We apply the Leggett-Garg inequalities (LGIs) to the cases of classical and quantum unstable systems. For classical systems the two assumptions of macroscopic realism and noninvasive measurements imply that the three-measurement string $K_3$ is identically equal to 1. Also, for quantum-mechanical systems, for which the two assumptions are, in general, not valid, we find that $K_3=1$ for purely exponential decays ($K_3\le 1$ is the general LGI). On the other hand, the necessary deviations from the exponential decay law at short and long times predicted by quantum mechanics lead to values of $K_3\ne 1$. Moreover, a strict violation $K_3>1$ typically occurs at short times. Thus, we conclude that experiments in which such deviations from the exponential decay law have been observed should also have in their data violations of the LGIs.

What compels quantum measurement to violate the Bell inequalities? Suppose that regardless of measurement, one can assign to a spin-$\frac{1}{2}$ particle (qubit) a definite value of spin, called $c$-valued spin variable, but it may take any continuous real number. Suppose further that measurement maps the $c$-valued spin variable from the continuous range of possible values onto the binary standard quantum spin values $\pm 1$ while preserving the bipartite correlation. Here, we show that such $c$-valued spin variables can indeed be constructed. In this model, one may therefore argue that it is the requirement of conservation of correlation which compels quantum measurement to violate the Bell inequalities when the prepared state is entangled. We then discuss a statistical game which captures the model of measurement, wherein two parties are asked to independently map a specific ensemble of pairs of real numbers onto pairs of binary numbers $\pm 1$, under the requirement that the correlation is preserved. The conservation of correlation forces the game to respect the Bell theorem, which implies that there is a class of games no classical (i.e., local and deterministic) strategy can ever win. On the other hand, a quantum strategy with an access to an ensemble of entangled spin-$\frac{1}{2}$ particles and circuits for local quantum spin measurement can be used to win the game.

In the conventional axiomatic formalism of quantum mechanics, a physical measurement of a quantum observable is mathematically represented by the spectral decomposition of the Hermitian operator associated with the observable. This constitutes the scenario of a Lüders (projective) measurement, which includes a von Neumann (rank-one projective) measurement as a special and prominent instance. In this context, the measurement is called sharp in the sense that each measurement operator is an orthogonal projection (eigenprojection of the observable). In the modern operational formalism, a measurement is represented by a positive-operator valued measure (POVM), which consists of a family of non-negative definite operators (measurement operators, effects) summing to the identity. In this scenario, a measurement is called unsharp (fuzzy) if some measurement operators are not orthogonal projections. A natural question arises as to how to quantify unsharpness of a measurement. In this work, we address this issue in terms of uncertainty. For this purpose, we study a family of observables associated with a measurement and their uncertainty. By exploiting the difference between the (total) measurement uncertainty and the observable uncertainty, we are led to some information-theoretic quantifiers of unsharpness. We reveal their basic properties and illustrate them through some important measurements. In particular, we characterize Lüders measurements and equiangular POVMs as extreme measurements in terms of unsharpness.

Einstein-Podolsky-Rosen (EPR) steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect the steerability both analytically and numerically. Supervised machine learning such as support vector machines and neural networks have also been trained to detect the EPR steerability. To implement supervised machine learning, one needs a lot of labeled quantum states by using the semidefinite programming, which is very time consuming. We present a semisupervised support vector machine method which only uses a small portion of labeled quantum states in detecting quantum steering. We show that our approach can significantly improve the accuracies by detailed examples.

Suppose we want to benchmark a quantum device held by a remote party, e.g., by testing its ability to carry out challenging quantum measurements outside of a free set of measurements $\mathcal{M}$. A very simple way to do so is to set up a binary-state discrimination task that cannot be solved efficiently by means of free measurements. If we can find pairs of orthogonal states that become arbitrarily indistinguishable under measurements in $\mathcal{M}$, in the sense that the error probability in discrimination approaches that of a random guess, we say that there are data hiding against $\mathcal{M}$. Here we investigate data hiding in the context of continuous-variable quantum systems. First, we look at the case where $\mathcal{M}$ denotes the set of measurements implementable with local operations and classical communication. While previous studies have placed upper bounds on the maximum efficiency of data hiding in terms of the local dimension and are thus not applicable to continuous-variable systems, we establish more general bounds that rely solely on the local mean photon number of the states employed. Along the way, we perform a rigorous quantitative analysis of the error introduced by the nonideal Braunstein-Kimble quantum teleportation protocol, determining how much squeezing and local detection efficiency are needed in order to teleport an arbitrary multimode local state of known mean energy with a prescribed accuracy. Finally, following a seminal proposal by Sabapathy and Winter, we look at data hiding against Gaussian operations assisted by the feedforward of measurement outcomes, providing an example of a relatively simple scheme that works with a single mode only.

An unknown quantum state can be teleported by using quantum measurements and the maximally entangled Einstein-Podolsky-Rosen (EPR) pair. It is well known that the usual nonclassical teleportation that cannot be simulated by the seminal classical measure-prepare strategy can be demonstrated with all entangled states. Herein, we propose a benchmark which reveals that not all such nonclassical teleportations are truly quantum mechanical. Rather, there exists a more robust classical-teleportation model, which includes the measure-prepare mimicry as a special case, that can describe certain nonclassical teleportations. Invalidating such a general classical model indicates genuine quantum teleportation wherein both the pair state and the measurement are truly quantum mechanical. We prove that EPR steering empowers genuine quantum teleportations, rather than entanglement. The benchmark can be readily used in practical experiments for ensuring that genuine quantum teleportation is implemented. The results presented herein provide strict criteria for implementing quantum-information processing where genuine quantum teleportation is indispensable.

Continuous-variable multipartite entanglement is a key resource for quantum technologies. This paper considers the multipartite entanglement generated in separated spatial modes of the same light beam by three different parametric sources: a standard ${\chi }^{\left(2\right)}$ medium pumped by two pumps, a single-pump nonlinear photonic crystal, and a doubly pumped nonlinear photonic crystal. These sources have in common the coexistence of several concurrent nonlinear processes in the same medium, which allows the generation of nonstandard three- and four-mode couplings. We test the genuine nature of the multipartite entangled states thereby generated in a common framework, using criteria based both on proper bounds for the variances of nonlocal observables and on the positive partial transpose criterion. The relative simplicity of these states allows a (hopefully) useful comparison of the different inseparability tests.

In this article, we introduce a framework for Hamiltonian tomography of multiqubit systems with random noise. We adopt the quantum quench protocol to reconstruct a many-body Hamiltonian by local measurements that are distorted by random unitary operators and time uncertainty. In particular, we consider a transverse-field Ising Hamiltonian describing interactions of two spins $1/2$ and three-qubit Hamiltonians of a heteronuclear system within the radio-frequency field. For a sample of random Hamiltonians, we report the fidelity of reconstruction versus the amount of noise quantified by two parameters. Furthermore, we discuss the correlation between the accuracy of Hamiltonian tomography and the number of pairs of quantum states involved in the framework. The results provide valuable insight into the robustness of the protocol against random noise.

We analyze the estimation of a time-dependent perturbation acting on a continuously monitored quantum system. We describe the temporal fluctuations of the perturbation by a hidden Markov model, and we combine quantum measurement theory and classical filter theory into a time-evolving hybrid quantum and classical trajectory. The forward-backward analysis that permits smoothed estimates of classical hidden Markov models has a counterpart in the theory of retrodiction and past quantum states. As a specific example, we apply our hybrid trajectory and past-quantum-state theory to the sensing of a fluctuating magnetic field by microwave interrogation of a single quantum spin.

Braess, while working on traffic modeling, noticed that traffic flow in a network can be worsened by adding extra edges to an existing network. This seemingly counterintuitive phenomenon is known as the Braess paradox. We consider a quantum network, where edges represent shared entangled states between spatially separated parties (nodes). The goal is to entangle two previously uncorrelated nodes using entanglement swappings. The amount of entanglement between the distant nodes is quantified by the average concurrence of the states established, as a result of the entanglement swappings. We then introduce an additional edge of maximally entangled Bell states in the network. We show that the introduction of the additional maximally entangled states to this network leads to lower concurrence between the two previously uncorrelated nodes. Thus we demonstrate the occurrence of a phenomenon in a quantum network that is analogous to the Braess paradox in traffic networks.

Several emerging quantum technologies, including quantum networks, and modular and fusion-based quantum computing, rely crucially on the ability to perform photonic Bell state measurements. Therefore, photon losses and the 50% success probablity upper bound of Bell state measurements pose a critical limitation to photonic quantum technologies. Here, we develop protocols that overcome these two key challenges through logical encoding of photonic qubits. Our approach uses a tree graph state logical encoding, which can be produced deterministically with a few quantum emitters, and achieves near-deterministic logical photonic Bell state measurements while also protecting against errors including photon losses, with a record loss-tolerance threshold.

Conversion of water nuclear-spin isomers by the quantum relaxation phenomenon is revisited and the high stability of water isomers is predicted. As an example, the isomer lifetime is found equal to 18 hours at room temperature and the water vapor pressure equal to $\simeq 5$ torr. Our calculations correct the prediction of rather short lifetime of water spin isomers, $\approx 1$ s, made by Cacciani et al. [Phys. Rev. A 85, 012521 (2012)].

Based on a theoretical model where the nuclear spins remain unchanged during a collision, we provide an analytical and general expression for the nuclear spin state-to-state distribution of an ultracold diatom-diatom chemical reaction in a magnetic field, for given rotational transitions of the molecules. It simply requires knowledge of the field-dependent eigenfunctions of the molecular reactants and products of the chemical reaction. The final state-to-state distribution drastically changes with the magnetic field. When the distribution is summed over all the final products, a simplified expression is found where only the knowledge of the eigenfunctions of the molecular reactants is required. The present theoretical formalism has been successfully used to explain the magnetic field behavior of the product-state distribution in chemical reactions of ultracold KRb molecules [Hu et al., Nat. Chem. 13, 435 (2021)].

Bicircular twisted Laguerre-Gaussian beams possess a definite torus knot angular momentum (TKAM) $\widehat{J_{\gamma }}=\widehat{L}+\gamma \widehat{S}$ as an alternative a form of angular momentum. TKAM is conserved in nonlinear atomic processes such as high harmonic generation and can be classified by a time delay parameter $\tau$ and a coordination parameter $\gamma$. These parameters are defined by the respective projected orbital angular momentum ${\ell }_i$ and the energy $\hslash p_i{\omega }_0$ of the two superimposed Laguerre-Gaussian beams. We derive a consistent geometric method to determine $\tau$ and $\gamma$ from the driving beam as well as from the high harmonic radiation. The method relates both invariance parameters ($\tau$ and $\gamma$) to a torus knot which can be constructed from the emitted high harmonic radiation. These knots are constructed from the spatiotemporal evolution of the electric field of the respective high harmonic radiation or the driving beam. We demonstrate the classification of the invariance parameters for a planar atomic gas target irradiated by bicircular Laguerre-Gaussian beams explicitly. In addition, we demonstrate that the respective torus knots determined by $\tau$ and $\gamma$ can be mapped onto each other within minor modifications. The numerical calculations are done within the strong-field approximation and the associated quantum orbit approach. Therefore, we also briefly review high harmonic generation by bicircular twisted light beams. This introduced geometric method is a different approach to interpret the invariance parameters $\tau$ and $\gamma$, as well as their underlying relations, compared to a purely formal derivation. The investigations presented in this work are in good agreement with previous findings and provide insight into the dynamical symmetry of TKAM in the context of high harmonic generation induced by bicircular twisted Laguerre-Gaussian beams.

We analyzed bichromatic and polychromatic stimulated forces for laser cooling and trapping of Yb atoms using only the narrow ${}^1{S}_0\to {}^3{P}_1$ transition. Our model is based on numerical solutions of optical Bloch equations for two-level atoms driven by multiple time-dependent fields combined with Monte Carlo simulations, which account for realistic experimental conditions such as atomic beam divergence, geometry, and Gaussian laser modes. Using 1 W of laser power, we predict a loading rate of $\approx {10}^8$ atoms/s into a 556-nm magneto-optical trap (MOT) with a slowing force of $\approx 60F_{\text{rad}}$. We show that a square-wave modulation can produce similar stimulated forces with almost twice the velocity range and improve the MOT loading rate of Yb atoms by up to 70%.

We investigate the combined effects of spin-orbit coupling and rotation on the topological vortical phase transition in $F=2$ Bose-Einstein condensates. We find that the spin-orbit coupling can precisely manipulate the canonical atom current which is generated in the opposite direction of the gauge atom current and causes both a continuous and a discontinuous canonical angular momentum. We apply the canonical angular momentum and magnetization to reveal the emergence of novel topological excitations, such as the Anderson-Toulouse vortex as well as the vortex–dipole lattice. Especially, strong spin-orbit coupling can induce two perpendicular vortex chains. We also find that both the first-order and the second-order phase transition can be characterized by the canonical angular momentum and the magnetization. Differently from the spin singlet-pairing interaction, the spin-exchange interaction can adjust the canonical angular momentum and control the phase transition well. The topological vortical phase transition in an $F=2$ cold-atom system is compatible with the current experiment and can be detected by the spin polarization procedure.

Two-component coupled Bose gas in a 1D optical lattice is examined. In addition to the postulated Mott insulator and superfluid phases, multiple bosonic components manifest spin degrees of freedom. Coupling of the components in the Bose gas leads to substantial changes in the previously observed spin phases, giving rise to a new effective spin Hamiltonian and unraveling remarkable spin correlations. The system in the absence of coupling exhibits ferromagnetic and nonferromagnetic spin phases for on-site intracomponent interaction stronger than intercomponent interaction. Upon introduction of coupling, the phase transition switches from first to second order. For comparable on-site inter- and intracomponent interactions, with coupling, instead of one, two spin phases emerge with a second-order phase transition. Exact diagonalization and variational Monte Carlo with stochastic minimization on the entangled-plaquette state bestow a unique and enhanced perspective on the system beyond the scope of a mean-field treatment.

Discrete time crystals (DTCs) refer to a novel many-body steady state that spontaneously breaks the discrete time-translational symmetry in a periodically driven quantum system. Here, we study DTCs in a Bose-Einstein condensate bouncing resonantly on an oscillating mirror, using a two-mode model derived from a standard quantum field theory. We investigate the validity of this model and apply it to study the long-time behavior of our system. A wide variety of initial states based on two Wannier modes are considered. We find that in previous studies the investigated phenomena in the evolution time window ($⪅2000$ driving periods) are actually “short-time” transient behavior though DTC formation signaled by the subharmonic responses is still shown if the interboson interaction is strong enough. After a much longer (about 20 times) evolution time, initial states with no “long-range” correlations relax to a steady state, where time-symmetry breaking can be unambiguously defined. Quantum revivals also eventually occur. This long-time behavior can be understood via the many-body Floquet quasieigenenergy spectrum of the two-mode model. A symmetry-breaking edge for DTC formation appears in the spectrum for strong enough interaction, where all quasieigenstates below the edge are symmetry breaking while those above the edge are symmetric. The late-time steady state's time-translational symmetry depends solely on whether the initial energy is above or below the symmetry-breaking edge. A phase diagram showing regions of symmetry-broken and symmetric phases for differing initial energies and interaction strengths is presented. We find that, according to this two-mode model, the discrete time crystal survives for times out to at least $250000$ driving periods.

Electromagnetic nonlinearity, combined with symmetry breaking, can be exploited to obtain highly nonreciprocal light transmission. In previous studies on nonreciprocity, it has been generally assumed that the material's nonlinearity responds instantaneously to the applied field. Here, we consider a noninstantaneous nonlinear response and study its influence on nonreciprocity. We show that the inclusion of such delayed effects can lead to highly nontrivial nonreciprocal dynamics, which are forbidden in the instantaneous regime. Particularly, when the characteristic delay time of the nonlinearity approaches the input pulse's duration, high-contrast nonreciprocal compression and reshaping of the pulse can be achieved. In the high-power regime, we show that it is possible to generate a nonreciprocal response in a single resonator, which, for instance, is chaotic when sourced along one direction and periodically modulated when sourced from the opposing direction. Indeed, by tuning the nonlinearity's memory time we can flexibly control the system's response, enabling a wide range of nonreciprocal functionalities in compact, passive, and bias-free devices.