2 research outputs found
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 and its dual
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