'Scuola Normale Superiore - Edizioni della Normale'
Abstract
This thesis studies the limits on the performances of inference tasks with quantum data
and quantum operations. Our results can be divided in two main parts.
In the first part, we study how to infer relative properties of sets of quantum states,
given a certain amount of copies of the states. We investigate the performance of optimal
inference strategies according to several figures of merit which quantifies the precision of
the inference. Since we are not interested in obtaining a complete reconstruction of the
states, optimal strategies do not require to perform quantum tomography. In particular,
we address the following problems:
- We evaluate the asymptotic error probabilities of optimal learning machines for
quantum state discrimination. Here, a machine receives a number of copies of a
pair of unknown states, which can be seen as training data, together with a test
system which is initialized in one of the states of the pair with equal probability.
The goal is to implement a measurement to discriminate in which state the test
system is, minimizing the error probability. We analyze the optimal strategies for
a number of different settings, differing on the prior incomplete information on the
states available to the agent.
- We evaluate the limits on the precision of the estimation of the overlap between two
unknown pure states, given N and M copies of each state. We find an asymptotic
expansion of a Fisher information associated with the estimation problem, which
gives a lower bound on the mean square error of any estimator. We compute the
minimum average mean square error for random pure states, and we evaluate the
effect of depolarizing noise on qubit states. We compare the performance of the
optimal estimation strategy with the performances of other intuitive strategies,
such as the swap test and measurements based on estimating the states.
- We evaluate how many samples from a collection of N d-dimensional states are
necessary to understand with high probability if the collection is made of identical
states or they differ more than a threshold according to a motivated closeness
measure. The access to copies of the states in the collection is given as follows:
each time the agent ask for a copy of the states, the agent receives one of the states with some fixed probability, together with a different label for each state in the collection. We prove that the problem can be solved with O(pNd=2) copies, and
that this scaling is optimal up to a constant independent on d;N; .
In the second part, we study optimal classical and quantum communication rates for
several physically motivated noise models.
- The quantum and private capacities of most realistic channels cannot be evaluated
from their regularized expressions. We design several degradable extensions
for notable channels, obtaining upper bounds on the quantum and private capacities
of the original channels. We obtain sufficient conditions for the degradability
of flagged extensions of channels which are convex combination of other channels.
These sufficient conditions are easy to verify and simplify the construction of
degradable extensions.
- We consider the problem of transmitting classical information with continuous variable
systems and an energy constraint, when it is impossible to maintain a shared
reference frame and in presence of losses. At variance with phase-insensitive noise
models, we show that, in some regimes, squeezing improves the communication
rates with respect to coherent state sources and with respect to sources producing
up to two-photon Fock states. We give upper and lower bounds on the optimal
coherent state rate and show that using part of the energy to repeatedly restore a
phase reference is strictly suboptimal for high energies