Quantum Tomography and Detection: Design via Convex Optimization
by
Robert Kosut
SC Solutions, Sunnyvale, CA 94085 USA
Coauthors: Ian Walmsley (Oxford University, Oxford, UK), Herschel Rabitz (Princeton University, Princeton, NJ)

In this paper we show that a number of problems in quantum state estimation (state tomography), quantum system identification (process tomography) and quantum state and system detection can be cast as convex optimization problems. The great advantage of convex optimization is a globally optimal solution can be found efficiently and reliably, and perhaps most importantly, can be computed to within any desired accuracy using an interior-point method.

Some of the problems addressed in this paper are already known to be convex but have not fully exploited the available convex solvers or duality theory. For example, it is known that Maximum likelihood Estimation (MLE) of the quantum state (density) is a convex optimization. What we also show is that a number of other MLE problems are convex, e.g., estimating the distribution of known states and quantum process tomography in the Kraus operator sum representation (OSR) in a fixed basis. One important problem which is not convex is MLE of Hamiltonian parameters. We show, however, how duality theory can help establish bounds on the parameter estimates for this problem.

Another problem which can be solved via convex optimization is experiment design. (Experiment design here means choosing the number of experiments to be performed in a particular system configuration; a configuration being any number of combinations of sample times, hardware settings, etc. For example, in quantum state photonic tomography, we can determine the optimum number of wave plate setting to achieve a desired estimation accuracy.) In this paper we will apply the experiment design procedure invoked by the Cramer-Rao Inequality to all the MLE problems mentioned above, including MLE of Hamiltonian parameters. We will show that in all these cases the optimum experiment design problem, although integer-combinatoric, can be relaxed to a convex optimization problem whose solution provides upper and lower bounds on the unknown optimal integer solution. The MLE of the state or process can be combined with the optimal experiment design in a ``bootstrapping'' iteration to make the estimation more efficient.

Finally, we also address the problem of designing a detector which is maximally sensitive to specific quantum states. We show that the design problem can be formulated as a convex optimization problem in the matrices of the POVM (positive operator valued measure) which characterize the measurement apparatus, or with a given POVM, the matrices which characterize the OSR in a fixed basis. We specifically address maximizing the posterior probability of detection and show that this is a quasiconvex optimization problem in either the POVM or OSR matrices. Previous work in this area has only considered the joint probability of detection over POVM matrices,

In all the cases described above we show how duality theory can be used in various special cases to give insight into the nature (and possible physical implementation) of the optimal solutions. In addition we will briefly comment on the numerical properties of the convex programming methods required. At present we have some experimental results for some of the above cases.

Date received: February 9, 2004