) matrix on . That is, it belongs to .
Next, if then we can compute by the triple product formula. So if , , and then Comparing the coefficients in , we conclude that and so we see that induces an orientation-preserving map on . This map is therefore in and so acts as a rotation.
To explicitly see the reverse mapping from to , consider the spherical coordinates for as and take Straighforward calculation shows that
This establishes the usual spherical coordinatization of the Bloch sphere.
A mixed state is represented by a density matrix, which is a positive semi-definite matrix with trace 1. The set of density matrices is convex and the extremal points of this convex set are the pure states, which can be shown to be the vector states, i.e., of the form . The pure states can also be recognized as the density matrices which satsfy .
Density matrices can be mapped to in exactly the same way as we mapped rank-1 projections, and in this case density matrices correspond to points with , and the points on the boundary (i.e., ) correspond to the pure states.
If is a density matrix then is a trace-0 self-adjoint matrix and so there is unique such that or, equivalently, . Thus we extend our mapping into to the density matrixes. Note that . Taking the trace we see that Now the eigenvalues of are and for some and so the eigenvalues of are and . Thus we can compute This quantity clearly varies between 0 and -1/2 for , and so and is in the unit ball of . Further if and only if which, as remarked, corresponds to a pure state.
Note also, that the correspondence is an affine map and that convex combinations of density matrices map to convex combinations of vectors in . Thus, again, extremal points of the sphere map to extramal points of the set of density matrices which we have seen are vector states. The centre of the sphere, on the other hand, corresponds to the matrix which, phyically, represents a mixed state of maximum uncertainty (e.g., the states and are equally likely).