This example is inspired from a lecture of John Watrous in the course on Theory of Quantum Information.
The Fidelity between two Hermitian semidefinite matrices P and Q is defined as:
$F(P,Q) = {||{P}^{1/2}{Q}^{1/2} ||}_{tr}$ = max $|trace({P}^{1/2}U{Q}^{1/2})|$
where the trace norm $||.||_{tr}$ is the sum of the singular values, and the maximization goes over the set of all unitary matrices U. This quantity can be expressed as the optimal value of the following complex-valued SDP:
maximize 1/2 trace(Z+Z*)
subject to $\left[\begin{array}{cc}P&Z\\{Z}^{*}&Q\end{array}\right] \succeq 0$
where $Z \in \mathbf {C}^{n \times n}$
using Convex
n = 20
P = randn(n,n) + im*randn(n,n)
P = P*P'
Q = randn(n,n) + im*randn(n,n)
Q = Q*Q'
Z = ComplexVariable(n,n)
objective = 0.5*real(trace(Z+Z'))
constraint = [P Z;Z' Q] ⪰ 0
problem = maximize(objective,constraint)
solve!(problem)
computed_fidelity = evaluate(objective)
---------------------------------------------------------------------------- SCS v1.1.8 - Splitting Conic Solver (c) Brendan O'Donoghue, Stanford University, 2012-2015 ---------------------------------------------------------------------------- Lin-sys: sparse-direct, nnz in A = 19 eps = 1.00e-04, alpha = 1.80, max_iters = 20000, normalize = 1, scale = 5.00 Variables n = 9, constraints m = 65 Cones: primal zero / dual free vars: 29 sd vars: 36, sd blks: 1 Setup time: 1.45e-04s ---------------------------------------------------------------------------- Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s) ---------------------------------------------------------------------------- 0| inf inf -nan -inf -nan inf 1.39e-04 100| 2.65e-04 2.64e-02 2.29e-03 -5.19e+00 -5.16e+00 1.37e-15 8.48e-03 200| 2.07e-06 2.08e-04 1.42e-05 -5.19e+00 -5.19e+00 1.52e-16 1.77e-02 220| 7.84e-07 7.89e-05 5.38e-06 -5.19e+00 -5.19e+00 2.30e-15 1.96e-02 ---------------------------------------------------------------------------- Status: Solved Timing: Solve time: 1.96e-02s Lin-sys: nnz in L factor: 93, avg solve time: 2.19e-06s Cones: avg projection time: 8.26e-05s ---------------------------------------------------------------------------- Error metrics: dist(s, K) = 1.1202e-09, dist(y, K*) = 1.7323e-09, s'y/m = -4.9828e-12 |Ax + s - b|_2 / (1 + |b|_2) = 7.8427e-07 |A'y + c|_2 / (1 + |c|_2) = 7.8923e-05 |c'x + b'y| / (1 + |c'x| + |b'y|) = 5.3752e-06 ---------------------------------------------------------------------------- c'x = -5.1910, -b'y = -5.1910 ============================================================================
5.191031665238333
# Verify that computer fidelity is equal to actual fidelity
P1,P2 = eig(P)
sqP = P2 * diagm([p1^0.5 for p1 in P1]) * P2'
Q1,Q2 = eig(Q)
sqQ = Q2 * diagm([q1^0.5 for q1 in Q1]) * Q2'
2×2 Array{Complex{Float64},2}: 1.73143-5.55112e-17im -0.047971+0.182073im -0.047971-0.182073im 0.951632+0.0im
actual_fidelity = sum(svd(sqP * sqQ)[2])
5.1910314597465606
We can see that the actual fidelity value is very close the computed fidelity value.