Vittorio Giovannetti

Quantum state tomography, aimed at deriving a classical description of an unknown state from measurement data, is a fundamental task in quantum physics. In this work, we analyse the ultimate achievable performance of tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of qudit systems: the minimum number of state copies needed for tomography not only scales exponentially with the number of modes but also exhibits a dramatic scaling with the trace-distance error, even for low-energy states. On a more positive note, we prove that tomography of Gaussian states is efficient. To accomplish this, we answer a fundamental question for the field of continuous-variable quantum information: if we know with a certain error the first and second moments of an unknown Gaussian state, what is the resulting trace-distance error that we make on the state? Lastly, we demonstrate that tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.

Quantum optical systems are typically affected by two types of noise: photon loss and dephasing. Despite extensive research on each noise process individually, a comprehensive understanding of their combined effect is still lacking. A crucial problem lies in determining the values of loss and dephasing for which the resulting loss-dephasing channel is anti-degradable, implying the absence of codes capable of correcting its effect or, alternatively, capable of enabling quantum communication. A conjecture in [Quantum 6, 821 (2022)] suggested that the bosonic loss-dephasing channel is not anti-degradable if the loss is below 50%. In this paper we refute this conjecture, specifically proving that for any value of the loss, if the dephasing is above a critical value, then the bosonic loss-dephasing channel is anti-degradable. While our result identifies a large parameter region where quantum communication is not possible, we also prove that if two-way classical communication is available, then quantum communication — and thus quantum key distribution — is always achievable, even for high values of loss and dephasing.

The two-way capacities of quantum channels determine the ultimate entanglement and secret-key distribution rates achievable by two distant parties that are connected by a noisy transmission line, in absence of quantum repeaters. Since repeaters will likely be expensive to build and maintain, a central open problem of quantum communication is to understand what performances are achievable without them. In this paper, we find a new lower bound on the energy-constrained and unconstrained two-way quantum and secret-key capacities of all phase-insensitive bosonic Gaussian channels, namely thermal attenuator, thermal amplifier, and additive Gaussian noise, which are realistic models for the noise affecting optical fibres or free-space links. Ours is the first nonzero lower bound on the two-way quantum capacity in the parameter range where the (reverse) coherent information becomes negative, and it shows explicitly that entanglement distribution is always possible when the channel is not entanglement breaking. This completely solves a crucial open problem of the field, namely, establishing the maximum excess noise which is tolerable in continuous-variable quantum key distribution. In addition, our construction is fully explicit, i.e.~we devise and optimise a concrete entanglement distribution and distillation protocol that works by combining recurrence and hashing protocols.

We study the problem of estimating distance measures between unknown quantum states, given M and N copies of each type. For pure states, any distance measure can be expressed in terms of the overlap between the states. Overlap estimation 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 overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies, in particular using weak Schur sampling. 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. We show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states. In the case of mixed states, distance measure cannot be expressed in terms of a single distance. With the swap test, one can estimate the Hilbert-Schmidt distance and the overlap, but it is not possible to obtain other very relevant distance measures. We show that simultaneous weak Schur sampling can instead estimate the spectra of each of the states and of their convex combination, with weight given by M/M+N and N/M+N. This estimate is much more informative than the outcome of swap tests and it can be used to estimate a wider class of properties and distinguishability measures. In particular we show how to measure the Holevo quantity of a an ensemble constructed with the states. We discuss how to get bounds on the trace distance using these estimates, and apply these bounds to closeness and remoteness testing, improving sample complexity with respect to the swap test.

A new bound for the quantum capacity of the d-dimensional depolarizing channels is presented. Our derivation makes use of a flagged extension of the map where the receiver obtains a copy of a state σ_0 whenever the messages are transmitted without errors, and a copy of a state σ_1 when instead the original state gets fully depolarized. By varying the overlap between the flag states, the resulting transformation nicely interpolates between the depolarizing map (when σ_0 = σ_1), and the d-dimensional erasure channel (when σ_0 and σ_1 have orthogonal support). We find sufficient conditions for degradability of the flagged channel, which let us to calculate its quantum capacity in a suitable parameter region. From this last result we get the upper bound for the depolarizing channel, which by a direct comparison appears to be tighter than previous available results for d > 2, and for d = 2 it is tighter in an intermediate regime of noise. In particular, in the limit of large d values, our findings presents a previously unnoticed O(1) correction.

The no-go theorem regarding unconditionally secure Quantum Bit Commitment protocols is a relevant result in quantum cryptography. The impossibility proof for Quantum Bit Commitment has been used to prove the impossibility of unditional security for other protocols, such as Quantum Oblivious Transfer or One-Sided Two Party Computation. In this paper, we extend the same proof to the non-deterministic version of Quantum Private Queries, a protocol addressing the Symmetric-Private Information Retrieval problem. Moreover, we prove the equivalence between Quantum Private Queries and Quantum Bit Commitment and One-Sided Two Party Computation protocols.