Abstract. We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information? We show that if the prior information restricts the possible states to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that the minimal number of measurement outcomes (POVM elements) necessary to identify all pure states in a d-dimensional Hilbert space is 4d − 3 − c(d)α(d) for some c(d) ∈ [1, 2] and α(d) being the number of ones appearing in the binary expansion of (d − 1).

This paper extends the consensus framework, widely studied in the literature on distributed computing and control algorithms, to networks of quantum systems. We define consensus situations on the basis of invariance and symmetry properties, finding four different probabilistic generalizations of classical consensus states. We then extend the gossip consensus algorithm to the quantum setting and prove its convergence properties, showing how it converges to symmetric states while preserving the expectation of permutation-invariant global observables.

Concerted efforts are underway to establish an infrastructure for a global quantum Internet to realise a spectrum of quantum technologies. This will enable more precise sensors, secure communications, and faster data processing. Quantum communications are a front-runner with quantum networks already implemented in several metropolitan areas. A number of recent proposals have modelled the use of space segments to overcome range limitations of purely terrestrial networks. Rapid progress in the design of quantum devices have enabled their deployment in space for in-orbit demonstrations. We review developments in this emerging area of space-based quantum technologies and provide a roadmap of key milestones towards a complete, global quantum networked landscape. Small satellites hold increasing promise to provide a cost effective coverage required to realise the quantum Internet. The state of art in small satellite missions is reviewed and the most current in-field demonstrations of quantum cryptography are collated. The important challenges in space quantum technologies that must be overcome and recent efforts to mitigate their effects are summarised. A perspective on future developments that would improve the performance of space quantum communications is included. The authors conclude with a discussion on fundamental physics experiments that could take advantage of a global, space-based quantum network.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Single-photon sources represent a fundamental building block for optical implementations of quantum information tasks ranging from basic tests of quantum physics to quantum communication and high-resolution quantum measurement. In this paper, in order to compare the effectiveness of different designs, we introduce a single-photon source performance index, based on the maximum probability of generating a single photon that still guarantees a given signal-to-noise ratio. We then investigate the performance of a multiplexed system based on asymmetric configuration of multiple heralded single-photon sources. The performance and scalability comparison with both currently existing multiple-source architectures and faint laser configurations reveals an advantage the proposed scheme offers in realistic scenarios. This analysis also provides insights on the potential of using such architectures for integrated implementation

Global-scale quantum communication links will form the backbone of the quantum internet. However, exponential loss in optical fibres precludes any realistic application beyond few hundred kilometres. Quantum repeaters and space-based systems offer solutions to overcome this limitation. Here, we analyse the use of quantum memory (QM)-equipped satellites for quantum communication focussing on global range repeaters and memory-assisted (MA-) QKD, where QMs help increase the key rate by synchronising otherwise probabilistic detection events. We demonstrate that satellites equipped with QMs provide three orders of magnitude faster entanglement distribution rates than existing protocols based on fibre-based repeaters or space systems without QMs. We analyse how entanglement distribution performance depends on memory characteristics, determine benchmarks to assess the performance of different tasks and propose various architectures for light-matter interfaces. Our work provides a roadmap to realise unconditionally secure quantum communications over global distances with near-term technologies.