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17 pages, 6 figures, In this work we propose and theoretically characterize optical isolators consisting of an all-dielectric and non-magnetic resonator featuring an intensity-dependent refractive index and a strong coherent field propagating in a single direction. Such devices can be straightforwardly realized in state-of-the-art integrated photonics platforms. The mechanism underlying optical isolation is based on the breaking of optical reciprocity induced by the asymmetric action of four-wave mixing processes coupling a strong propagating pump field with co-propagating signal/idler modes but not with reverse-propagating ones. Taking advantage of a close analogy with fluids of light, our proposed isolation mechanism is physically understood in terms of the Bogoliubov dispersion of collective excitations on top of the strong pump beam. A few most relevant set-ups realizing our proposal are specifically investigated, such as a coherently illuminated passive ring resonator and unidirectionally lasing ring or Taiji resonators., We acknowledge financial support from the European Union FET-Open grant “MIR-BOSE” (No. 737017), from the H2020-FETFLAG-2018-2020 project “PhoQuS” (No. 820392), from the Provincia Autonoma di Trento, and from the Q@TN initiative. A.M. acknowledges financial support from the Comunidad de Madrid via Proyecto Sinérgico CAM 2020 Y2020/TCS-6545 (NanoQuCo-CM).

In this work, we present a lower bound on the quantum Fisher information (QFI) which is efficiently computable on near-term quantum devices. This bound itself is of interest, as we show that it satisfies the canonical criteria of a QFI measure. Specifically, it is essentially a QFI measure for subnormalized states, and hence it generalizes the standard QFI in this sense. Our bound employs the generalized fidelity applied to a truncated state, which is constructed via the $m$ largest eigenvalues and their corresponding eigenvectors of the probe quantum state $\rho_{\theta}$. Focusing on unitary families of exact states, we analyze the properties of our proposed lower bound, and demonstrate its utility for efficiently estimating the QFI., Comment: v4: close to published version

We study several variations of the question of minimum-error discrimination of thermal states. Besides of providing the optimal values for the probability of error, we also characterize the optimal measurements. For the case of a fixed Hamiltonian, we show that for a general discrimination problem the optimal measurement is the measurement in the energy basis of the Hamiltonian. We identify a critical temperature, determining whether the given temperature is best distinguishable from thermal state of very high or very low temperatures. Further, we investigate the decision problem of whether the thermal state is above or below some threshold value of the temperature. Also, in this case, the minimum-error measurement is the measurement in the energy basis. This is no longer the case once the thermal states to be discriminated have different Hamiltonians. We analyze a specific situation when the temperature is fixed but the Hamiltonians are different. For the considered case, we show the optimal measurement is independent of the fixed temperature and also of the strength of the interaction.

We consider an approach in which the usual wave function in the quadrature representation of mode j of the electromagnetic field is further quantized to produce a field operator. Since the electromagnetic field is already second quantized, this corresponds to an additional or third quantization. The third-quantization approach can be used to perform certain quantum optics calculations in the Heisenberg picture that could only be performed in the Schrodinger picture when using the conventional second-quantized theory. This approach also allows an interesting generalization of quantum optics and quantum electrodynamics that is analogous to symmetry breaking in elementary particle theory. The predictions of the generalized theory could be tested using a proposed photon scattering experiment., Comment: 14 pages, 8 figures

Despite significant progress in theoretical and laboratory quantum control, engineering quantum systems remains principally challenging due to manifestation of noise and uncertainties associated with the field and Hamiltonian parameters. In this paper, we extend and generalize the asymptotic quantum control robustness analysis method -- which provides more accurate estimates of quantum control objective moments than standard leading order techniques -- to diverse quantum observables, gates and moments thereof, and also introduce the Pontryagin Maximum Principle for quantum robust control. In addition, we present a Pareto optimization framework for achieving robust control via evolutionary open loop (model-based) and closed loop (model-free) approaches with the mechanisms of robustness and convergence described using asymptotic quantum control robustness analysis. In the open loop approach, a multiobjective genetic algorithm is used to obtain Pareto solutions in terms of the expectation and variance of the transition probability under Hamiltonian parameter uncertainty. The set of numerically determined solutions can then be used as a starting population for model-free learning control in a feedback loop. The closed loop approach utilizes real-coded genetic algorithm with adaptive exploration and exploitation operators in order to preserve solution diversity and dynamically optimize the transition probability in the presence of field noise. Together, these methods provide a foundation for high fidelity adaptive feedback control of quantum systems wherein open loop control predictions are iteratively improved based on data from closed loop experiments.

Quantum gates are typically vulnerable to imperfections in the classical control fields applied to physical qubits to drive the gates. One approach to reduce this source of error is to break the gate into parts, known as composite pulses (CPs), that typically leverage the constancy of the error over time to mitigate its impact on gate fidelity. Here we extend this technique to suppress secular drifts in Rabi frequency by regarding them as sums of power-law drifts whose first-order effects on over- or under-rotation of the state vector add linearly. Power-law drifts have the form $t^p$ where $t$ is time and the constant $p$ is its power. We show that composite pulses that suppress all power-law drifts with $p \leq n$ are also high-pass filters of filter order $n+1$ arXiv:1410.1624. We present sequences that satisfy our proposed power-law amplitude criteria, $\text{PLA}(n)$, obtained with this technique, and compare their simulated performance under time-dependent amplitude errors to some traditional composite pulse sequences. We find that there is a range of noise frequencies for which the $\text{PLA}(n)$ sequences provide more error suppression than the traditional sequences, but in the low frequency limit, non-linear effects become more important for gate fidelity than frequency roll-off. As a result, the previously known $F_1$ sequence, which is one of the two solutions to the $\text{PLA}(1)$ criteria and furnishes suppression of both linear secular drift and the first order nonlinear effects, is a sharper noise filter than any of the other $\text{PLA}(n)$ sequences in the low frequency limit.

The Bhatia-Davis theorem provides a useful upper bound for the variance in mathematics, and in quantum mechanics, the variance of a Hamiltonian is naturally connected to the quantum speed limit due to the Mandelstam-Tamm bound. Inspired by this connection, we construct a formula, referred to as the Bhatia-Davis formula, for the characterization of the quantum speed limit in the Bloch representation. We first prove that the Bhatia-Davis formula is an upper bound for a recently proposed operational definition of the quantum speed limit, which means it can be used to reveal the closeness between the timescale of certain chosen states to the systematic minimum timescale. In the case of the largest target angle, the Bhatia-Davis formula is proved to be a valid lower bound for the evolution time to reach the target when the energy structure is symmetric. Regarding few-level systems, it is also proved to be a valid lower bound for any state in two-level systems with any target, and for most mixed states with large target angles in equally spaced three-level systems., 9+8 pages, 8 figures

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 semi-supervised 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.

Quantum teleportation is a method for utilizing quantum measurements and the maximally entangled Einstein-Podolsky-Rosen (EPR) pair to transmit an unknown quantum state. It is well known that all entangled states demonstrate so-called "nonclassical teleportation" that cannot be simulated by the seminal classical measure-prepare strategy. Herein, we propose a new benchmark which reveals that not all 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 new 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.

In this article, we introduce a framework for Hamiltonian tomography of multi-qubit 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 Hamiltonians 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.

Continuous variables multipartite entanglement is a key resource for quantum technologies. This works considers the multipartite entanglement generated in separated spatial modes of the same light beam by three different parametric sources: a standard $\chi^{(2)}$ 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 non-standard 3 and 4-mode couplings. We test the genuine nature of the multipartite entangled states thereby generated in a common framework, using both criteria based 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., Comment: Accepted for publication in Phys. Rev. A

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 (BEC) 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 ($\lessapprox$2000 driving periods) are actually "short-time" transient behavior though DTC formation signaled by the sub-harmonic responses is still shown if the inter-boson 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 quasi-eigenenergy spectrum of the two-mode model. A symmetry-breaking edge for DTC formation appears in the spectrum for strong enough interaction, where all quasi-eigenstates 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 250,000 driving periods.

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 one 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, one says that there is 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 $\mathcal{M}=\mathrm{LOCC}$, 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 non-ideal Braunstein-Kimble quantum teleportation protocol, determining how much squeezing and local detection efficiency is needed in order to teleport an arbitrary multi-mode 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 feed-forward of measurement outcomes, providing the first example of a relatively simple scheme that works with a single mode only., 25 pages; v2 has a generally improved presentation and is close to the published version

Textbook quantum mechanics treats time as a classical parameter, and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as $\Delta {t}\Delta E\gtrsim\hbar$, the term $\Delta t$ must be interpreted as a time interval, and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the "time at which an event happens" by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework, we derive here two {\em true} uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty., Comment: Comments are welcome

We investigate the groups generated by the sets of $CP$, $CNOT$ and $SWAP^\alpha$ (power-of-SWAP) quantum gate operations acting on $n$ qubits. Isomorphisms to standard groups are found, and using techniques from representation theory, we are able to determine the invariant subspaces of the $n-$qubit Hilbert space under the action of each group. For the $CP$ operation, we find isomorphism to the direct product of $n(n-1)/2$ cyclic groups of order $2$, and determine $2^n$ $1$-dimensional invariant subspaces corresponding to the computational state-vectors. For the $CNOT$ operation, we find isomorphism to the general linear group of an $n$-dimensional space over a field of $2$ elements, $GL(n,2)$, and determine two $1$-dimensional invariant subspaces and one $(2^n-2)$-dimensional invariant subspace. For the $SWAP^\alpha$ operation we determine a complex structure of invariant subspaces with varying dimensions and occurrences and present a recursive procedure to construct them. As an example of an application for our work, we suggest that these invariant subspaces can be used to construct simple formal verification procedures to assess the operation of quantum computers of arbitrary size.

We examine how non-destructive measurements generate spin squeezing in an atomic Bose-Einstein condensate confined in a double-well trap. The condensate in each well is monitored using coherent light beams in a Mach-Zehnder configuration that interacts with the atoms through a quantum nondemolition Hamiltonian. We solve the dynamics of the light-atom system using an exact wavefunction approach, in the presence of dephasing noise, which allows us to examine arbitrary interaction times and a general initial state. We find that monitoring the condensate at zero detection current and with identical coherent light beams minimizes the backaction of the measurement on the atoms. In the weak atom-light interaction regime, we find the mean spin direction is relatively unaffected, while the variance of the spins is squeezed along the axis coupled to the light. Additionally, squeezing persists in the presence of tunneling and dephasing noise.

We investigate the use of programmable optical lattices for quantum simulation of Hubbard models, determining analytic expressions for the hopping and Hubbard U, finding that they are suitable for emulating strongly correlated systems with arbitrary structures, including those with a multiple site basis and impurities. Programmable potentials are highly flexible, with the ability to control the depth and shape of individual sites in the optical lattice dynamically. Quantum simulators of Hubbard models with (1) an arbitrary basis are required to represent many real materials of contemporary interest, (2) broken translational symmetry are needed to study impurity physics, and (3) dynamical lattices are needed to investigate strong correlations out of equilibrium. We derive analytic expressions for Hubbard Hamiltonians in programmable potential systems. We find experimental parameters for quantum simulation of Hubbard models with an arbitrary basis, concluding that programmable optical lattices are suitable for this purpose. Finally, we demonstrate how programmable optical lattices can be used for quantum simulation of ionic Hubbard models, and thus the onset of charge transfer insulating states, in parameter regimes that are currently inaccessible to experiment.

Quantum emitters interacting with photonic band-gap materials lead to the appearance of qubit-photon bound states that mediate decoherence-free, tunable emitter-emitter interactions. Recently, it has been shown that when these band-gaps have a topological origin, like in the photonic SSH model, these qubit-photon bound states feature chiral shapes and certain robustness to disorder. In this work, we consider a more general situation where the emitters interact with an extended SSH photonic model with longer range hoppings that displays a richer phase diagram than its nearest-neighbour counterpart, e.g., phases with larger winding numbers. In particular, we first study the features of the qubit-photon bound states when the emitters couple to the bulk modes in the different phases, discern its connection with the topological invariant, and show how to further tune their shape through the use of giant atoms, i.e., non-local couplings. Then, we consider the coupling of emitters to the edge modes appearing in the different topological phases. Here, we show that giant atoms dynamics can distinguish between all different topological phases, as compared to the case with local couplings. Finally, we provide a possible experimental implementation of the model based on periodic modulations of circuit QED systems. Our work enriches the understanding of the interplay between topological photonics and quantum optics., Comment: 18 pages, 19 figures. Improved discussion in Appendix B. Code is available at: https://github.com/Carlos-Vega-97/Extended_SSH

The acclaimed Maxwell-Bloch (or Arecchi-Bonifacio) equations are a valid dynamical model, effectively describing wave propagation in nonlinear optical media: from the amplification in input-output devices to multimode instabilities arising in laser systems. However, the inherent spatial variability of the physical observables represents an obstacle to fast simulations and analysis, especially whenever networks of active elements have to be considered. In this paper, we propose an approach which, stripping the spatial dependence of its role as a generator of dynamical richness, allows for a compelling simple portrait. It leads to (a few) ordinary differential equations in input-output configurations, complemented by a time-delayed feedback in closed-loop setups. Such scheme reproduces accurately the dynamics, paving the way to a plain treatment of the wealth of phenomena described by the Maxwell-Bloch equations., Comment: 3 figures

We develop two approximate theories to describe the miscible and immiscible droplets that can occur in a binary mixture of highly magnetic bosonic atoms. In addition to allowing simpler calculations, the approximate theories provide insight into the role of quantum fluctuations in the two regimes. Results are validated by comparison to those from the extended Gross-Pitaevskii equation. As an application we solve for the ground state droplets crossing the miscible-immiscible transition as function of the short-ranged interspecies interaction parameter. We consider regimes where the transition occurs suddenly or as a smooth cross-over. Using dynamical calculations we show that the character of the transition is revealed in the number of domains produced when ramping the droplet into the immiscible regime., Comment: 10 pages, 6 figures

We present a closed-form analytical description for the metrological performance of a generic superposition of spin coherent states (SSCS) used as probes for quantum phase estimation. Working in this framework, we derive transparent analytical expressions for the pertinent quantum Fisher information and identify a general class of spin coherent states enabling quantum metrology with Heisenberg-limit precision. Bloch-sphere analysis of antipodal SSCS shows that the phase-estimation precision attainable with such states increases with the distance from the equator of the Bloch sphere, growing from the shot-noise limit level for the equatorial states all the way up to the Heisenberg limit for SSCS that can be represented as superpositions of the poles of the Bloch sphere.

Certain aspects of some unitary quantum systems are well-described by evolution via a non-Hermitian effective Hamiltonian, as in the Wigner-Weisskopf theory for spontaneous decay. Conversely, any non-Hermitian Hamiltonian evolution can be accommodated in a corresponding unitary system + environment model via a generalization of Wigner-Weisskopf theory. This demonstrates the physical relevance of novel features such as exceptional points in quantum dynamics, and opens up avenues for studying many body systems in the complex plane of coupling constants. In the case of lattice field theory, sparsity lends these channels the promise of efficient simulation on standardized quantum hardware. We thus consider quantum operations that correspond to Suzuki-Lee-Trotter approximation of lattice field theories undergoing non-Hermitian time evolution, with potential applicability to studies of spin or gauge models at finite chemical potential, with topological terms, to quantum phase transitions - a range of models with sign problems. We develop non-Hermitian quantum circuits and explore their promise on a benchmark, the quantum one-dimensional Ising model with complex longitudinal magnetic field, showing that observables can probe the Lee-Yang edge singularity. The development of attractors past critical points in the space of complex couplings indicates a potential for study on near-term noisy hardware., 17 pages, 14 figures