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1) Derivation of the gradient of |f(U)|^{2}
Let Mat(m) designate the space of complex m x m matrices equipped with the scalar product (9), U(m) the space of all unitary m x m matrices U, and f resp. F functions defined by
By definition, a matrix TMat(m) is tangent at the group U(m) in the point UU(m) if there exists a curve tÆV(t) U(m) which is continuously differentiable and satisfies
A useful relation is TU^{†} = UT^{†} which follows from
The gradientF(U) Mat(m) of F is by definition a matrix which is tangent at UU(m) and for all curves tÆV(t) defined above satisfies
Using the product rule of calculus we obtain:
d/dt|_{t=t0}F(V(t)) | = d/dt|_{t=t0} { f(V(t)) f^{*}(V(t))} |
= 2 Re{f^{*}(U) d/dt|_{t=t0}f(V(t))} | |
= 2 Re{f^{*}(U) tr{TA^{†}U^{†}C + UA^{†}T^{†}C}} | |
= 2 Re{f^{*}(U) tr{ U^{†} [UA^{†}U^{†}, C] T}} | |
= {(f*(U) [ UA^{†}U^{†},C])^{†}f*(U) [ UA^{†}U^{†},C] } U | T |
Hence it follows that F(U) ={(f*(U) [ UA^{†}U^{†},C])^{†}f*(U) [ UA^{†}U^{†},C] } U. F(U) is the tangent in U of the curve tÆW(t)= exp{(tt_{0}) F(U)U^{†}} UU(m) since W(t_{0}) = U and d/dt|_{t=t0}W(t) = F(U).
2) An Algorithm to trace the boundary of W_{C}(A^{†})
The boundary of the C-numerical range W_{C}(A^{†}) can be traced by the following strategy based on the fact that W_{C}(A^{†}) is always star-shaped (17). (A subset of the complex plane is said to be star-shaped with respect to a center, if the respective line connecting any point of the subset with the center is entirely in the subset.) The star center is given by tr(A^{†}) tr(C)/m if the operators A and C are represented by m x m matrices (17). For the traceless auxiliary matrix A_{t}={A tr(A)/m}, the C-numerical range W_{C}(A_{t}^{†}) is shifted in the complex plane by tr(A^{†}) tr(C)/m relative to W_{C}(A^{†}) such that the star center of W_{C}(A_{t}^{†}) is located at the origin of the complex plane. Hence the boundary of the C-numerical range of A_{t}^{†} intersects the positive real axis at one and only one point. If this point can be found, the entire boundary of W_{C}(A_{t}^{†}) can be traced if the intersection between W_{C}(A^{†}) and the positive real axis is determined for 0 �ϕ� 2 π with A=e^{i ϕ}A_{t}, because a multiplication of A_{t} by e^{i ϕ} rotates W_{C}(A_{t}^{†}) by ϕ about the origin of the complex plane. Finally, the boundary of W_{C}(A^{†}) is obtained by shifting the boundary of W_{C}(A_{t}^{†}) by tr(A^{†}) tr(C)/m.
The point where the positive real axis intersects the boundary of W_{C}(A^{†}) can be approached by optimizing the real part Re{f (U)} subject to the constraint that the imaginary part Im{f (U)} = 0 with f(U):= UAU^{†}| C.
For the cases studied in this report (compare Figures 2 - 4), this was achieved by the following preliminary iterative algorithm which consists of two steps:
Starting from an arbitrary unitary operator U_{0}, steps (a) and (b) are repeated until convergence is reached. The gradients of Re{f(U)} and ( Im{f(U)} )^{2} are given by
Re{f(U)}= 1/2([UA^{†}U^{†}, C]^{†} [UA^{†}U^{†}, C])U and
(Im{f(U)} )^{2} = i Im{f(U)} ([ UA^{†}U^{†},C]^{†} + [ UA^{†}U^{†},C]) U, respectively.
If W_{C}(A^{†}) is convex, a simplified algorithm can be applied. In this case, for each angle ϕ the maximum of Re{f(U)} is also well defined and can be found using Re{f(U)}.