Hermitian operators
2 Feb 2019 (Happy closed-timelike-curve day!)
2 Feb 2019 (Happy closed-timelike-curve day!)
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The simplest kind of calculation performed by a node on a vector of complex numbers (a state) is called a hermitian operator. This operator is also unitary, but to distinguish it from other unitary operators it is usually just called hermitian. We start with vectors of two complex numbers, which represent 1-bit states on the computational sphere. A hermitian operator represents a half-rotation of the computational sphere about some axis. The eigenvectors of an operator are those points that do not move under the rotation: the axes, which are antipodal (orthogonal) pairs of points on the sphere. Below are three different axes of rotation (in yellow), and a half-rotation of the figure at left:

In general, a hermitian operator is a set of n orthogonal vectors that form a basis for all other vectors in the space, plus a set of binary labels {+1, -1} that are attached to each vector so that there is at least one +1 labeled vector and at least one -1 labeled vector in the basis. This can be visualized in normal 3d space, although in all our cases the coordinates of each vector are complex, rather than real, numbers, and we usually deal with a number of dimensions equal to 2^n (or 2^(2^n)). In any number of dimensions, every state vector can be represented as the sum of two other vectors:

  1. Its projection into the +1 labeled eigenspace, and
  2. Its projection into the -1 labeled eigenspace.

It might be the case that a particular state vector lies completely within the +1 eigenspace or completely within the -1 eigenspace, but in general a vector will lie partially in both subspaces and will therefore be the sum of the two "closest" vectors from each space. Each of these "closest" vectors, called projections, consists of the weighted sum of some of the basis vectors (all from the +1 or -1 eigenspaces), and therefore they are also orthogonal to one another. This can be represented symbolically by:

S = S+ + S-     ; where S is a state vector and S+ and S- are its projections 
                ; in the binary eigenbasis of a hermitian operator

In the following figure of different examples, gray = xyz axes, yellow = operator basis, white = S, green = S+, red = S-:

Application of the operator multiplies all eigenvectors by their corresponding eigenvalues (this is what it means to be an eigenvector). All +1 eigenvectors stay the same, while all -1 eigenvectors are multiplied by -1. This is also true of vectors composed of different eigenvectors, on a part-by-part basis, so that:

h(S) = (+1) * (S+) + (-1) * (S-) 
     = S+ - S-
     = (S - S-) - S-                ; using the equation above
     = S - 2 * S-

The application of h() on S is equal to S minus twice its projection in the -1 eigenspace of h. This is true of all states and hermitian operators, of any dimensionality. The effect of repeated applications of h() is to:

  1. Keep the S+ projection of S the same, while
  2. Alternately negating the S- projection of S.
h(h(S)) = (S - 2 * S-) - 2 * (S - 2 * S-)-    ; two applications of the previous equation
        = S - 2 * S- - 2 * (S- - 2 * S--)     ; projection is linear and distributes
        = S - 2 * S- - 2 * (S- - 2 * S-)      ; S- is unchanged by further projection into the -1 eigenspace
        = S - 2 * S- - 2 * S- + 4 * S-
        = S                                   ; h^2 = identity
        = S+ + S-

This makes points simply "bounce back and forth" across the computational sphere as their -1 projection changes sign. Here is a single axis full rotation from different angles:

Although this might seem trivial, it is the basis of all other, more complicated, behavior of states and operators. In particular, two similar operators h() and g() can be composed as g(h()) or h(g()) in order to create small rotations of the sphere. The order of composition determines the direction of rotation, and the angle between the two operator axes determines the angle of rotation for each application of the pair. This composition is the topic of the next page.

You can see some of these ideas put into use via analog electronics at the following site:

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