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.

We investigate the time-optimal solution of the selective control of two uncoupled spin 1/2 particles. Using the Pontryagin Maximum Principle, we derive the global time-optimal pulses for two spins with different offsets. We show that the Pontryagin Hamiltonian can be written as a one-dimensional effective Hamiltonian. The optimal fields can be expressed analytically in terms of elliptic integrals. The time-optimal control problem is solved for the selective inversion and excitation processes. A bifurcation in the structure of the control fields occurs for a specific offset threshold. In particular, we show that for small offsets, the optimal solution is the concatenation of regular and singular extremals.

Exploiting a well-established mapping from a d-dimensional quantum Hamiltonian to a d+1-dimensional classical Hamiltonian that is commonly used in software quantum Monte Carlo algorithms, we propose a scalable hardware emulator where quantum circuits are emulated with room temperature p-bits. The proposed emulator operates with probabilistic bits (p-bit) that fluctuate between logic 0 and 1, that are suitably interconnected with a crossbar of resistors or conventional CMOS devices. One particularly compact hardware implementation of a p-bit is based on the standard 1 transistor/1 Magnetic Tunnel Junction (1T/1MTJ) cell of the emerging Embedded Magnetoresistive RAM (eMRAM) technology, with a simple modification: The free layer of the MTJ uses a thermally unstable nanomagnet so that the resistance of the MTJ fluctuates in the presence of thermal noise. Using established device models for such p-bits and interconnects simulated in SPICE, we demonstrate a faithful mapping of the Transverse Ising Hamiltonian to its classical counterpart, by comparing exact calculations of averages and correlations. Even though we focus on the Transverse Ising Hamiltonian, many other "stoquastic" Hamiltonians - avoiding the sign problem - can be mapped to the hardware emulator. For such systems, large scale integration of the eMRAM technology can enable the intriguing possibility of emulating a very large number of q-bits by room temperature p-bits. The compact and low-level representation of the p-bit offers the possibility of greater efficiency and scalability compared to standard software implementations of quantum Monte Carlo methods.

Trapped-ion quantum information processors offer many advantages for achieving high-fidelity operations on a large number of qubits, but current experiments require bulky external equipment for classical and quantum control of many ions. We demonstrate the cryogenic operation of an ion-trap that incorporates monolithically-integrated high-voltage CMOS electronics ($\pm 8\mathrm{V}$ full swing) to generate surface-electrode control potentials without the need for external, analog voltage sources. A serial bus programs an array of 16 digital-to-analog converters (DACs) within a single chip that apply voltages to segmented electrodes on the chip to control ion motion. Additionally, we present the incorporation of an integrated circuit that uses an analog switch to reduce voltage noise on trap electrodes due to the integrated amplifiers by over $50\mathrm{dB}$. We verify the function of our integrated electronics by performing diagnostics with trapped ions and find noise and speed performance similar to those we observe using external control elements.

We experimentally demonstrate a variation on a Sisyphus cooling technique that was proposed for cooling antihydrogen. In our implementation, atoms are selectively excited to an electronic state whose energy is spatially modulated by an optical lattice, and the ensuing spontaneous decay completes one Sisyphus cooling cycle. We characterize the cooling efficiency of this technique on a continuous beam of Sr, and compare it with the case of a Zeeman slower. We demonstrate that this technique provides similar atom number for lower end temperatures, provides additional cooling per scattering event and is compatible with other laser cooling methods.

Role of entanglement is yet to be fully understood in quantum thermodynamics. We shed some light upon that direction by considering the role of entanglement for a single temperature quantum heat engine without feedback, introduced recently by J. Yi, P. Talkner and Y. W. Kim (Phys. Rev. E 96, 022108 (2017)). We take the working medium of the engine to be a 1-dim Heisenberg model of two spins. We calculate the efficiency of the engine undergoing a cyclic process at a single temperature and show that for a coupled working medium the efficiency can be higher than that of an uncoupled one.

The emergence of a special type of fluid-like behavior at large scales in one-dimensional (1d) quantum integrable systems, theoretically predicted in 2016, is established experimentally, by monitoring the time evolution of the in situ density profile of a single 1d cloud of $^{87}{\rm Rb}$ atoms trapped on an atom chip after a quench of the longitudinal trapping potential. The theory can be viewed as a dynamical extension of the thermodynamics of Yang and Yang, and applies to the whole range of repulsion strength and temperature of the gas. The measurements, performed on weakly interacting atomic clouds that lie at the crossover between the quasicondensate and the ideal Bose gas regimes, are in very good agreement with the 2016 theory. This contrasts with the previously existing 'conventional' hydrodynamic approach---that relies on the assumption of local thermal equilibrium---, which is unable to reproduce the experimental data.

We present the experimental generation of light with directly observable close-to ideal thermal statistical properties. The thermal light state is prepared using a spontaneous Raman emission in a warm atomic vapor. The photon number statistics is evaluated by both the measurement of second-order correlation function and by the detailed analysis of the corresponding photon number distribution, which certifies the quality of the Bose-Einstein statistics generated by natural physical mechanism. We further demonstrate the extension of the spectral bandwidth of the generated light to hundreds of MHz domain while keeping the ideal thermal statistics, which suggests a direct applicability of the presented source in a broad range of applications including optical metrology, tests of robustness of quantum communication protocols, or quantum thermodynamics.

We study a two-level impurity coupled locally to a quantum gas on an optical lattice. For state-dependent interactions between the impurity and the gas, we show that its evolution encodes information on the local excitation spectrum of gas at the coupling site. Based on this, we design a nondestructive method to probe the system's excitations in a broad range of energies by measuring the state of the probe using standard atom optics methods. We illustrate our findings with numerical simulations for quantum lattice systems, including realistic dephasing noise on the quantum probe, and discuss practical limits on the probe dephasing rate to fully resolve both regular and chaotic spectra.

Coherent superposition is a key feature of quantum mechanics that underlies the advantage of quantum technologies over their classical counterparts. Recently, coherence has been recast as a resource theory in an attempt to identify and quantify it in an operationally well-defined manner. Here we study how the coherence present in a state can be used to implement a quantum channel via incoherent operations and, in turn, to assess its degree of coherence. We introduce the robustness of coherence of a quantum channel---which reduces to the homonymous measure for states when computed on constant-output channels---and prove that: i) it quantifies the minimal rank of a maximally coherent state required to implement the channel; ii) its logarithm quantifies the amortized cost of implementing the channel provided some coherence is recovered at the output; iii) its logarithm also quantifies the zero-error asymptotic cost of implementation of many independent copies of a channel. We also consider the generalized problem of imperfect implementation with arbitrary resource states. Using the robustness of coherence, we find that in general a quantum channel can be implemented without employing a maximally coherent resource state. In fact, we prove that \textit{every} pure coherent state in dimension larger than $2$, however weakly so, turns out to be a valuable resource to implement \textit{some} coherent unitary channel. We illustrate our findings for the case of single-qubit unitary channels.