Quantum statistical inference and communication

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

Quantum-capacity bounds in spin-network communication channels

Using the Lieb-Robinson inequality and the continuity property of the quantum capacities in terms of the diamond norm, we derive an upper bound on the values that these capacities can attain in spin-network communication i.i.d. models of arbitrary topology. Differently from previous results we make no assumptions on the encoding mechanisms that the sender of the messages adopts in loading information on the network.Comment: 9 pages, 1 figur

Optimal Quantum Subtracting Machine

The impossibility of undoing a mixing process is analysed in the context of quantum information theory. The optimal machine to undo the mixing process is studied in the case of pure states, focusing on qubit systems. Exploiting the symmetry of the problem we parametrise the optimal machine in such a way that the number of parameters grows polynomially in the size of the problem. This simplification makes the numerical methods feasible. For simple but non-trivial cases we computed the analytical solution, comparing the performance of the optimal machine with other protocols.Comment: 13 pages, 2 figure

Bounding the quantum capacity with flagged extensions

In this article we consider flagged extensions of channels that can be written as convex combination of other channels, and find general sufficient conditions for the degradability of the flagged extension. An immediate application is a bound on the quantum and private capacities of any channel being a mixture of a unitary operator and another channel, with the probability associated to the unitary operator being larger than $1/2$. We then specialize our sufficient conditions to flagged Pauli channels, obtaining a family of upper bounds on quantum and private capacities of Pauli channels. In particular, we establish new state-of-the-art upper bounds on the quantum and private capacities of the depolarizing channel, BB84 channel and generalized amplitude damping channel. Moreover, the flagged construction can be naturally applied to tensor powers of channels with less restricting degradability conditions, suggesting that better upper bounds could be found by considering a larger number of channel uses.Comment: 18 pages, 4 figure

Low-ground/High ground capacity regions analysis for Bosonic Gaussian Channels

We present a comprehensive characterization of the interconnections between single-mode, phaseinsensitive Gaussian Bosonic Channels resulting from channel concatenation. This characterization enables us to identify, in the parameter space of these maps, two distinct regions: low-ground and high-ground. In the low-ground region, the information capacities are smaller than a designated reference value, while in the high-ground region, they are provably greater. As a direct consequence, we systematically outline an explicit set of upper bounds for the quantum and private capacity of these maps, which combine known upper bounds and composition rules, improving upon existing results.Comment: 18 pages; 7 figure

Quantum theory in finite dimension cannot explain every general process with finite memory

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory state is a finite dimensional quantum state, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.Comment: 18 pages, 0 figure

Testing identity of collections of quantum states: sample complexity analysis

We study the problem of testing identity of a collection of unknown quantum states given sample access to this collection, each state appearing with some known probability. We show that for a collection of $d$-dimensional quantum states of cardinality $N$, the sample complexity is $O(\sqrt{N}d/\epsilon^2)$, which is optimal up to a constant. The test is obtained by estimating the mean squared Hilbert-Schmidt distance between the states, thanks to a suitable generalization of the estimator of the Hilbert-Schmidt distance between two unknown states by B\u{a}descu, O'Donnell, and Wright (https://dl.acm.org/doi/10.1145/3313276.3316344).Comment: 20+6 pages, 0 figures. Typos corrected, improved presentatio

Beyond the swap test: optimal estimation of quantum state overlap

We study the estimation of the overlap between two unknown pure quantum states of a finite dimensional system, given $M$ and $N$ copies of each type. This is a fundamental primitive in quantum information processing that is commonly accomplished from the outcomes of $N$ swap-tests, a joint measurement on one copy of each type whose outcome probability is a linear function of the squared overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies. We derive the statistics of the optimal measurement and compute the optimal mean square error in the asymptotic pointwise and finite Bayesian estimation settings. Besides, we consider two strategies relying on the estimation of one or both the states, and show that, although they are suboptimal, they outperform the swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.Comment: 5+19 pages, 5 figures, references adde