A promising route to novel quantum technologies are hybrid quantum systems, which combine the advantages of several individual quantum systems. We have realized a hybrid atomic-mechanical experiment consisting of a SiN membrane oscillator cryogenically precooled to 500 mK and optically coupled to a cloud of laser cooled Rb atoms. Here, we demonstrate active feedback cooling of the oscillator to a minimum mode occupation of n = 16 corresponding to a mode temperature of T = 200 {\mu}K. Furthermore, we characterize in detail the coupling of the membrane to the atoms by means of sympathetic cooling. By simultaneously applying both cooling methods we demonstrate the possibility of preparing the oscillator near the motional ground state while it is coupled to the atoms. Realistic modifications of our setup will enable the creation of a ground state hybrid quantum system, which opens the door for coherent quantum state transfer, teleportation and entanglement as well as quantum enhanced sensing applications.

A new non-perturbative approach is proposed to solve time-independent Schr\"{o}dinger equations in quantum mechanics and chromodynamics (QCD). It is based on the homotopy analysis method (HAM), which was developed by the author for highly nonlinear equations since 1992 and has been widely applied in many fields. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters. Besides, convergent series solution can be obtained even if the disturbance is far from the known status. A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be more than hundreds larger than the possible superior limit of the perturbative approach. This HAM-based approach could provide us rigorous theoretical results in quantum mechanics and chromodynamics (QCD), which can be directly compared with experimental data. Obviously, this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.

We propose a protocol for state transfer and entanglement generation between two distant spin qubits (sender and receiver) that have different energies. The two qubits are permanently coupled to a far off-resonant spin-chain, and the qubit of the sender is driven by an external field, which provides the energy required to bridge the energy gap between the sender and the receiver. State transfer and entanglement generation are achieved via virtual single-photon and multi-photon transitions to the eigenmodes of the channel.

We present a joint experimental and theoretical study of spin dynamics of a single $^{88}$Sr$^+$ ion colliding with an ultracold cloud of Rb atoms in various hyperfine states. While spin-exchange between the two species occurs after 9.1(6) Langevin collisions on average, spin-relaxation of the Sr$^+$ ion Zeeman qubit occurs after 48(7) Langevin collisions which is significantly slower than in previously studied systems due to a small second-order spin-orbit coupling. Furthermore, a reduction of the endothermic spin-exchange rate was observed as the magnetic field was increased. Interestingly, we found that, while the phases acquired when colliding on the spin singlet and triplet potentials vary largely between different partial waves, the singlet-triplet phase difference, which determines the spin-exchange cross-section, remains locked to a single value over a wide range of partial-waves which leads to quantum interference effects.

Measurement-device-independent quantum key distribution (MDI-QKD) is proved to be able to eliminate all potential detector side channel attacks. Combining with the reference frame independent (RFI) scheme, the complexity of practical system can be reduced because of the unnecessary alignment for reference frame. Here, based on polarization multiplexing, we propose a time-bin encoding structure, and experimentally demonstrate the RFI-MDI-QKD protocol. Thanks to this, two of the four Bell states can be distinguished, whereas only one is used to generate the secure key in previous RFI-MDI-QKD experiments. As far as we know, this is the first demonstration for RFI-MDI-QKD protocol with clock rate of 50 MHz and distance of more than hundred kilometers between legitimate parties Alice and Bob. In asymptotic case, we experimentally compare RFI-MDI-QKD protocol with the original MDI-QKD protocol at the transmission distance of 160 km, when the different misalignments of the reference frame are deployed. By considering observables and statistical fluctuations jointly, four-intensity decoy-state RFI-MDI-QKD protocol with biased bases is experimentally achieved at the transmission distance of 100km and 120km. The results show the robustness of our scheme, and the key rate of RFI-MDI-QKD can be improved obviously under a large misalignment of the reference frame.

We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality arguments, quantum maximum entropy principles and the generalised Gibbs ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches and present the most important results in a unified language.

Classical information encoded in composite quantum states can be completely hidden from the reduced subsystems and may be found only in the correlations. Can the same be true for quantum information? If quantum information is hidden from subsystems and spread over quantum correlation, we call it as masking of quantum information. We show that while this may still be true for some restricted sets of non-orthogonal quantum states, it is not possible for arbitrary quantum states. This result suggests that quantum qubit commitment -- a stronger version of the quantum bit commitment is not possible in general. Our findings may have potential applications in secret sharing and future quantum communication protocols.

We compare the roles of the Bures-Helstrom (BH) and Bogoliubov-Kubo-Mori (BKM) metrics in the subject of quantum information geometry. We note that there are two limits involved in state discrimination, which we call the "thermodynamic" limit (of $N$, the number of realizations going to infinity) and the infinitesimal limit (of the separation of states tending to zero). We show that these two limits do not commute in the quantum case. Taking the infinitesimal limit first leads to the BH metric and the corresponding Cram\'er-Rao bound, which is widely accepted in this subject. Taking limits in the opposite order leads to the BKM metric, which results in a weaker Cram\'er-Rao bound. This lack of commutation of limits is a purely quantum phenomenon arising from quantum entanglement. We can exploit this phenomenon to gain a quantum advantage in state discrimination and get around the limitation imposed by the Bures-Helstrom Cram\'er-Rao (BHCR) bound. We propose a technologically feasible experiment with cold atoms to demonstrate the quantum advantage in the simple case of two qubits.

We propose a hybrid quantum repeater based on ancillary coherent field states and material qubits coupled to optical cavities. For this purpose, resonant qubit-field interactions and postselective field measurements are determined which are capable of realizing all necessary two-qubit operations for the actuation of the quantum repeater. We explore both theoretical and experimental possibilities of generating near-maximally-entangled qubit pairs ($F>0.999$) over long distances. It is shown that our scheme displays moderately low repeater rates, between $5 \times 10^{-4}$ and $23$ pairs per second, over distances up to $900$ km, and it relies completely on current technology of cavity quantum electrodynamics.

We improve the number of T gates needed to perform an n-bit adder from 8n + O(1) to 4n + O(1). We do so via a "temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of 8n + O(1), a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.

We study numerically the onset of higher-level excitations and resonance frequency shifts in the generalized multilevel Rabi model with dispersive coupling under strong driving. The response to a weak probe is calculated using the Floquet method, which allows us to calculate the probe spectrum and extract the resonance frequency. We test our predictions using a superconducting circuit consisting of transmon coupled capacitively to a coplanar waveguide resonator. This system is monitored by a weak probe field, and at the same time driven at various powers by a stronger microwave tone. We show that the transition from the quantum to the classical regime is accompanied by a rapid increase of the transmon occupation and, consequently that the qubit approximation is valid only in the extreme quantum limit.

We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the ETH density matrix, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theories with the same value of central charge. We argue that the ETH density matrix is close in trace distance to the reduced density matrix of the (micro)canonical ensemble. We support the argument in higher dimensions by comparing the Von Neumann entropy of the ETH density matrix with the entropy of a black hole in holographic systems in the low temperature limit. Finally, we generalize our analysis to the coherent states with energy density that varies slowly in space, and show that locally such states are well described by the ETH density matrix.

Recently generalizations of the harmonic lattice model has been introduced as a discrete approximation of bosonic field theories with Lifshitz symmetry with a generic dynamical exponent z. In such models in (1+1) and (2+1)-dimensions, we study logarithmic negativity in the vacuum state and also finite temperature states. We investigate various features of logarithmic negativity such as the universal term, its z-dependence and also its temperature dependence in various configurations. We present both analytical and numerical evidences for linear z-dependence of logarithmic negativity in almost all range of parameters both in (1+1) and (2+1)-dimensions. We also investigate the validity of area law behavior of logarithmic negativity in these generalized models and find that this behavior is still correct for small enough dynamical exponents.

The problem of estimating the frequency of a two-level atom in a noisy environment is studied. Our interest is to minimise both the energetic cost of the protocol and the statistical uncertainty of the estimate. In particular, we prepare a probe in a "GHZ-diagonal" state by means of a sequence of qubit gates applied on an ensemble of $ n $ atoms in thermal equilibrium. Noise is introduced via a phenomenological time-nonlocal quantum master equation, which gives rise to a phase-covariant dissipative dynamics. After an interval of free evolution, the $ n $-atom probe is globally measured at an interrogation time chosen to minimise the error bars of the final estimate. We model explicitly a measurement scheme which becomes optimal in a suitable parameter range, and are thus able to calculate the total energetic expenditure of the protocol. Interestingly, we observe that scaling up our multipartite entangled probes offers no precision enhancement when the total available energy $ \mathcal{E} $ is limited. This is at stark contrast with standard frequency estimation, where larger probes---more sensitive but also more "expensive" to prepare---are always preferred. Replacing $ \mathcal{E} $ by the resource that places the most stringent limitation on each specific experimental setup, would thus help to formulate more realistic metrological prescriptions.

We apply the formalism of quantum estimation theory to obtain information about the value of the nonlinear optomechanical coupling strength. In particular, we discuss the minimum mean-square error estimator and a quantum Cram\'er--Rao-type inequality for the estimation of the coupling strength. Our estimation strategy reveals some cases where quantum statistical inference is inconclusive and merely result in the reinforcement of prior expectations. We show that these situations also involve the highest expected information losses. We demonstrate that interaction times in the order of one time period of mechanical oscillations are the most suitable for our estimation scenario, and compare situations involving different photon and phonon excitations.

The current shift in the quantum optics community towards large-size experiments -- with many modes and photons -- necessitates new classical simulation techniques that go beyond the usual phase space formulation of quantum mechanics. To address this pressing demand we formulate linear quantum optics in the language of tensor network states. As a toy model, we extensively analyze the quantum and classical correlations of time-bin interference in a single fiber loop. We then generalize our results to more complex time-bin quantum setups and identify different classes of architectures for high-complexity and low-overhead boson sampling experiments.

It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature's signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.

Imaginary time evolution is a powerful tool for studying quantum systems. While it is simple to simulate with a classical computer, the time and memory requirements scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a hybrid, variational algorithm for simulating imaginary time evolution on a quantum computer. We use this algorithm to find the ground state energy of many-particle systems; specifically molecular Hydrogen and Lithium Hydride. Our algorithm finds the ground state with high probability, outperforming the variational quantum eigensolver (VQE). Our approach requires no costly classical optimisation subroutine, unlike the VQE. Our method can also be applied to general optimisation problems, Gibbs state preparation, and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation methods, and can exploit shallow quantum circuits, it can be implemented with current quantum computers.

The `Bohrification" program in the foundations of quantum mechanics implements Bohr's doctrine of classical concepts through an interplay between commutative and non-commutative operator algebras. Following a brief conceptual and mathematical review of this program, we focus on one half of it, called "exact" Bohrification, where a (typically noncommutative) unital C*-algebra A is studied through its commutative unital C*-subalgebras, organized into a poset C(A). This poset turns out to be a rich invariant of A. To set the stage, we first give a general review of symmetries in elementary quantum mechanics (i.e., on Hilbert space) as well as in algebraic quantum theory, incorporating C(A) as a new kid in town. We then give a detailed proof of a deep result due to Hamhalter (2011), according to which C(A) determines A as a Jordan algebra (at least for a large class of C*-algebras). As a corollary, we prove a new Wigner-type theorem to the effect that order isomorphisms of C(B(H)) are (anti) unitarily implemented. We also show how C(A) is related to the orthomodular poset P(A) of projections in A. These results indicate that C(A) is a serious player in C*-algebras and quantum theory.

The Classical and Quantum mechanical correspondence for constant mass settings is used, along with some nonlocal point transformation, to find the position-dependent mass (PDM) Classical and Quantum Hamiltonians. The comparison between the resulting Quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction/identification of the PDM-momentum operator. The same recipe is followed to identify the form of the minimal coupling of electromagnetic interactions for the Classical and Quantum PDM-Hamiltonians. It turned out that the minimal coupling of the electromagnetic interactions inherits the usual forms but with PDM-momentum (operator) rather than the usual momentum (operator). Two commonly used vector potential are considered as Illustrative examples.