The von Neumann entropy of a quantum state is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state $\rho$ can be formulated as the difference of the number of physical qubits and the "relative entropy distance" between $\rho$ and the maximally mixed state, here we define the entropy of a channel $\mathcal{N}$ as the difference of the number of physical qubits of the channel output with the "relative entropy distance" between $\mathcal{N}$ and the completely depolarizing channel. We prove that this definition satisfies all of the axioms, recently put forward in [Gour, arXiv:1808.02607], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. We define R\'{e}nyi and min-entropies of a channel and prove that they satisfy the axioms required for a channel entropy function. Among other results, we also prove that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.

Modern applications require a robust and theoretically solid tool for the realistic modeling of electronic states in low dimensional nanostructures. The $k \cdot p$ theory has fruitfully served this role for the long time since its establishment. During the last three decades several problems have been detected in connection with the application of the $k \cdot p$ approach to such nanostructures. These problems are closely related to the violation of the ellipticity conditions for the underlying model, the fact that has been largely overlooked in the literature. We derive ellipticity conditions for $6 \times 6$, $8\times 8$ and $14 \times 14$ Hamiltonians obtained by the application of Luttinger-Kohn theory to the bulk zinc blende (ZB) crystals, and demonstrate that the corresponding models are non-elliptic for many common crystalline materials. With the aim to obtain the admissible (in terms of ellipticity) parameters, we further develop and justify a parameter rescaling procedure for $8\times 8$ Hamiltonians. This allows us to calculate the admissible parameter sets for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, InN. The newly obtained parameters are then optimized in terms of the bandstructure fit by changing the value of the inversion asymmetry parameter $B$ that is proved to be essential for ellipticity of $8\times 8$ Hamiltonian. The consecutive analysis, performed here for all mentioned $k \cdot p$ Hamiltonians, indicates the connection between the lack of ellipticity and perturbative terms describing the influence of out-of-basis bands on the structure of the Hamiltonian. This enables us to quantify the limits of models' applicability material-wise and to suggest a possible unification of two different $14 \times 14$ models, analysed in this work.

We present the first multi-node room-temperature memory-assisted quantum communication network using polarization qubits. Our quantum network combines several independent quantum nodes in an elementary configuration: (i) two independent polarization qubit generators working at rubidium transitions, (ii) two ultra-low noise room-temperature quantum memories, and (iii) a Bell-state qubit decoder and reading station. After storage and retrieval in the two table-top quantum memories, we measure a high-visibility Hong-Ou-Mandel interference of $V=46.8\%\pm3.4\%$ between the outputs. We envision this realization to become the backbone of future memory-assisted cryptographic and entanglement distribution networks.

We apply the measurement reduction technique to optimally reconstruct an object image from multiplexed ghost images (GI) while taking into account both GI correlations and object image sparsity. We show that one can reconstruct an image in that way even if the object is illuminated by a small photon number. We consider frequency GI multiplexing using coupled parametric processes. We revealed that the imaging condition depends on the type of parametric process, namely, whether down- or up-conversion is used. Influence of information about sparsity in discrete cosine transform and Haar transform bases on reconstruction quality is studied. In addition, we compared ordinary and ghost images when the detectors are additionally illuminated by noise photons in a computer experiment, which showed increased noise immunity of GI, especially with processing via the proposed technique.

We propose a method to generate nonclassical states of light in multimode microwave cavities. Our approach considers two-photon processes that take place in a system composed of two extended cavities and an ultrastrongly coupled light-matter system. Under specific resonance conditions, our method generates, in a deterministic manner, product states of uncorrelated photon pairs, Bell states, and W states. We demonstrate improved generation times when increasing the number of multimode cavities, and prove the generation of genuine multipartite entangled states when coupling an ancillary system to each cavity. Finally, we discuss the feasibility of our proposal in circuit quantum electrodynamics.

We introduce the Simulator for Quantum Networks and Channels ($\texttt{SQUANCH}$), an open-source Python library for creating parallelized simulations of distributed quantum information processing. The framework includes many features of a general-purpose quantum computing simulator, but it is optimized specifically for simulating quantum networks. It includes functionality to allow users to easily design complex multi-party quantum networks, extensible classes for modeling noisy quantum channels, and a multiprocessed NumPy backend for performant simulations. We present an overview of the structure of the library, describing how the various API elements represent the underlying physics and providing simple usage examples for each module. Finally, we present several demonstrations of canonical quantum information protocols implemented using this framework.

Boson sampling, thought to be intractable classically, can be solved by a quantum machine composed of merely generation, linear evolution and detection of single photons. Such an analog quantum computer for this specific problem provides a shortcut to boost the absolute computing power of quantum computers to beat classical ones. However, the capacity bound of classical computers for simulating boson sampling has not yet been identified. Here we simulate boson sampling on the Tianhe-2 supercomputer which occupied the first place in the world ranking six times from 2013 to 2016. We computed the permanent of the largest matrix using up to 312,000 CPU cores of Tianhe-2, and inferred from the current most efficient permanent-computing algorithms that an upper bound on the performance of Tianhe-2 is one 50-photon sample per ~100 min. In addition, we found a precision issue with one of two permanent-computing algorithms.

We examine important properties of the difference between the variance and the quantum Fisher information over four, i.e., $(\Delta A)^2-F_{\rm Q}[\varrho,A]/4.$ We find that it is equal to a generalized variance defined in Petz [J. Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. Theory 55, 439 (2009)]. We present an upper bound on this quantity that is proportional to the linear entropy. As expected, our relation shows that for states that are close to being pure, the quantum Fisher information over four is close to the variance. We also obtain the variance and the quantum Fisher information averaged over all Hermitian operators, and examine its relation to the von Neumann entropy. Apart from the usual quantum Fisher information, we also consider the Kubo-Mori-Bogoliubov quantum Fisher information.

How do isolated quantum systems approach an equilibrium state? We experimentally and theoretically address this question for a prototypical spin system formed by ultracold atoms prepared in two Rydberg states with different orbital angular momenta. By coupling these states with a resonant microwave driving we realize a dipolar XY spin-1/2 model in an external field. Starting from a spin-polarized state we suddenly switch on the external field and monitor the subsequent many-body dynamics. Our key observation is density dependent relaxation of the total magnetization much faster than typical decoherence rates. To determine the processes governing this relaxation we employ different theoretical approaches which treat quantum effects on initial conditions and dynamical laws separately. This allows us to identify an intrinsically quantum component to the relaxation attributed to primordial quantum fluctuations.

The wave-function Monte-Carlo method, also referred to as the use of "quantum-jump trajectories", allows efficient simulation of open systems by independently tracking the evolution of many pure-state "trajectories". This method is ideally suited to simulation by modern, highly parallel computers. Here we show that Krotov's method of numerical optimal control, unlike others, can be modified in a simple way, so that it becomes fully parallel in the pure states without losing its effectiveness. This provides a highly efficient method for finding optimal control protocols for open quantum systems and networks. We apply this method to the problem of generating entangled states in a network consisting of systems coupled in a unidirectional chain. We show that due to the existence of a dark-state subspace in the network, nearly-optimal control protocols can be found for this problem by using only a single pure-state trajectory in the optimization, further increasing the efficiency.

Photonic time-frequency entanglement is a promising resource for quantum information processing technologies. We investigate swapping of continuous-variable entanglement in the time-frequency degree of freedom using three-wave mixing in the low-gain regime with the aim of producing heralded biphoton states with high purity and low multi-pair probability. Heralding is achieved by combining one photon from each of two biphoton sources via sum-frequency generation to create a herald photon. We present a realistic model with pulsed pumps, investigate the effects of resolving the frequency of the herald photon, and find that frequency-resolving measurement of the herald photon is necessary to produce high-purity biphotons. We also find a trade-off between the rate of successful entanglement swapping and both the purity and quantified entanglement resource (negativity) of the heralded biphoton state.

We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young's inequality for convolutions, and the theorem "Gaussian kernels have only Gaussian maximizers." Quantum Gaussian channels play a key role in quantum communication theory: they are the quantum counterpart of Gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space. The restriction of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides with the thinning, which is the analog of the rescaling for positive integer random variables. Quantum Gaussian channels provide then a bridge between functional analysis and discrete probability.

We isolate a novel four-wave mixing process, enabled by Coherent Population Trapping (CPT), leading to efficient phase sensitive amplification. This process is permitted by the exploitation of two transitions starting from the same twofold degenerate ground state. One of the transitions is used for CPT, defining bright and dark states from which ultra intense four-wave mixing is obtained via the other transition. This leads to the measurement of a strong phase sensitive gain even for low optical densities and out-of-resonance excitation. The enhancement of four-wave mixing is interpreted in the framework of the dark-state polariton formalism.

The multinomial model is one of the simplest statistical models. When constraints are placed on the possible values for the probabilities, however, it becomes much more difficult to deal with. Model checking and checking for prior-data conflict is considered here for such models. A theorem is proved that establishes the consistency of the check on the prior. Applications are presented to models that arise in quantum state estimation as well as the Bayesian analysis of models for ordered probabilities.

To implement fault-tolerant quantum computation with continuous variables, the Gottesman--Kitaev--Preskill (GKP) qubit has been recognized as an important technological element. We have proposed a method to reduce the required squeezing level to realize large scale quantum computation with the GKP qubit [Phys. Rev. X. {\bf 8}, 021054 (2018)], harnessing the virtue of analog information in the GKP qubits. In the present work, to reduce the number of qubits required for large scale quantum computation, we propose the tracking quantum error correction, where the logical-qubit level quantum error correction is partially substituted by the single-qubit level quantum error correction. In the proposed method, the analog quantum error correction is utilized to make the performances of the single-qubit level quantum error correction almost identical to those of the logical-qubit level quantum error correction in a practical noise level. The numerical results show that the proposed tracking quantum error correction reduces the number of qubits during a quantum error correction process by the reduction rate $\left\{{2(n-1)\times4^{l-1}-n+1}\right\}/({2n \times 4^{l-1}})$ for $n$-cycles of the quantum error correction process using the Knill's $C_{4}/C_{6}$ code with the concatenation level $l$. Hence, the proposed tracking quantum error correction has great advantage in reducing the required number of physical qubits, and will open a new way to bring up advantage of the GKP qubits in practical quantum computation.

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the $(1+1)-$dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators---differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.

We propose a quantum simulator based on driven superconducting qubits where the interactions are generated parametrically by a polychromatic magnetic flux modulation of a tunable bus element. Using a time-dependent Schrieffer-Wolff transformation, we analytically derive a multi-qubit Hamiltonian which features independently tunable $XX$ and $YY$-type interactions as well as local bias fields over a large parameter range. We demonstrate the adiabatic simulation of the ground state of a hydrogen molecule using two superconducting qubits and one tunable bus element. The time required to reach chemical accuracy lies in the few microsecond range and therefore could be implemented on currently available superconducting circuits. Further applications of this technique may also be found in the simulation of interacting spin systems.

Quantum theory allows direct measurement of the average of a non-Hermitian operator using the weak value of the positive semidefinite part of the non-Hermitian operator. Here, we experimentally demonstrate the measurement of weak value and average of non-Hermitian operators by a novel interferometric technique. Our scheme is unique as we can directly obtain the weak value from the interference visibility and the phase shift in a Mach Zehnder interferometer without using any weak measurement or post selection. Both the experiments discussed here were performed with laser sources, but the results would be the same with average statistics of single photon experiments. Thus, the present experiment opens up the novel possibility of measuring weak value and the average value of non-Hermitian operator without weak interaction and post-selection, which can have several technological applications.

The Continuous Spontaneous Localization (CSL) model is the best known and studied among collapse models, which modify quantum mechanics and identify the fundamental reasons behind the unobservability of quantum superpositions at the macroscopic scale. Albeit several tests were performed during the last decade, up to date the CSL parameter space still exhibits a vast unexplored region. Here, we study and propose an unattempted non-interferometric test aimed to fill this gap. We show that the angular momentum diffusion predicted by CSL heavily constrains the parametric values of the model when applied to a macroscopic object.

This is a brief review on the theoretical interpretation of the Aharonov-Bohm effect, which also contains our new insight into the problem. A particular emphasis is put on the unique role of electron orbital angular momentum, especially viewed from the novel concept of the physical component of the gauge field, which has been extensively discussed in the context of the nucleon spin decomposition problem as well as the photon angular momentum decomposition problem. Practically, we concentrate on the frequently discussed idealized setting of the Aharonov-Bohm effect, i.e. the interference phenomenon of the electron beam passing around the infinitely-long solenoid. One of the most puzzling observations in this Aharonov-Bohm solenoid effect is that the pure-gauge potential outside the solenoid appears to carry non-zero orbital angular momentum. Through the process of tracing its dynamical origin, we try to answer several fundamental questions of the Aharonov-Bohm effect, which includes the question about the reality of the electromagnetic potential, the gauge-invariance issue, and the non-locality interpretation, etc.