Entanglement measures find frequent application in the study of topologically ordered systems, where the presence of topological order is reflected in an additional contribution to the entanglement of the system. Obtaining this topological entropy from analytical calculations or numerical simulations is generally difficult due to the fact that it is an order-one correction to leading terms that scale with the size of the system. In order to distill the topological entropy, one resorts to extrapolation as a function of system size or to clever subtraction schemes that allow one to cancel out the leading terms. Both approaches have the disadvantage of requiring multiple (accurate) calculations of the entanglement of the system. Here we propose a modification of conventional entanglement calculations that allows one to obtain the topological entropy of a system from a single measurement of entanglement. In our approach, we replace the conventional trace over the degrees of freedom of a partition of the system with a projection onto a given state (which need not be known). We show that a proper choice of partition and projective measurement allows one to rid the entanglement measures of the typical boundary terms, thus exposing the topological contribution alone. We consider specifically the measures known as von Neumann entropy and entanglement negativity and we discuss their application to both models that exhibit quantum as well as classical topological order.

We consider a surface code suffering decoherence due to coupling to a bath of bosonic modes at finite temperature and study the time available before the unavoidable breakdown of error correction occurs as a function of coupling and bath parameters. We derive an exact expression for the error rate on each individual qubit of the code, taking spatial and temporal correlations between the errors into account. We investigate numerically how different kinds of spatial correlations between errors in the surface code affect its threshold error rate. This allows us to derive the maximal duration of each quantum error-correction period by studying when the single-qubit error rate reaches the corresponding threshold. At the time when error correction breaks down, the error rate in the code can be dominated by the direct coupling of each qubit to the bath, by mediated subluminal interactions, or by mediated superluminal interactions. For a two-dimensional Ohmic bath, the time available per quantum error-correction period vanishes in the thermodynamic limit of a large code size $L$ due to induced superluminal interactions, although it does so only like $1/\sqrt{lnL}$. For all other bath types considered, this time remains finite as $L\to \infty$.

Measurement-device-independent quantum key distribution (MDI-QKD), leaving the detection procedure to the third partner and thus being immune to all detector side-channel attacks, is very promising for the construction of high-security quantum information networks. We propose a scheme to implement MDI-QKD, but with continuous variables instead of discrete ones, i.e., with the source of Gaussian-modulated coherent states, based on the principle of continuous-variable entanglement swapping. This protocol not only can be implemented with current telecom components but also has high key rates compared to its discrete counterpart; thus it will be highly compatible with quantum networks.

We investigate the performance of quantum Fisher information (QFI) under the Unruh-Hawking effect, where one of the observers (e.g., Rob) is uniformly accelerated with respect to other partners. In the context of relativistic quantum information theory, we demonstrate that quantum Fisher information, as an important measure of the information content of quantum states, has a rich and subtle physical structure compared with entanglement or Bell nonlocality. In this work, we mainly focus on the parametrized (and arbitrary) pure two-qubit states, where the weight parameter $\theta$ and phase parameter $\varphi$ are naturally introduced. Intriguingly, we prove that QFI with respect to $\theta$ (${\mathcal{F}}_{\theta }$) remains unchanged for both scalar and Dirac fields. Meanwhile, we observe that QFI with respect to $\varphi$ (${\mathcal{F}}_{\varphi }$) decreases with the increase of acceleration $r$ but remains finite in the limit of infinite acceleration. More importantly, our results show that the symmetry of ${\mathcal{F}}_{\varphi }$ (with respect to $\theta =\pi /4$) has been broken by the influence of the Unruh effect for both cases.

The quantum Fourier transform (QFT) is one of the most widely used quantum algorithms, ranging from its primary role in finding the periodicity hidden in a quantum state to its use in constructing a quantum adder. Testing how the QFT performs under more realistic conditions, we find that the QFT, when used for period finding, shows extraordinary robustness with respect to static gate defects. For instance, replacing all rotation angles $\pi /2^j$ of the controlled rotation gates in the QFT circuit by $\pi (1+r)/2^j$, where $r$ is a uniformly distributed random variable taking values in the range $[-1,1]$, effectively resulting in a QFT with random gates, the QFT performs well above the expected random result. However, it is important to keep the $2^j$ terms in the denominators of the rotation angles, resulting in random, but hierarchically random, gates. Relaxing this hierarchical structure of the QFT circuit, we find that the performance of the QFT deteriorates significantly. This observation indicates that the hierarchical structure of the quantum circuit of the QFT is more important for the observed robustness in performance than the precise actions of individual gates. In addition to the specific example of the QFT circuit studied here, this observation also corroborates our experience with more general and more complex quantum circuits. Thus, backed by our detailed numerical and analytical results, we may condense the results of our research into the following general principle: The topology of a quantum circuit is more important than the precise actions of its gates.

Since observation on a quantum system may cause the system state collapse, it is usually hard to find a way to monitor a quantum process, which is a quantum system that continuously evolves. We propose a protocol that can debug a quantum process by monitoring, but not disturb the evolution of the system. This protocol consists of an error detector and a debugging strategy. The detector is a projection operator that is orthogonal to the anticipated system state at a sequence of time points, and the strategy is used to specify these time points. As an example, we show how to debug the computational process of quantum search using this protocol. By applying the Skolem-Mahler-Lech theorem in algebraic number theory, we find an algorithm to construct all of the debugging protocols for quantum processes of time-independent Hamiltonians.

Metastable pionic helium ($\pi {\mathrm{He}}^{+}$) is a three-body atom composed of a helium nucleus, an electron occupying the 1$s$ ground state, and a negatively charged pion ${\pi }^{-}$ in a Rydberg state with principal and orbital angular momentum quantum numbers of $n\sim \ell +1\sim 16$. We calculate the spin-independent energies of the $\pi {{}^3\mathrm{He}}^{+}$ and $\pi {{}^4\mathrm{He}}^{+}$ isotopes in the region $n=15$–19. These include relativistic and quantum electrodynamics corrections of orders $R_{\infty }{\alpha }^2$ and $R_{\infty }{\alpha }^3$ in atomic units, where $R_{\infty }$ and $\alpha$ denote the Rydberg and fine structure constants. The fine-structure splitting due to the coupling between the electron spin and the orbital angular momentum of the ${\pi }^{-}$ and the radiative and Auger decay rates of the states are also calculated. Some states $(n,\ell )=(16,15)$ and $(17,16)$ retain nanosecond-scale lifetimes against ${\pi }^{-}$ absorption into the helium nucleus. We propose the use of laser pulses to induce ${\pi }^{-}$ transitions from these metastable states to states with large ($\sim {10}^{11}$ s${}^{-1}$) Auger rates. The $\pi {\mathrm{He}}^{2+}$ ion that remains after Auger emission of the 1$s$ electron undergoes Stark mixing with the $s$, $p$, and $d$ states during collisions with the helium atoms in the experimental target. This leads to immediate nuclear absorption of the ${\pi }^{-}$. The resonance condition between the laser beam and the atom is thus revealed as a sharp spike in the rates of neutrons, protons, deuterons, and tritons that emerge. A resonance curve is obtained from which the $\pi {\mathrm{He}}^{+}$ transition frequency can in principle be determined with a fractional precision of ${10}^{-8}–{10}^{-6}$ provided the systematic uncertainties can be controlled. By comparing the measured $\pi {\mathrm{He}}^{+}$ frequencies with the calculated values, the ${\pi }^{-}$ mass may be determined with a similar precision. The $\pi {\mathrm{He}}^{+}$ will be synthesized by allowing a high-intensity ($>{10}^8$ s${}^{-1}$) beam of ${\pi }^{-}$produced by a cyclotron to come to rest in a helium target. The precise time structure of the ${\pi }^{-}$ beam is used to ensure a sufficient rate of coincidence between the resonant laser pulses and the $\pi {\mathrm{He}}^{+}$ atoms.

The present numerical, time-dependent density-functional study of a He atom interacting with strong, oscillating magnetic fields shows that this scenario is quite different from the case of a laser electric field–He atom interaction. Signatures of sluggish electron dynamics are found in this study, while through a mechanical analogy the flow of electron density under such conditions has been explained. These calculations take into account both exchange and correlation. Through several calculated dynamical quantities, we have shown that, in contrast to the case of the (one-electron) H atom studied earlier, the nonlinear dependence of interelectronic repulsions (a combination of Coulomb, exchange and correlation terms) on the magnetic field plays a significant role in this strong-field electron dynamics in the He atom, which cannot be explained by a perturbative approach.

We calculate the left-right asymmetry of the photoelectron momentum distributions generated in a hydrogen atom exposed to an intense few-cycle laser pulse as a function of both the carrier-envelope phase and the laser intensity. We present results of the numerical solution of the three-dimensional time-dependent Schrödinger equation, semiclassical simulations accounting for both laser and Coulomb fields, and the strong-field approximation. We predict pronounced oscillations of the asymmetry parameter as a function of the intensity for a particular range of the carrier-envelope phase. In order to reveal the mechanism underlying these oscillations, we investigate in detail the electron momentum distributions in the one-dimensional case. We show that quantum interference among a large set of both bound and continuum field-free states is responsible for the oscillatory behavior of the left-right asymmetry.

High-resolution spectroscopy has been carried out on the ${}^{199}\mathrm{Hg}$ ${}^1{S}_0-{}^3{P}_0$ spin and dipole forbidden transition, where the atoms are confined in a vertical one-dimensional optical lattice trap using light at the magic wavelength. We describe various characteristics of the resulting line spectra and assess the strength of the Lamb-Dicke confinement. Through a series of absolute frequency measurements of the ${}^{199}\mathrm{Hg}$ clock transition with respect to the LNE-SYRTE primary frequency standard, recorded over a 3-month period, we demonstrate a statistical fractional uncertainty of $2.5\times {10}^{-16}$. We include details relating to the generation of ultraviolet light at three wavelengths necessary for the experiment: 253.7 nm for cooling and detection, 265.6 nm for clock transition probing, and 362.570 nm for lattice trapping, along with further aspects related to the magic wavelength evaluation.

We study the rotational (yrast) spectra of dilute two-component atomic Bose gases in the low angular momentum regime, assuming equal interspecies and intraspecies interaction. Our analysis employs the composite fermion (CF) approach including a pseudospin degree of freedom. While the CF approach is not a priori expected to work well in this angular momentum regime, we show that composite fermion diagonalization gives remarkably accurate approximations to low-energy states in the spectra. For angular momenta $0<L<M$ (where $N$ and $M$ denote the numbers of particles of the two species, and $M\ge N$), we find that the CF states span the full Hilbert space and provide a convenient set of basis states which, by construction, are eigenstates of the symmetries of the Hamiltonian. Within this CF basis, we identify a subset of the basis states with the lowest $\Lambda$-level kinetic energy. Diagonalization within this significantly smaller subspace constitutes a major computational simplification and provides very close approximations to ground states and a number of low-lying states within each pseudospin and angular momentum channel.

For strongly interacting bosons in optical lattices, the standard description using the Bose-Hubbard model becomes questionable. The role of excited bands becomes important. In such a situation, we compare results of simulations using the multiband Bose-Hubbard model with a recent proposition based on a time-dependent variational approach. It is shown that the latter, in its original formulation, uses a too small variational space, often leading to spurious effects. Possible expansion of the variational approach is discussed.

Motivated by recent experiments reporting Bose-Einstein condensation of light coupled to incoherent dye molecules in a microcavity, we show that due to a dimensionality mismatch between the two-dimensional cavity photons and the three-dimensional arrangement of molecules, the system permits superfluid regimes where the relevant molecular degrees of freedom are collective Dicke states rather than individual excitations. For sufficiently high dye concentration, Dicke states become robust against local decoherence. In the limit when all dye molecules become excited, this system also shows Mott-Hubbard physics, despite the absence of an underlying lattice.

By examination of the exerted electromagnetic (EM) force on boundary of an object in a few examples, we look into the compatibility of the stress tensors corresponding to different formulas of the EM force density with special relativity. Ampere-Lorentz's formula of the EM force density is physically justifiable in that the electric field and the magnetic flux density act on the densities of the total charges and the total currents, unlike Minkowski's formula which completely excludes the densities of the bounded charges and the bounded currents inside homogeneous media. Abraham's formula is fanciful and devoid of physical meaning. Einstein-Laub's formula seems to include the densities of the total charges and the total currents at first sight, but grouping the bounded charges and the bounded currents into pointlike dipoles erroneously results in the hidden momentum being omitted, hence the error in [Phys. Rev. Lett. 108, 193901 (2012)]. Naturally, the Ampere-Lorentz stress tensor accords with special relativity. The Minkowski sress tensor is also consistent with special relativity. It is worth noting that the mathematical expression of the Minkowski stress tensor can be quite different from the well-known form of this stress tensor in the literature. We show that the Einstein-Laub stress tensor is incompatible with special relativity, and therefore we rebut the Einstein-Laub force density. Since the Abraham momentum density of the EM fields is inherently corresponding to the Einstein-Laub force density [Phys. Rev. Lett. 111, 043602 (2013)], our rebuttal may also shed light on the controversy over the momentum of light.

Recently, the initial longitudinal momentum at the tunnel exit has attracted considerable interest in the strong-field community. We have measured the photoelectron momentum distribution of Kr atoms at an intensity of $1.2\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ and the ellipticity-resolved angular distribution of Ne atoms at an intensity of $3.6\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (790-nm, 25-fs laser pulses). To shed more light on the debated issue, we have introduced an initial longitudinal momentum spread at the tunnel exit to compare with the measured photoelectron angular distributions using a dedicated semiclassical model. The simulation indicates that the value of this longitudinal momentum spread is in the range 0.0–0.1 a.u., which is less than the initial transverse momentum spread predicted by the tunneling theory.

How much of the uncertainty in predicting measurement outcomes for noncommuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two noncommuting observables into a classical component and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits “purity-based” lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a nontrivial interplay between quantum and classical randomness in any finite-dimensional state.

Multipartite quantum correlations, in spite of years of intensive research, still leave many questions unanswered. While bipartite entanglement is relatively well understood for Gaussian states, the complexity of mere qualitative characterization grows rapidly with increasing number of parties. Here, we present two schemes for transformations of multipartite permutation invariant Gaussian states by Gaussian local operations and classical communication. To this end, we use a scheme for possible experimental realization, making use of the fact that in this picture the whole $N$-partite state can be described by specifying the states of two separable modes. Numerically, we study entanglement transformations of tripartite states. Finally, we look at the effect our protocols have on fidelity of assisted quantum teleportation and find that, while adding correlated noise does not affect the fidelity at all, there is strong evidence that partial nondemolition measurement leads to a drop in teleportation fidelity.

We identify a pair of near-degenerate states of opposite parity in atomic Xe, the $5p^{5}10s{}^2{[3/2]}_2^o$ at $E=94759.927$ cm${}^{-1}$ and $5p^{5}6f{}^2{[5/2]}_2$ at $E=94759.935$ cm${}^{-1}$, for which parity- and time-odd effects are expected to be enhanced by the small energy separation. We present theoretical calculations which indicate narrow widths for both states and we report a calculated value for the weak matrix element, arising from configuration mixing, of $\left|W\right|=2.1$ Hz for ${}^{132}\mathrm{Xe}$. In addition, we measured the Stark effect of the $5p^{5}6f$ ${}^2{[5/2]}_2$ and $5p^{5}6f{}^2{[3/2]}_2$ ($E=94737.121{\mathrm{cm}}^{-1}$) states. The Stark shift of the $6f$ states is observed to be negative, revealing the presence of nearby $6g$ states at higher energies, which have not been observed before. The Stark-shift measurements imply an upper limit on the weak matrix element of $\left|W\right|<5$ Hz for the near-degenerate states ($10s{}^2{[3/2]}_2^o$ and $6f{}^2{[5/2]}_2$), which is in agreement with the presented calculations.