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We study the dynamics of out-of-time-ordered correlators (OTOCs) and entanglement of entropy as quantitative measures of information propagation in disordered many-body systems exhibiting Floquet time-crystal (FTC) phases. We find that OTOC spreads in the FTC with different characteristic timescales due to the existence of a preferred ``quasi-protected'' direction - denoted as $\ell$-bit direction - along which the spins stabilize their period-doubling magnetization for exponentially long times. While orthogonal to this direction the OTOC thermalizes as an usual MBL time-independent system (at stroboscopic times), along the $\ell$-bit direction the system features a more complex structure. The scrambling appears as a combination of an initially frozen dynamics (while in the stable period doubling magnetization time window) and a later logarithmic slow growth (over its decoherence regime) till full thermalization. Interestingly, in the late time regime, since the wavefront propagation of correlations has already settled through the whole chain, scrambling occurs at the same rate regardless of the distance between the spins, thus resulting in an overall envelope-like structure of all OTOCs, independent of their distance, merging into a single growth. Alongside, the entanglement entropy shows a logarithmic growth over all time, reflecting the slow dynamics up to a thermal volume-law saturation., Comment: 9 pages, 4 figures

We study the emergence of prethermal Floquet Time Crystal (pFTC) in disordered chiral multiferroic chains. The model is an extension of the usual periodically driven nearest-neighbor disordered Heisenberg chain, with additional next-nearest-neighbor Heisenberg couplings and DMI interactions due to external magnetic and electric couplings. We derive the phase diagram of the model, characterizing the magnetization, entanglement, and coherence dynamics of the system along the extended interactions. In addition, we explore the application of the pFTC as quantum sensors of AC fields. The sensor performance to estimate small AC fields is quantified through the quantum Fisher information (QFI) measure. The sensor offers several advantages as compared to those composed of non-interacting spins due to its intrinsic robustness, long coherent interrogation time, and many-body correlations. Specifically, the sensor can overcome the standard quantum limit ($\rm{SQL} \sim N t^2$) during the prethermal regime, reaching an optimum performance at the pFTC lifetime $t^*$, where the $\rm{QFI}/Nt^{*^2} \sim N^\alpha$ with $\alpha > 0$, scaling superlinarly with the number of spins. Different from \text{full} FTCs, the prethermal lifetime does not diverge in the thermodynamic limit, nevertheless it can be increasingly long with tuning system parameters., Comment: 11 pages, 7 figures

The study of open system dynamics is of paramount importance both from its fundamental aspects as well as from its potential applications in quantum technologies. In the simpler and most commonly studied case, the dynamics of the system can be described by a Lindblad master equation. However, identifying the Lindbladian that leads to general non-equilibrium steady states (NESS) is usually a non-trivial and challenging task. Here we introduce a method for reconstructing the corresponding Lindbaldian master equation given any target NESS, i.e., a Lindbladian Reverse Engineering ($\mathcal{L}$RE) approach. The method maps the reconstruction task to a simple linear problem. Specifically, to the diagonalization of a correlation matrix whose elements are NESS observables and whose size scales linearly (at most quadratically) with the number of terms in the Hamiltonian (Lindblad jump operator) ansatz. The kernel (null-space) of the correlation matrix corresponds to Lindbladian solutions. Moreover, the map defines an iff condition for $\mathcal{L}$RE, which works as both a necessary and a sufficient condition; thus, it not only defines, if possible, Lindbaldian evolutions leading to the target NESS, but also determines the feasibility of such evolutions in a proposed setup. We illustrate the method in different systems, ranging from bosonic Gaussian to dissipative-driven collective spins. We also discuss non-Markovian effects and possible forms to recover Markovianity in the reconstructed Lindbaldian., Comment: 11 pages, 7 figures

We investigate the use of a boundary time crystals (BTCs) as sensors of AC fields. Boundary time crystals are non-equilibrium phases of matter in contact to an environment, for which a macroscopic fraction of the many-body system breaks the time translation symmetry. We find an enhanced sensitivity of the BTC when its spins are resonant with the applied AC field, as quantified by the quantum Fisher information (QFI). The QFI dynamics in this regime is shown to be captured by a relatively simple ansatz consisting of an initial power-law growth and late-time exponential decay. We study the scaling of the ansatz parameters with resources (encoding time and number of spins) and identify a moderate quantum enhancement in the sensor performance through comparison with classical QFI bounds. Investigating the precise source of this performance, we find that despite of its long coherence time and multipartite correlations (advantageous properties for quantum metrology), the entropic cost of the BTC (which grows indefinitely in the thermodynamic limit) hinders an optimal decoding of the AC field information. This result has implications for future candidates of quantum sensors in open system and we hope it will encourage future study into the role of entropy in quantum metrology., Comment: 15 pages, 5 figures

We investigate the consequences of introducing non-uniform initial conditions in the dynamics of spin ensembles characterized by all-to-all interactions. Specifically, our study involves the preparation of a set of semi-classical spin ensembles with varying orientations. Through this setup, we explore the influence of such non-uniform initial states on the disruption of permutational invariance. Comparing this approach to the traditional scenario of initializing with spins uniformly aligned, we find that the dynamics of the spin ensemble now spans a more expansive effective Hilbert space. This enlargement arises due to the inclusion of off-diagonal coherences between distinct total angular momentum subspaces - an aspect typically absent in conventional treatments of all-to-all spin dynamics. Conceptually, the dynamic evolution can be understood as a composite of multiple homogeneous sub-ensembles navigating through constrained subspaces. Notably, observables that are sensitive to the non-uniformity of initial conditions exhibit discernible signatures of these off-diagonal coherences. We adopt this fresh perspective to reexamine the relaxation phenomena exhibited by the Dicke model, as well as a prototypical example of a boundary time crystal. Intriguingly, ensembles initialized with inhomogeneous initial conditions can show distinctive behaviors when contrasted with canonical instances of collective dynamics. These behaviors encompass the emergence of novel gapless excitations, the manifestation of limit-cycles featuring dressed frequencies due to superradiance, instances of frequency locking or beating synchronizations, and even the introduction of ``extra'' dimensions within the dynamics. In closing, we provide a brief overview of the potential implications of our findings in the context of modern cavity quantum electrodynamics (QED) platforms., Comment: 12 pages, 8 figures

The long range spatial and temporal ordering displayed by discrete time crystals, can become advantageous properties when used for sensing extremely weak signals. Here, we investigate their performance as quantum sensors of weak AC-fields and demonstrate, using the quantum Fisher information measure, that they can overcome the shot noise limit while allowing long interrogation times. In such systems, collective interactions stabilize their dynamics against noise making them robust enough to protocol imperfections., Comment: 6 + 6 pages, 5 + 5 figures

Quantum computational models can be approached via the lens of resources needed to perform computational tasks, where a computational advantage is achieved by consuming specific forms of quantum resources, or, conversely, resource-free computations are classically simulable. Can we similarly identify quantum computational resources in the setting of more general quasi-particle statistics? In this work, we develop a framework to characterize the separability of a specific type of one-dimensional quasiparticle known as a fermionic anyon. As we evince, the usual notion of partial trace fails in this scenario, so we build the notion of separability through a fractional Jordan-Wigner transformation, leading to an entanglement description of fermionic-anyon states. We apply this notion of fermionic-anyon separability, and the unitary operations that preserve it, mapping it to the free resources of matchgate circuits. We also identify how entanglement between two qubits encoded in a dual-rail manner, as standard for matchgate circuits, corresponds to the notion of entanglement between fermionic anyons., Comment: 5+18 pages, 2 figures. Addition of a new Theorem about fermionic-anyons single-particle entanglement, as well as minor text revisions. Comments are welcome

We discuss a sufficient condition for gapless excitations in the Lindbladian master equation for collective spin-boson systems and permutationally invariant systems. The condition relates a nonzero macroscopic cumulant correlation in the steady state to the presence of gapless modes in the Lindbladian. In phases arising from competing coherent and dissipative Lindbladian terms, we argue that such gapless modes, concomitant with angular momentum conservation, can lead to persistent dynamics in the spin observables with the possible formation of dissipative time-crystals. We study different models within this perspective, from Lindbladians with Hermitian jump operators, to non-Hermitian ones composed by collective spins and Floquet spin-boson systems. We also provide a simple analytical proof for the exactness of mean-field semiclassical approach in such systems based on a cumulant expansion., Comment: 18 pages, 7 figures

Non-Hermitian quantum system recently have attracted a lots of attentions theoretically and experimentally. However, the results based on the single-particle picture may not apply to understand the property of non-Hermitian many-body system. How the property of quantum many-body system especially the phase transition will be affected by the non-hermiticity remains unclear. Here we study non-Hermitian quantum contact process (QCP) model, whose effective Hamiltonian is derived from Lindbladian master equation. We show that there is a continuous phase transition induced by the non-hermiticity in QCP. We also determine the critical exponents $\beta$ of order parameter, $\gamma$ of susceptibility and study the correlation and entanglement near phase transition. We observe that the order parameter and susceptibility display infinitely singularity even for finite size system, since non-hermiticity endow many-body system with different singular behaviour from classical phase transition. Moreover our results show that the phase transition have no counterpart in Hermitian case and belongs to completely different universality class.

In this work we study genuine multipartite correlations (GMC's) in a boundary time crystal (BTC). Boundary time crystals are nonequilibrium quantum phases of matter in contact to an environment, for which a macroscopic fraction of the many-body system breaks the time-translation symmetry. We analyze both (i) the structure (orders) of GMC's among the subsystems, as well as (ii) their build-up dynamics for an initially uncorrelated state. We find that, in the thermodynamic limit (and only in such a limit), multipartite correlations of all orders grow indefinitely in time in the BTC phase, further displaying a persistent oscillatory behavior around their mean growth. The orders of the correlations show a power-law decaying hierarchy among its $k$-partitions. Moreover, in the long-time limit the GMC's are shown extensive with the system size, contrasting to the subextensive scaling in the non time-crystal (ferromagnetic) phase of the model. We also discuss the classical and quantum nature of these correlations with basis on multipartite entanglement witnesses, specifically, the analysis of the Quantum Fisher Information (QFI). Both GMC and QFI are able to capture and distinguish the different phases of the model. Our work highlights the genuine many-body properties of these peculiar non-equilibrium phases of matter., Comment: 15 pages, 9 figures

We study the ground state properties of the one-dimensional extended Hubbard model at half-filling from the perspective of its particle reduced density matrix. We focus on the reduced density matrix of $2$ fermions and perform an analysis of its quantum correlations and coherence along the different phases of the model. Specifically, we study its (i) entanglement entropy, (ii) $\ell_1$ norm of coherence, (iii) irreducible two-body cumulant matrix and (iv) entanglement spectrum. Our results show that these different properties are complementary to each other depending on the phase of the system, exhibiting peculiar behaviors such as discontinuities, maximum or minimum values at the quantum phase transitions, thus providing a qualitative view of the phase diagram of the model. In particular, in the superconducting region, we obtain that the entanglement spectrum signals a transition between a dominant singlet (SS) to triplet (TS) pairing ordering in the system. Moreover, from the analysis of the dominant eigenvector in the reduced state, we can relate the SS-TS transition to the spatial separation between the fermion pairs in the two different pairing orderings. The entanglement gap is also able to highlight a transition - at a few-body level - in the groundstate wavefunction, not discussed previously in the literature. While other quantifiers are less sensitive to few-body defects in the wavefunction, the entanglement gap can work as a magnifying glass for these, capturing such small fluctuations., Comment: 14 pages and 7 figures

Recent predictions for quantum-mechanical enhancements in the operation of small heat engines have raised renewed interest in their study from both a fundamental perspective and in view of applications. One essential question is whether collective effects may help to carry enhancements over larger scales, when increasing the number of systems composing the working substance of the engine. Such enhancements may consider not only power and efficiency, that is its performance, but, additionally, its constancy, i.e. the stability of the engine with respect to unavoidable environmental fluctuations. We explore this issue by introducing a many-body quantum heat engine model composed by spin pairs working in continuous operation. We study how power, efficiency and constancy scale with the number of spins composing the engine and introduce a well-defined macroscopic limit where analytical expressions are obtained. Our results predict power enhancements, both in finite-size and macroscopic cases, for a broad range of system parameters and temperatures, without compromising the engine efficiency, accompanied by coherence-enhanced constancy for finite sizes. We discuss these quantities in connection to Thermodynamic Uncertainty Relations (TUR)., Comment: 20 pages, 10 figures

We propose a generalization of the coherent anomaly method to extract the critical exponents of a phase transition occurring in the steady-state of an open quantum many-body system. The method, originally developed by Suzuki [J. Phys. Soc. Jpn. {\bf 55}, 4205 (1986)] for equilibrium systems, is based on the scaling properties of the singularity in the response functions determined through cluster mean-field calculations. We apply this method to the dissipative transverse-field Ising model and the dissipative XYZ model in two dimensions obtaining convergent results already with small clusters., Comment: Accepted version, 9 pages, 7 figures

Boundary time crystals (BTC's) are non-equilibrium phases of matter occurring in quantum systems in contact to an environment, for which a macroscopic fraction of the many body system breaks time translation symmetry. We study BTC's in collective $d$-level systems, focusing in the cases with $d=2$, $3$ and $4$. We find that BTC's appear in different forms for the different cases. We first consider the model with collective $d=2$-level systems [presented in Phys. Rev. Lett. $121$, $035301$ ($2018$)], whose dynamics is described by a Lindblad master equation, and perform a throughout analysis of its phase diagram and Jacobian stability for different interacting terms in the coherent Hamiltonian. In particular, using perturbation theory for general (non Hermitian) matrices we obtain analytically how a specific $\mathbb{Z}_2$ symmetry breaking Hamiltonian term destroys the BTC phase in the model. Based on these results we define a $d=4$ model composed of a pair of collective $2$-level systems interacting with each other. We show that this model support richer dynamical phases, ranging from limit-cycles, period-doubling bifurcations and a route to chaotic dynamics. The BTC phase is more robust in this case, not annihilated by the former symmetry breaking Hamiltonian terms. The model with collective $d=3$-level systems is defined similarly, as competing pairs of levels, but sharing a common collective level. The dynamics can deviate significantly from the previous cases, supporting phases with the coexistence of multiple limit-cycles, closed orbits and a full degeneracy of zero Lyapunov exponents., Comment: 18 pages, 12 figures

We generalize the theory of flow equations to open quantum systems focusing on Lindblad master equations. We introduce and discuss three different generators of the flow that transform a linear non-Hermitian operator into a diagonal one. We first test our dissipative flow equations on a generic matrix and on a physical problem with a driven-dissipative single fermionic mode. We then move to problems with many fermionic modes and discuss the interplay between coherent (disordered) dynamics and localized losses. Our method can also be applied to non-Hermitian Hamiltonians., Comment: 22 pages plus appendices, 9 figures. Typos in Sec. 2.2 and appendix A corrected

We study the phase diagram, both at zero and finite temperature, in a class of $\mathbb{Z}_q$ models with infinite range interactions. We are able to identify the transitions between a symmetry-breaking and a trivial phase by using a mean-field approach and a perturbative expansion. We perform our analysis on a Hamiltonian with $2p$-body interactions and we find first-order transitions for any $p>1$; in the case $p=1$, the transitions are first-order for $q=3$ and second-order otherwise. In the infinite-range case there is no trace of gapless incommensurate phase but, when the transverse field is maximally chiral, the model is in a symmetry-breaking phase for arbitrarily large fields. We analytically study the transtion in the limit of infinite $q$, where the model possesses a continuous $U(1)$ symmetry.

Quantum teleportation is considered a basic primitive in many quantum information processing tasks and has been experimentally confirmed in various photonic and matter-based setups. Here, we consider teleportation of quantum information encoded in modes of a fermionic field. In fermionic systems, superselection rules lead to a more differentiated picture of entanglement and teleportation. In particular, one is forced to distinguish between single-mode entanglement swapping, and qubit teleportation with or without authentication via Bell inequality violation, as we discuss here in detail. We focus on systems subject to parity superselection where the particle number is not fixed, and contrast them with systems constrained by particle number superselection which are relevant for possible practical implementations. Finally, we analyze the consequences for the operational interpretation of fermionic mode entanglement and examine the usefulness of so-called mixed maximally entangled states for teleportation., Comment: 15 pages, 3 figures, 3 tables; see also the related work by Olofsson et al.: arXiv:2002.08134

We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signalling function of quantum order-disorder transitions. We discuss the possibility for such transitions to characterize interesting physical phenomena, as quantum phase transitions, or abrupt variations in the correlation distributions. We apply our measure to two exactly solvable Hamiltonian models, namely: the $1D$-Quantum Ising Model and the Heisenberg XXZ spin-$1/2$ chain. We also compute this measure for one-qubit and two-qubit reduced states for the considered models, and analyse its behaviour across its quantum phase transitions for finite system sizes as well as in the thermodynamic limit by using Bethe ansatz., Comment: 12 pages, 5 figures

Quantum systems are always subject to interactions with an environment, typically resulting in decoherence and distortion of quantum correlations. It has been recently shown that a controlled interaction with the environment may actually help to create a state, dubbed as {\em ``dark''}, which is immune to decoherence. To encode quantum information in the dark states, they need to span a space with a dimensionality larger than one, so different orthogonal states act as a computational basis. We devise a symmetry-based conceptual framework to engineer such degenerate dark spaces (DDS), protected from decoherence by the environment. We illustrate this construction with a model protocol, inspired by the fractional quantum Hall effect, where the DDS basis is isomorphic to a set of degenerate Laughlin states. The long-time steady state of our driven-dissipative model exhibits thus all the characteristics of degenerate vacua of a unitary topological system. This approach offers new possibilities for storing, protecting and manipulating quantum information in open systems., Comment: 14 pages. Main text: 6 pages and 5 figures. Supplemental: 8 pages and 3 figures. References and extended discussion of Lindblad properties added

We construct a class of period-$n$-tupling discrete time crystals based on $\mathbb{Z}_n$ clock variables, for all the integers $n$. We consider two classes of systems where this phenomenology occurs, disordered models with short-range interactions and fully connected models. In the case of short-range models we provide a complete classification of time-crystal phases for generic $n$. For the specific cases of $n=3$ and $n=4$ we study in details the dynamics by means of exact diagonalisation. In both cases, through an extensive analysis of the Floquet spectrum, we are able to fully map the phase diagram. In the case of infinite-range models, the mapping onto an effective bosonic Hamiltonian allows us to investigate the scaling to the thermodynamic limit. After a general discussion of the problem, we focus on $n=3$ and $n=4$, representative examples of the generic behaviour. Remarkably, for $n=4$ we find clear evidence of a new crystal-to-crystal transition between period $n$-tupling and period $n/2$-tupling., Comment: 32 pages, 30 figures

We introduce a framework for the construction of completely positive maps for subsystems of indistinguishable fermionic particles. In this scenario, the initial global state is always correlated, and it is not possible to tell system and environment apart. Nonetheless, a reduced map in the operator sum representation is possible for some sets of states where the only non-classical correlation present is exchange., Comment: 9 pages, 2 figures. In this new version we eliminated a section with a mistaken definition of mutual information for indistinguishable particles, which was not relevant for the main argument of our paper

We study scrambling in connection to multipartite entanglement dynamics in regular and chaotic long-range spin chains, characterized by a well defined semi-classical limit. For regular dynamics, scrambling and entanglement dynamics are found to be very different: up to the Ehrenfest time they rise side by side departing only afterwards. Entanglement saturates and becomes extensively multipartite, while scrambling continues its growth up to the recurrence time. Remarkably, the exponential behaviour of scrambling emerges not only in the chaotic case, but also in the regular one, when the dynamics occurs at a dynamical critical point., Comment: 11 pages, 8 figures and Supplementary Material

We discuss a one-dimensional fermionic model with a generalized $\mathbb{Z}_{N}$ even multiplet pairing extending Kitaev $\mathbb{Z}_{2}$ chain. The system shares many features with models believed to host localized edge parafermions, the most prominent being a similar bosonized Hamiltonian and a $\mathbb{Z}_{N}$ symmetry enforcing an $N$-fold degenerate ground state robust to certain disorder. Interestingly, we show that the system supports a pair of parafermions but they are non-local instead of being boundary operators. As a result, the degeneracy of the ground state is only partly topological and coexists with spontaneous symmetry breaking by a (two-particle) pairing field. Each symmetry-breaking sector is shown to possess a pair of Majorana edge modes encoding the topological twofold degeneracy. Surrounded by two band insulators, the model exhibits for $N=4$ the dual of an $8 \pi$ fractional Josephson effect highlighting the presence of parafermions., Comment: 12 pages, 3 figures

We study the effect of many-body quantum interference on the dynamics of coupled periodically kicked systems whose classical dynamics is chaotic and shows an unbounded energy increase. We specifically focus on a $N$ coupled kicked rotors model: we find that the interplay of quantumness and interactions dramatically modifies the system dynamics inducing a transition between energy saturation and unbounded energy increase. We discuss this phenomenon both numerically and analytically, through a mapping onto a $N$-dimensional Anderson model. The thermodynamic limit $N\to\infty$, in particular, always shows unbounded energy growth. This dynamical delocalization is genuinely quantum and very different from the classical one: using a mean field approximation we see that the system self-organizes so that the energy per site increases in time as a power law with exponent smaller than one. This wealth of phenomena is a genuine effect of quantum interference: the classical system for $N\geq 2$ always behaves ergodically with an energy per site linearly increasing in time. Our results show that quantum mechanics can deeply alter the regularity/ergodicity properties of a many body driven system., Comment: 25 pages, 17 figures

In this work we discuss the existence of time-translation symmetry breaking in a kicked infinite-range-interacting clean spin system described by the Lipkin-Meshkov-Glick model. This Floquet time crystal is robust under perturbations of the kicking protocol, its existence being intimately linked to the underlying $\mathbb{Z}_2$ symmetry breaking of the time-independent model. We show that the model being infinite-range and having an extensive amount of symmetry breaking eigenstates is essential for having the time-crystal behaviour. In particular we discuss the properties of the Floquet spectrum, and show the existence of doublets of Floquet states which are respectively even and odd superposition of symmetry broken states and have quasi-energies differing of half the driving frequencies, a key essence of Floquet time crystals. Remarkably, the stability of the time-crystal phase can be directly analysed in the limit of infinite size, discussing the properties of the corresponding classical phase space. Through a detailed analysis of the robustness of the time crystal to various perturbations we are able to map the corresponding phase diagram. We finally discuss the possibility of an experimental implementation by means of trapped ions., Comment: 14 pages, 12 figures

We present a new approach for the quantification of quantumness of correlations in fermionic systems. We study the Multipartite Relative Entropy of Quantumness in such systems, and show how the symmetries in the states can be used to obtain analytical solutions. Numerical evidences about the uniqueness of such solutions are also presented. Supported by these results, we show that the minimization of the Multipartite Relative Entropy of Quantumness, over certain choices of its modes multipartitions, reduces to the notion of Quantumness of Indistinguishable Particles. By means of an activation protocol, we characterize the class of states without quantumness of correlations. As an example, we calculate the dynamics of quantumness of correlations for a purely dissipative system, whose stationary states exhibit interesting topological non-local correlations., Comment: 13 pages, 5 figures

Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this paper we present a non-trivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave-functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wavefunctions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the non-approximate study of topological one-dimensional parafermionic chains with spatial-inversion and time-reversal symmetry in the absence of strong edge modes., Comment: 6 + 9 pages, 3 figures

We are able to detect clear signatures of dephasing -- a distinct trait of Many-Body Localisation (MBL) -- via the dynamics of two-sites entanglement, quantified through the concurrence. Using the protocol implemented in [Science {\bf 349}, 842 (2015)] we show that -- in the MBL phase -- the average two-site entanglement decays in time as a power law, while in the Anderson localised phase it tends to a plateau. The exponent of the power law is not universal and shows a clear dependence on the strength of the interaction. This behaviour is also qualitatively different in the ergodic phase where the two-site entanglement decays exponentially. All the results are obtained by means of time-dependent density matrix renormalisation group simulations; they are corroborated by analytical calculations on an effective model. Two-site entanglement has been already measured in cold atoms: Our analysis paves the way for the first direct experimental test of many-body dephasing in the MBL phase., Comment: 16 pages, 11 figures. References updated. See also the related work by S. Campbell et al., arXiv:1608.08897

We discuss the dissipative preparation of p-wave superconductors in number-conserving one-dimensional fermionic systems. We focus on two setups: the first one entails a single wire coupled to a bath, whereas in the second one the environment is connected to a two-leg ladder. Both settings lead to stationary states which feature the bulk properties of a p-wave superconductor, identified in this number-conserving setting through the long-distance behavior of the proper p-wave correlations. The two schemes differ in the fact that the steady state of the single wire is not characterized by topological order, whereas the two-leg ladder hosts Majorana zero modes, which are decoupled from damping and exponentially localized at the edges. Our analytical results are complemented by an extensive numerical study of the steady-state properties, of the asymptotic decay rate and of the robustness of the protocols., Comment: 14 pages, 7 figures, 2 appendices

We obtain the analytical expression for the Kraus decomposition of the quantum map of an environment modeled by an arbitrary quadratic fermionic Hamiltonian acting on one or two qubits, and derive simple functions to check the non-positivity of the intermediate map. These functions correspond to two different sufficient criteria for non-Markovianity. In the particular case of an environment represented by the Ising Hamiltonian, we discuss the two sources of non-Markovianity in the model, one due to the finite size of the lattice, and another due to the kind of interactions., Comment: 11 pages, 10 figures

In this letter we present, in a number conserving framework, a model of interacting fermions in a two-wire geometry supporting non-local zero-energy Majorana-like edge excitations. The model has an exactly solvable line, on varying the density of fermions, described by a topologically non-trivial ground state wave-function. Away from the exactly solvable line we study the system by means of the numerical density matrix renormalization group. We characterize its topological properties through the explicit calculation of a degenerate entanglement spectrum and of the braiding operators which are exponentially localized at the edges. Furthermore, we establish the presence of a gap in its single particle spectrum while the Hamiltonian is gapless, and compute the correlations between the edge modes as well as the superfluid correlations. The topological phase covers a sizeable portion of the phase diagram, the solvable line being one of its boundaries., Comment: Improved version, in particular for the braiding and superfluid discussions. 5 pages, 4 figures, supplemental material (7 pages)

We investigate the quantum phase transitions of the extended Hubbard model at half-filling with periodic boundary conditions employing the entanglement of particles, as opposed to the more traditional entanglement of modes. Our results show that the entanglement has either discontinuities or local minima at the critical points. We associate the discontinuities to first order transitions, and the minima to second order ones. Thus we show that the entanglement of particles can be used to derive the phase diagram, except for the subtle transitions between the phases SDW-BOW, and the superconductor phases TS-SS., Comment: Improved text and dicussions. Added finite size scaling analysis. 9 pages, 8 figures. Accepted for publication in Phys. Rev. B

We discuss a general notion of quantum correlations in fermionic or bosonic indistinguishable particles. Our approach is mainly based on the identification of the algebra of single-particle observables, which allows us to devise an activation protocol in which the \textit{quantumness of correlations} in the system leads to a unavoidable creation of entanglement with the measurement apparatus. Using the distillable entanglement, or the relative entropy of entanglement, as entanglement measure, we show that our approach is equivalent to the notion of minimal disturbance in a single-particle von Neumann measurement, also leading to a geometrical approach for its quantification., Comment: 7 pages, 2 figures

We discuss particle entanglement in systems of indistinguishable bosons and fermions, in finite Hilbert spaces, with focus on operational measures of quantum correlations. We show how to use von Neumann entropy, Negativity and entanglement witnesses in these cases, proving interesting relations. We obtain analytic expressions to quantify quantum correlations in homogeneous D-dimensional Hamiltonian models with certain symmetries., Comment: 9 pages

We present a method to quantify quantum correlations in arbitrary systems of indistinguishable fermions using witness operators. The method associates the problem of finding the optimal entan- glement witness of a state with a class of problems known as semidefinite programs (SDPs), which can be solved efficiently with arbitrary accuracy. Based on these optimal witnesses, we introduce a measure of quantum correlations which has an interpretation analogous to the Generalized Robust- ness of entanglement. We also extend the notion of quantum discord to the case of indistinguishable fermions, and propose a geometric quantifier, which is compared to our entanglement measure. Our numerical results show a remarkable equivalence between the proposed Generalized Robustness and the Schliemann concurrence, which are equal for pure states. For mixed states, the Schliemann con- currence presents itself as an upper bound for the Generalized Robustness. The quantum discord is also found to be an upper bound for the entanglement., Comment: 7 pages, 6 figures, Accepted for publication in Quantum Information Processing