Dark states are eigenstates or steady-states of a system that are decoupled from the radiation. Their use, along with associated technique such as Stimulated Raman Adiabatic Passage, has extended from atomic physics where it is an essential cooling mechanism, to more recent versions in condensed phase where it has been demonstrated to be capable of increasing coherence times of qubits. These states are often discussed in the context of unitary evolution and found with elegant methods exploiting symmetries, or via the Bruce-Shore transformation. However, the link with dissipative systems is not always transparent, and distinctions between classes of CPT are not always clear. We present a detailed overview of the arguments to find stationary dark states in dissipative systems, and examine their dependence on the Hamiltonian parameters, their multiplicity and purity.

The creation of delocalized coherent superpositions of quantum systems experiencing different relativistic effects is an important milestone in future research at the interface of gravity and quantum mechanics. This could be achieved by generating a superposition of quantum clocks that follow paths with different gravitational time dilation and investigating the consequences on the interference signal when they are eventually recombined. Light-pulse atom interferometry with elements employed in optical atomic clocks is a promising candidate for that purpose, but suffers from major challenges including its insensitivity to the gravitational redshift in a uniform field. All these difficulties can be overcome with a novel scheme presented here which is based on initializing the clock when the spatially separate superposition has already been generated and performing a doubly differential measurement where the differential phase shift between the two internal states is compared for different initialization times. This can be exploited to test the universality of the gravitational redshift with delocalized coherent superpositions of quantum clocks and it is argued that its experimental implementation should be feasible with a new generation of 10-meter atomic fountains that will soon become available. Interestingly, the approach also offers significant advantages for more compact set-ups based on guided interferometry or hybrid configurations. Furthermore, in order to provide a solid foundation for the analysis of the various interferometry schemes and the effects that can be measured with them, a general formalism for a relativistic description of atom interferometry in curved spacetime is developed. It can deal with freely falling atoms, but also include the effects of external forces and guiding potentials, and can be applied to a very wide range of situations.

In a Quantum Walk (QW) the "walker" follows all possible paths at once through the principle of quantum superposition, differentiating itself from classical random walks where one random path is taken at a time. This facilitates the searching of problem solution spaces faster than with classical random walks, and holds promise for advances in dynamical quantum simulation, biological process modelling and quantum computation. Current efforts to implement QWs have been hindered by the complexity of handling single photons and the inscalability of cascading approaches. Here we employ a versatile and scalable resonator configuration to realise quantum walks with bright classical light. We experimentally demonstrate the versatility of our approach by implementing a variety of QWs, all with the same experimental platform, while the use of a resonator allows for an arbitrary number of steps without scaling the number of optics. Our approach paves the way for practical QWs with bright classical light and explicitly makes clear that quantum walks with a single walker do not require quantum states of light.

In a recent publication in Nature Communications by Frauchiger and Renner (Nat. Commun. 9, 3711 (2018)), a Gedankenexperiment was proposed, which was claimed to be able to lead to inconsistent conclusions with a self-referential use of quantum theory. Thus it seems to prove that quantum theory cannot consistently describe the use of itself. Shortly after, Chen and Zhang suggested an improvement (arXiv:1810.01080) which can made the explanation of the Gedankenexperiment become consistent. Here we show that from the viewpoint of any quantum interpretation theory which agrees that there is collapse of the wavefunction when being measured (e.g., the Copenhagen interpretation), the original conclusions of Frauchiger and Renner actually came from an incorrect description of some quantum states. With the correct description there will be no inconsistent results, even without modifying the original Gedankenexperiment.

We report on numerical calculations of the spontaneous emission rate of a Rydberg-excited sodium atom in the vicinity of an optical nanofibre. In particular, we study how this rate varies with the distance of the atom to the fibre, the fibre's radius, the symmetry s or p of the Rydberg state as well as its principal quantum number. We find that a fraction of the spontaneously emitted light can be captured and guided along the fibre. This suggests that such a setup could be used for networking atomic ensembles, manipulated in a collective way due to the Rydberg blockade phenomenon.

Choosing the right first quantization basis in quantum optics is critical for the interpretation of experimental results. The usual frequency basis is, for instance, inappropriate for short, subcycle waveforms. We derive first quantization in time domain, and apply the results to ultrashort pulses propagating along unidimensional waveguides. We show how to compute the statistics of the photon counts, or that of their times of arrival. We also extend the concept of quadratures to the time domain, making use of the Hilbert transform.

Given a general $d$-dimensional unitary operation $U_d$ for which, apart from the dimension, its description is unknown, is it possible to implement its inverse operation $U_d^{-1}$ with a universal protocol that works for every unitary $U_d$? How does the situation change when $k$ uses of unitary operation $U_d$ are allowed? In this paper we show that any universal protocol implementing the inverse of a general unitary $U_d$ with a positive heralded probability requires at least $d-1$ uses of $U_d$. For the cases where $k\geq d-1$ uses are accessible, we construct a parallel and sequential protocol, whose respective probability of failure decreases linearly and exponentially. We then analyse protocols with indefinite causal order. These more general protocols still cannot yield the inverse of a general $d$-dimensional unitary operation with $k<d-1$ uses. However we show via a general semidefinite programming that protocols with indefinite causal order attain a higher success probability when $k> d-1$. This paper also introduces the notion of delayed input-state protocols and provides a one-to-one correspondence between the unitary learning (unitary store and retrieve) problem and universal parallel protocols for unitary transposition.

When compared to quantum mechanics, classical mechanics is often depicted in a specific metaphysical flavour: spatio-temporal realism or a Newtonian "background" is presented as an intrinsic fundamental classical presumption. However, the Hamiltonian formulation of classical analytical mechanics is based on abstract generalized coordinates and momenta: It is a mathematical rather than a philosophical framework. If the metaphysical assumptions ascribed to classical mechanics are dropped, then there exists a presentation in which little of the purported difference between quantum and classical mechanics remains. This presentation allows to derive the mathematics of relativistic quantum mechanics on the basis of a purely classical Hamiltonian phase space picture. It is shown that a spatio-temporal description is not a condition for but a consequence of objectivity. It requires no postulates. This is achieved by evading spatial notions and assuming nothing but time translation invariance.

This paper proposes a brain-inspired approach to quantum machine learning with the goal of circumventing many of the complications of other approaches. The fact that quantum processes are unitary presents both opportunities and challenges. A principal opportunity is that a large number of computations can be carried out in parallel in linear superposition, that is, quantum parallelism. The challenge is that the process is linear, and most approaches to machine learning depend significantly on nonlinear processes. Fortunately, the situation is not hopeless, for we know that nonlinear processes can be embedded in unitary processes, as is familiar from the circuit model of quantum computation. This paper explores an approach to the quantum implementation of machine learning involving nonlinear functions operating on information represented topographically (by computational maps), as common in neural cortex.

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

A century after the discovery of quantum mechanics, the meaning of quantum mechanics still remains elusive. This is largely due to the puzzling nature of the wave function, the central object in quantum mechanics. If we are realists about quantum mechanics, how should we understand the wave function? What does it represent? What is its physical meaning? Answering these questions would improve our understanding of what it means to be a realist about quantum mechanics. In this survey article, I review and compare several realist interpretations of the wave function. They fall into three categories: ontological interpretations, nomological interpretations, and the \emph{sui generis} interpretation. For simplicity, I will focus on non-relativistic quantum mechanics.

The fractional Laplacian $(- \Delta)^{\alpha /2}$, $\alpha \in (0,2)$ has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of $\alpha $-stable stochastic processes in $R^n$. On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data respecting fractional Laplacian should actually be. This ambiguity holds true not only for each specific choice of the process behavior at the boundary (like e.g. absorbtion, reflection, conditioning or boundary taboos), but extends as well to its particular technical implementation (Dirchlet, Neumann, etc. problems). The inferred jump-type processes are inequivalent as well, differing in their spectral and statistical characteristics. In the present paper we focus on L\'evy flight-induced jump-type processes which are constrained to stay forever inside a finite domain. That refers to a concept of taboo processes (imported from Brownian to L\'evy - stable contexts), to so-called censored processes and to reflected L\'evy flights whose status still remains to be settled on both physical and mathematical grounds.

We report on the first experimental reconstruction of an entanglement quasiprobability. In contrast to related techniques, the negativities in our distributions are a necessary and sufficient identifier of entanglement and enable a full characterization of the quantum state. A reconstruction algorithm is developed, a polarization Bell state is prepared, and its entanglement is certified based on the reconstructed entanglement quasiprobabilities, with a high significance and without correcting for imperfections.

A free-falling nanodiamond containing a nitrogen vacancy centre in a spin superposition should experience a superposition of forces in an inhomogeneous magnetic field. We propose a practical design that brings the internal temperature of the diamond to under 10 K. This extends the expected spin coherence time from 2 ms to 500 ms, so the spatial superposition distance could be increased from 0.05 nm to over 1 $\mu$m, for a 1 $\mu$m diameter diamond and a magnetic inhomogeneity of only 10$^4$ T/m. The low temperature allows single-shot spin readout, reducing the number of nanodiamonds that need to be dropped by a factor of 10,000. We also propose solutions to a generic obstacle that would prevent such macroscopic superpositions.

We show that the quantum description of measurement based on decoherence fixes the bug in quantum theory discussed in [D. Frauchiger and R. Renner, {\em Quantum theory cannot consistently describe the use of itself}, Nat. Comm. {\bf 9}, 3711 (2018)]. Assuming that the outcome of a measurement is determined by environment-induced superselection rules, we prove that different agents acting on a particular system always reach the same conclusions about its actual state.

Recently, there has been a surge of interest in using R\'enyi entropies as quantifiers of correlations in many-body quantum systems. However, it is well known that in general these entropies do not satisfy the strong subadditivity inequality, which is a central property ensuring the positivity of correlation measures. In fact, in many cases they do not even satisfy the weaker condition of subadditivity. In the present paper we shed light on this subject by providing a detailed survey of R\'enyi entropies for bosonic and fermionic Gaussian states. We show that for bosons the R\'enyi entropies always satisfy subadditivity, but not necessarily strong subadditivity. Conversely, for fermions both do not hold in general. We provide the precise intervals of the R\'enyi index $\alpha$ for which subadditivity and strong subadditivity are valid in each case.

We introduce the generalized equidistant Chebyshev polynomials T(k,h) of kind k of hyperkind h, where k,h are positive integers. They are obtained by a generalization of standard and monic Chebyshev polynomials of the first and second kinds. This generalization is fulfilled in two directions. The horizontal generalization is made by introducing hyperkind h and expanding it to infinity. The vertical generalization proposes expanding kind k to infinity with the help of the method of equidistant coefficients. Some connections of these polynomials with the Alexander knot and link polynomial invariants are investigated.

Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations.

Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are focused on a vertex amplitude of 3+1 D gravity, associated with a pentagram spin-network. Furthermore, all spin labels of the spin network are assumed to be equal $j=1/2$, which is crucial for the introduction of the \emph{intertwiner qubits}. A procedure of determining modulus squares of vertex amplitudes on universal quantum computers is proposed. Utility of the approach is tested with the use IBM's \emph{ibmqx4} 5-qubit quantum computer as well as on simulator of quantum computer provided by the same company. Finally, upper bound on the value of the vertex probability is determined employing the IBM simulator with 20-qubit quantum register.

We construct efficient deterministic dynamical decoupling schemes protecting continuous variable degrees of freedom. Our schemes target decoherence induced by quadratic system-bath interactions with analytic time-dependence. We show how to suppress such interactions to $N$-th order using only $N$~pulses. Furthermore, we show to homogenize a $2^m$-mode bosonic system using only $(N+1)^{2m+1}$ pulses, yielding - up to $N$-th order - an effective evolution described by non-interacting harmonic oscillators with identical frequencies. The decoupled and homogenized system provides natural decoherence-free subspaces for encoding quantum information. Our schemes only require pulses which are tensor products of single-mode passive Gaussian unitaries and SWAP gates between pairs of modes.