Maximum Likelihood, Minimum Effort

Details Maximum Likelihood, Minimum Effort Maximum Likelihood, Minimum Effort

2011-06-27

arxiv.org/abs/1106.5458v2

Phys. Rev. Lett. 108, 070502 (2012)

Physics

Quantum Physics

4 pages, 5 pdf figures. Replaced with corrections and expanded figures

Scientific paper

10.1103/PhysRevLett.108.070502

We provide an efficient method for computing the maximum likelihood mixed quantum state (with density matrix rho) given a set of measurement outcome in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix mu which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst O(d^4) for the basis change plus O(d^3) for finding rho where d is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only O(d^3) as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.

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