Optimal networks for Quantum Metrology: semidefinite programs and product rules

We investigate the optimal estimation of a quantum process that can possibly consist of multiple time steps. The estimation is implemented by a quantum network that interacts with the process by sending an input and processing the output at each time step. We formulate the search of the optimal network as a semidefinite program and use duality theory to give an alternative expression for the maximum payoff achieved by estimation. Combining this formulation with a technique devised by Mittal and Szegedy we prove a general product rule for the joint estimation of independent processes, stating that the optimal joint estimation can achieved by estimating each process independently, whenever the figure of merit is of a product form. We illustrate the result in several examples and exhibit counterexamples showing that the optimal joint network may not be the product of the optimal individual networks if the processes are not independent or if the figure of merit is not of the product form. In particular, we show that entanglement can reduce by a factor K the variance in the estimation of the sum of K independent phase shifts.Comment: 19 pages, no figures, published versio

The Hilbertian Tensor Norm and Entangled Two-Prover Games

We study tensor norms over Banach spaces and their relations to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm $\gamma_2$ and its dual $\gamma_2^*$ that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.Comment: 33 pages, some of the results have been obtained independently in arXiv:1007.3043v2, v2: an error in Theorem 4 has been corrected; Section 6 rewritten, v3: completely rewritten in order to improve readability; title changed; references added; published versio