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A convenient way to represent quantum optical states is through the quadrature basis of single-modes of the field. This framework provides intuitive definitions for quasi-classical states, their phase-space representations, and a robust toolbox for quantum state manipulation using universal gates. In this widely adopted representation of quantum optics, most pure states consist of coherent superpositions of photon-number states. However, this approach neglects the particle-number superselection rule - which prohibits coherence between states of differing photon numbers - and implicitly assumes a common phase reference - even though global phases are unmeasurable in quantum optics. We adopt a representation of quantum optical states that respects the superselection rule and revisit key tools and results in quantum optics and information encoding within quantum optics. In the introduced framework, general pure states are described using two orthogonal modes in the Fock basis. We show that this approach preserves the intuitive aspects of the traditional quadrature representation while unifying insights from quantum optics with those from symmetric spin-like and angular momentum systems. Moreover, the superselection rule-compliant representation provides a consistent definition of non-purity and coherence for optical modes and states. It offers a clearer and general perspective on quantum universality, non-classicality, and the (im)possibility of efficient classical simulation across various quantum information encoding schemes involving quantum optical modes and states.

We introduce a framework for the study of multiparty entanglement by analyzing the behavior of collective variables. Throughout the manuscript, we explore a specific type of multiparty entanglement which can be detected through the fluctuation of a collective observable. We thoroughly analyze its properties and how it can be extended to mixed states while placing it within the context of the existing literature. The novelty of our approach also lies in the fact that we present a graphical point of view. This is done by introducing a spectral space on which the various properties of our entanglement quantifier have a direct pictorial interpretation. Notably, this approach proves particularly effective for assessing $k$-entanglement, as we show its ability to extend previously established inequalities. To enhance understanding, we also demonstrate how this framework applies to specific scenarios, encompassing both finite-dimensional cases and infinite-dimensional systems, the latter being exemplified by the time-frequency modal degree of freedom of co-propagating single photons., Comment: 15 pages, 5 figures

We present a method to systematically identify and classify quantum optical non-classical states as classical/non-classical based on the resources they create on a bosonic quantum computer. This is achieved by converting arbitrary bosonic states into multiple modes, each occupied by a single photon, thereby defining qubits of a bosonic quantum computer. Starting from a bosonic classical-like state in a representation that explicitly respects particle number super-selection rules, we apply universal gates to create arbitrary superpositions of states with the same total particle number. The non-classicality of the corresponding states can then be associated to the operations they induce in the quantum computer. We also provide a correspondence between the adopted representation and the more conventional one in quantum optics, where superpositions of Fock states describe quantum optical states, and we identify how multi-mode states can lead to quantum advantage. Our work contributes to establish a seamless transition from continuous to discrete properties of quantum optics while laying the grounds for a description of non-classicality and quantum computational advantage that is applicable to spin systems as well.

GKP states, introduced by Gottesman, Kitaev, and Preskill, are continuous variable logical qubits that can be corrected for errors caused by phase space displacements. Their experimental realization is challenging, in particular using propagating fields, where quantum information is encoded in the quadratures of the electromagnetic field. However, travelling photons are essential in many applications of GKP codes involving the long-distance transmission of quantum information. We introduce a new method for encoding GKP states in propagating fields using single photons, each occupying a distinct auxiliary mode given by the propagation direction. The GKP states are defined as highly correlated states described by collective continuous modes, as time and frequency. We analyze how the error detection and correction protocol scales with the total photon number and the spectral width. We show that the obtained code can be corrected for displacements in time-frequency phase space - which correspond to dephasing, or rotations, in the quadrature phase space - and to photon losses. Most importantly, we show that generating two-photon GKP states is relatively simple, and that such states are currently produced and manipulated in several photonic platforms where frequency and time-bin biphoton entangled states can be engineered.

In quantum mechanics, the precision achieved in parameter estimation using a quantum state as a probe is determined by the measurement strategy employed. The ultimate quantum limit of precision is bounded by a value set by the state and its dynamics. Theoretical results have revealed that in interference measurements with two possible outcomes, this limit can be reached under ideal conditions of perfect visibility and zero losses. However, in practice, this cannot be achieved, so precision {\it never} reaches the quantum limit. But how do experimental setups approach precision limits under realistic circumstances? In this work we provide a general model for precision limits in two-photon Hong-Ou-Mandel interferometry for non-perfect visibility. We show that the scaling of precision with visibility depends on the effective area in time-frequency phase space occupied by the state used as a probe, and we find that an optimal scaling exists. We demonstrate our results experimentally for different states in a set-up where the visibility can be controlled and reaches up to $99.5\%$. In the optimal scenario, a ratio of $0.97$ is observed between the experimental precision and the quantum limit, establishing a new benchmark in the field.

We use time-frequency continuous variables as the standard framework to describe states of light in the subspace of individual photons occupying distinguishable auxiliary modes. We adapt to this setting the interplay between metrological properties and the phase space picture already extensively studied for quadrature variables. We also discuss in details the Hong-Ou-Mandel interferometer, which was previously shown to saturate precision limits, and provide a general formula for the coincidence probability of a generalized version of this experiment. From the obtained expression, we systematically analyze the optimality of this measurement setting for arbitrary unitary transformations applied to each one of the input photons. As concrete examples, we discuss transformations which can be represented as translations and rotations in time-frequency phase space for some specific states.

We study the role of the electromagnetic field's frequency in time precision measurements using single photons as a paradigmatic system. For such, we independently identify the contributions of intensity and spectral resources and show that both can play a role on the scaling of the precision of parameter estimation with the number of probes. We show in particular that it is possible to observe a quadratic scaling using quantum mode correlations only and explicit the mathematical expression of states saturating the Heisenberg limit. We also provide a geometrical and phase space interpretation of our results, and observe a curious quantum-to-classical-like transition on scaling by modifying the spectral variance of states. Our results connect discrete and continuous aspects of single photons and quantum optics by considering from a quantum mechanical perspective the role of frequency., Comment: 15 pages, 3 figures

We extensively discuss the Hong-Ou-Mandel experiment taking an original phase-space-based perspective. For this, we analyze time and frequency variables as quantum continuous variables in perfect analogy with position and momentum of massive particles or with the electromagnetic field's quadratures. We discuss how this experiment can be used to directly measure the time-frequency Wigner function and implement logical gates in these variables. We also briefly discuss the quantum/classical aspects of this experiment providing a general expression for intensity correlations that explicit the differences between a classical Hong-Ou-Mandel like dip and a quantum one. Throughout the manuscript, we will often focus and refer to a particular system based on AlGaAs waveguides emitting photon pairs via spontaneous parametric down-conversion, but our results can be extended to other analogous experimental systems and to different degrees of freedom.

Stimulated emission tomography is a powerful and successful technique to both improve the resolution and experimentally simplify the task of determining the modal properties of biphotons. In the present manuscript we provide a theoretical description of SET valid for any quadratic coupling regime between a non-linear medium and pump fields generating photons by pairs. We use our results to obtain not only information about the associated modal function modulus but also its phase, for any mode, and we discuss the specific case of time-frequency variables as well as the quantities and limitations involved in the measurement resolution.

We present a second quantization description of frequency-based continuous variables quantum computation in the subspace of single photons. For this, we define frequency and time operators using the free field Hamiltonian and its Fourier transform, and show that these observables, when restricted to the one photon per mode subspace, reproduce the canonical position-momentum commutation relations. As a consequence, frequency and time operators can be used to define a universal set of gates in this particular subspace. We discuss the physical implementation of these gates as well as their effect on single photon states, and show that frequency and time variables can also be used to implement continuous variables quantum information protocols, in the same way than polarization is currently used as a two-dimensional quantum variable.

In the context of phase estimation with Gaussian states, we introduce a quantifiable definition of metrological advantage that takes into account thermal noise in the preparation procedure. For a broad set of states, \textit{isotropic non-pure Gaussian states}, we show that squeezing is not only necessary, but sufficient, to achieve metrological advantage. We interpret our results in the framework of resource theory, and discuss possible sources of advantage other than squeezing. Our work is a step towards using phase estimation with pure and mixed state to define and quantify nonclassicality. This work is complementary with studies that defines nonclassicality using quadrature displacement estimation., Comment: Changes to wording, figures replaced

When the coupling rate between two quantum systems becomes as large as their characteristic frequencies, it induces dramatic effects on their dynamics and even on the nature of their ground state. The case of a qubit coupled to a harmonic oscillator in this ultrastrong coupling regime has been investigated theoretically and experimentally. Here, we explore the case of two harmonic oscillators in the ultrastrong coupling regime. Specifically, we realize an analog quantum simulation of this coupled system by dual frequency pumping a nonlinear superconducting circuit. The pump amplitudes directly tune the effective coupling rate. We observe spectroscopic signature of a mode hybridization that is characteristic of the ultrastrong coupling. Further we experimentally demon- strate a key property of the ground state of this simulated ultrastrong coupling between modes by observing simultaneous single-mode and two-mode squeezing of the radiated field below vacuum fluctuations.

The connection between the Leggett-Garg inequality and optimal scenarios from the point of view of quantum metrology is investigated for perfect and noisy general dichotomic measurements. In this context, we show that the Fisher information can be expressed in terms of quantum temporal correlations. This connection allows us to associate scenarios with relatively high Fisher information to scenarios in which the Leggett-Garg inequality is violated. We thus demonstrate a qualitative and, to some extent, quantitative link between measurement invasiveness and metrological performance. Finally, we illustrate our results by using a specific model for spin systems., Comment: 6 pages, 3 figures; close to published version

Particle number fluctuations, no matter how small, are present in experimental set-ups. One should rigorously take these fluctuations into account, especially, for entanglement detection. In this context, we generalize the spin squeezing inequalities introduced by T\'oth et al. in Phys. Rev. Lett. 99, 250405 (2007). These new inequalities are fulfilled by all separable states even when the number of particle is not constant, and may present quantum fluctuations. These inequalities are useful for detecting entanglement in many-body systems when the super-selection rule does not apply, or when only a subspace of the total systems Hilbert space is considered. We also define general dichotomic observables for which we obtain a coordinate independent form of the generalized spin squeezing inequalities. We give an example where our generalized coordinate independent spin squeezing inequalities present a clear advantage over the original ones., Comment: 11 pages, 2 figures

We present a general strategy to derive entanglement criteria which consists in performing a mapping from qudits to qubits that preserves the separability of the parties and SU(2) rotational invariance. Consequently, it is possible to apply the well known positive partial transpose criterion to reveal the existence of quantum correlations between qudits. We discuss some examples of entangled states that are detected using the proposed strategy. Finally, we demonstrate, using our scheme, how some variance-based entanglement witnesses can be generalized from qubits to higher dimensional spin systems.

We present a theoretical proposal to couple a single Nitrogen-Vacancy (NV) center to a superconducting flux qubit (FQ) in the regime where both systems are off resonance. The coupling between both quantum devices is achieved through the strong driving of the flux qubit by a classical microwave field that creates dressed states with an experimentally controlled characteristic frequency. We discuss several applications such as controlling the NV center's state by manipulation of the flux qubit, performing the NV center full tomography and using the NV center as a quantum memory. The effect of decoherence and its consequences to the proposed applications are also analyzed. Our results provide a theoretical framework describing a promising hybrid system for quantum information processing, which combines the advantages of fast manipulation and long coherence times., Comment: 8 pages, 9 figures

We provide a toolbox for continuous variables quantum state engineering and characterization of biphoton states produced by spontaneous parametric down conversion in a transverse pump configuration. We show that the control of the pump beam's incidence spot and angle corresponds to phase space displacements of conjugate collective continuous variables of the biphoton. In particular, we illustrate with numerical simulations on a semiconductor device how this technique can be used to engineer and characterize arbitrary states of the frequency and time degrees of freedom.

The Leggett-Garg inequality is a widely used test of the "quantumness" of a system, and involves correlations between measurements realized at different times. According to its widespread interpretation, a violation of the Legget-Garg inequality disproofs macroscopic realism and non-invasiveness. Nevertheless, recent results point out that macroscopic realism is a model dependent notion and that one should always be able to attribute to invasiveness a violation of a Legget-Garg inequality. This opens some natural questions: how to provide such an attribution in a systematic way? How can apparent macroscopic realism violation be recast into a dimensional independent invasiveness model? The present work answers these questions by introducing an operational model where the effects of invasiveness are controllable through a parameter associated with what is called the {\it measurability} of the physical system. Such a parameter leads to different generalized measurements that can be associated with the dimensionality of a system, to measurement errors or to back action., Comment: 5 pages, 4 figures, published version

Single photon interference due to passage through a periodic grating is considered in a novel proposal for processing D-dimensional quantum systems (quDits) encoded in the spatial degrees of freedom of light. We show that free space propagation naturally implements basic single quDit gates by means of the Talbot effect: an intricate time-space carpet of light in the near field diffraction regime. Adding a diagonal phase gate, we show that a complete set of single quDit gates can be implemented. We then introduce a spatially-dependent beam splitter that allows implementation of controlled operations between two quDits. A new form of universal quantum information processing can then be implemented with linear optics and ancilla photons. Though we consider photons, our scheme should be directly applicable to a number of other physical systems. Interpretation of the Talbot effect as a quantum logic operation provides a beautiful and interesting way to visualize quantum computation through wave propagation and interference., Comment: 11 pages, 8 figures

The Hong-Ou-Mandel (HOM) experiment was a benchmark in quantum optics, evidencing the quantum nature of the photon. In order to go deeper, and obtain the complete information about the quantum state of a system, for instance, composed by photons, the direct measurement or reconstruction of the Wigner function or other quasi--probability distribution in phase space is necessary. In the present paper, we show that a simple modification in the well-known HOM experiment provides the direct measurement of the Wigner function. We apply our results to a widely used quantum optics system, consisting of the biphoton generated in the parametric down conversion process. In this approach, a negative value of the Wigner function is a sufficient condition for non-gaussian entanglement between two photons. In the general case, the Wigner function provides all the required information to infer entanglement using well known necessary and sufficient criteria. We analyze our results using two examples of parametric down conversion processes taken from recent experiments. The present work offers a new vision of the HOM experiment that further develops its possibilities to realize fundamental tests of quantum mechanics involving decoherence and entanglement using simple optical set-ups., Comment: 5 pages, 2 figures plus supplementary information; submitted; comments welcome

We demonstrate the generation of polarization-entangled photon pairs at room temperature and telecom wavelength in a AlGaAs semiconductor waveguide. The source is based on spontaneous parametric down conversion with a counterpropagating phase-matching scheme. The quality of the two-photon state is assessed by the reconstruction of the density matrix giving a raw fidelity to a Bell state of 0.83; a theoretical model, taking into account the experimental parameters, provides ways to understand and control the amount of entanglement. Its compatibility with electrical injection, together with the high versatility of the generated two-photon state, make this source an attractive candidate for completely integrated quantum photonics devices.

We consider bipartite quantum systems characterized by a continuous angular variable \theta \in [-\pi, \pi[, representing, for instance, the position of a particle on a circle. We show how to reveal non-locality on this type of system using inequalities similar to CHSH ones, originally derived for bipartite spin 1/2 like systems. Such inequalities involve correlated measurement of continuous angular functions and are equivalent to the continuous superposition of CHSH inequalities acting on bidimensional subspaces of the infinite dimensional Hilbert space. As an example, we discuss in detail one application of our results, and we derive inequalities based on orientation correlation measurements. The introduced Bell-type inequalities open the perspective of new and simpler experiments to test non locality on a variety of quantum systems described by continuous variables.

We make a detailed theoretical description of the two-dimensional nature of a dc-SQUID, analyzing the coupling between its two orthogonal phase oscillation modes. While it has been shown that the mode defined as "longitudinal" can be initialized, manipulated and measured, so as to encode a quantum bit of information, the mode defined as "transverse" is usually repelled at high frequency and does not interfere in the dynamics. We show that, using typical parameters of existing devices, the transverse mode energy can be made of the order of the longitudinal one. In this regime, we can observe a strong coupling between these modes, described by an Hamiltonian providing a wide range of interesting effects, such as conditional quantum operations and entanglement. This coupling also creates an atomic-like structure for the combined two mode states, with a V-like scheme., Comment: 5 pages

We show that it is possible to initialize and manipulate in a deterministic manner protected qubits using time varying Hamiltonians. Taking advantage of the symmetries of the system, we predict the effect of the noise during the initialization and manipulation. These predictions are in good agreement with numerical simulations. Our study shows that the topological protection remains efficient under realistic experimental conditions., Comment: To be published in Phys. Rev. Lett

We investigate entanglement in a linear chain of $N$ polar molecules coupled by dipole interaction. In our model, nearest neighbour interaction predominate, and we compute entanglement with the help of a two-party correlation entanglement measure. We find that, in this system, only excited states are entangled. Moreover, when an electrostatic field is applied, energy levels crossings occur, changing significantly the system's entanglement properties. We make a systematic study of the entanglement dependency on the inter molecular distance separating pairs of molecules, different partitions of the chain and physical parameters as the temperature and the electrostatic field's intensity, showing that it persists for relatively high temperatures and changes its nature with varying field.

We show that trapped ions can be used to simulate a highly symmetrical Hamiltonian with eingenstates naturally protected against local sources of decoherence. This Hamiltonian involves long range coupling between particles and provides a more efficient protection than nearest neighbor models discussed in previous works. Our results open the perspective of experimentally realizing in controlled atomic systems, complex entangled states with decoherence times up to nine orders of magnitude longer than isolated quantum systems., Comment: 4 pages

We introduce Bell-type inequalities allowing for non-locality and entanglement tests with two cold heteronuclear molecules. The proposed inequalities are based on correlations between each molecule spatial orientation, an observable which can be experimentally measured with present day technology. Orientation measurements are performed on each subsystem at diferent times. These times play the role of the polarizer angles in Bell tests realized with photons. We discuss the experimental implementations of the proposed tests, which could also be adapted to other high dimensional quantum angular momenta systems., Comment: 4 pages

We propose an alternative scenario for the generation of entanglement between rotational quantum states of two polar molecules. This entanglement arises from dipole-dipole interaction, and is controlled by a sequence of laser pulses simultaneously exciting both molecules. We study the efficiency of the process, and discuss possible experimental implementations with cold molecules trapped in optical lattices or in solid matrices. Finally, various entanglement detection procedures are presented, and their suitability for these two physical situations is analyzed.

We make a geometric study of the phases acquired by a general pure bipartite two level system after a cyclic unitary evolution. The geometric representation of the two particle Hilbert space makes use of Hopf fibrations. It allows for a simple description of the dynamics of the entangled state's phase during the whole evolution. The global phase after a cyclic evolution is always an entire multiple of $\pi$ for all bipartite states, a result that does not depend on the degree of entanglement. There are three different types of phases combining themselves so as to result in the $n \pi$ global phase. They can be identified as dynamical, geometrical and topological. Each one of them can be easily identified using the presented geometric description. The interplay between them depends on the initial state and on its trajectory and the results obtained are shown to be in connection to those on mixed states phases., Comment: 9 figures, slightly different version from the accepted one

We generalize the quantum random walk protocol for a particle in a one-dimensional chain, by using several types of biased quantum coins, arranged in aperiodic sequences, in a manner that leads to a rich variety of possible wave function evolutions. Quasiperiodic sequences, following the Fibonacci prescription, are of particular interest, leading to a sub-ballistic wavefunction spreading. In contrast, random sequences leads to diffusive spreading, similar to the classical random walk behaviour. We also describe how to experimentally implement these aperiodic sequences., Comment: 4 pages and 4 figures