We present detailed derivations, various improvements and application to concrete experimental data of spin squeezing inequalities formulated recently by some of us [Phys. Rev. Lett. {\bf 95}, 120502 (2005)]. These inequalities generalize the concept of the spin squeezing parameter, and provide necessary and sufficient conditions for genuine 2-, or 3- qubit entanglement for symmetric states, and sufficient entanglement condition for general $N$-qubit states. We apply our method to theoretical study of Dicke states, and, in particular, to $W$-states of $N$ qubits. Then, we analyze the recently experimentally generated 7- and 8-ion $W$-states [Nature {\bf 438}, 643 (2005)]. We also present some novel details concerning this experiment. Finally, we improve criteria for detection of genuine tripartite entanglement based on entanglement witnesses.
We derive spin squeezing inequalities that generalize the concept of the spin squeezing parameter and provide necessary and sufficient conditions for genuine 2-, or 3- qubit entanglement for symmetric states, and sufficient condition for general states of $N$ qubits. Our inequalities have a clear physical interpretation as entanglement witnesses, can be relatively easy measured, and are given by complex, but {\it elementary} expressions.
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 Tóth 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 onl…
The manuscript investigates the entropy squeezing of a qubit (two-level atom) interacting with the cavity field in the Schrödinger cat states. The effects of the Schrödinger cat states of the field on the squeezing are examined. Our results suggest that entropy squeezing is not sensitive to the superposition features of two coherent states of the Schrödinger cat states. Furthermore, it is found that entropy squeezing occurs as the qubit initially close to its maximally coherent state, and the qubit initially in the maximally coherent state can generate optimal squeezing. This also demonstrates…
We study entanglement and spin squeezing in the ground state of three qubits interacting via the transverse Ising model. We give analytical results for the entanglement and spin squeezing, and a quantitative relation between the concurrence, quantifying the entanglement of two spins, and the spin squeezing parameter, measuring the degree of squeezing. Finally, by appropriately choosing the exchange interaction and strengths of the transverse field, we propose a scheme for generating entangled W state from an unentangled initial state with all spins down.
In this study we investigate the spin squeezing in superposition of a Biaxial state [1,2] and Bell state. Numerical and analytical solutions for the length of mean spin, mean spin direction and spin squeezing are given. It is shown that both the mean spin direction and spin squeezing parameter are determined by the coefficients of superposition and the relative phase.
We propose a new spin squeezing criterion for arbitrary multi-qubit states that is invariant under local unitary operations. We find that, for arbitrary pure two-qubit states, spin squeezing is equivalent to entanglement, and multi-qubit states are entangled if this new spin squeezing parameter is less than 1.
By identifying non-local effects in systems of identical Bosonic qubits through correlations of their commuting observables, we show that entanglement is not necessary to violate certain squeezing inequalities that hold for distinguishable qubits and that spin squeezing may not be necessary to achieve sub-shot noise accuracies in ultra-cold atom interferometry.
Quantum decoherence happens when the system interacts with the environment. Quantum correlation behaviours in the two-qubit spin squeezing model are studied under the influence of intrinsic decoherence. Quantitative results were determined, which depend on parameters of the physical system by checking different quantifiers of quantum correlation such as entanglement, local quantum uncertainty, trace distance discord and uncertainty-induced quantum nonlocality. We show that the entanglement suffers from intrinsic decoherence and exhibits sudden death, whereas the other measures are more robust …
While the squeezing of a propagating field can, in principle, be made arbitrarily strong, the cavity-field squeezing is subject to the well-known 3 dB limit, and thus has limited applications. Here, we propose the use of a fully quantum degenerate parametric amplifier (DPA) to beat this squeezing limit. Specifically, we show that by {\it simply} applying a two-tone driving to the signal mode, the pump mode can, {\it counterintuitively}, be driven by the photon loss of the signal mode into a squeezed steady state with, in principle, an {\it arbitrarily high} degree of squeezing. Furthermore, we…
We implement the squeezing operation as a genuine quantum gate, deterministically and reversibly acting `online' upon an input state no longer restricted to the set of Gaussian states. More specifically, by applying an efficient and robust squeezing operation for the first time to non-Gaussian states, we demonstrate a two-way conversion between a particle-like single-photon state and a wave-like superposition of coherent states. Our squeezing gate is reliable enough to preserve the negativities of the corresponding Wigner functions. This demonstration represents an important and necessary step…
We show that two definitions of spin squeezing extensively used in the literature [M. Kitagawa and M. Ueda, Phys. Rev. A {\bf 47}, 5138 (1993) and D.J. Wineland {\it et al.}, Phys. Rev. A {\bf 50}, 67 (1994)] give different predictions of entanglement in the two-atom Dicke system. We analyze differences between the definitions and show that the Kitagawa and Ueda's spin squeezing parameter is a better measure of entanglement than the commonly used spectroscopic spin squeezing parameter. We illustrate this relation by examining different examples of a driven two-atom Dicke system in which spin s…
Microwave squeezing represents the ultimate sensitivity frontier for superconducting qubit measurement. However, observation of enhancement has remained elusive, in part because integration with conventional dispersive readout pollutes the signal channel with antisqueezed vacuum. Here we induce a stroboscopic light-matter coupling with superior squeezing compatibility, and observe an increase in the room-temperature signal-to-noise ratio of 24%. Squeezing the orthogonal phase controls measurement backaction, slowing dephasing by a factor of 1.8. This protocol enables the practical use of micro…
We investigate a hybrid system consisting of an atomic ensemble trapped inside a dissipative optomechanical cavity assisted with the perturbative oscillator-qubit coupling. It is shown that such a hybrid system is very suitable for generating stationary squeezing of the mirror motion in the long-time limit under the unresolved sideband regime. Based on the approaches of master equation and covariance matrix, we discuss the respective squeezing effects in detail and find that in both approaches, simplifying the system dynamics with adiabatic elimination of the highly dissipative cavity mode is …
We theoretically investigate the implementation of the two-mode squeezing operator in circuit quantum electrodynamics. Inspired by a previous scheme for optical cavities [Phys. Rev. A $\textbf{73}$, 043803(2006)], we employ a superconducting qubit coupled to two nondegenerate quantum modes and use a driving field on the qubit to adequately control the resonator-qubit interaction. Based on the generation of two-mode squeezed vacuum states, firstly we analyze the validity of our model in the ideal situation and then we investigate the influence of the dissipation mechanisms on the generation of …
We investigate nonlinear effects in an electromechanical system consisting of a superconducting charge qubit coupled to transmission line resonator and a nanomechanical oscillator, which in turn is coupled to another transmission line resonator. The nonlinearities induced by the superconducting qubit and the optomechanical coupling play an important role in creating optomechanical entanglement as well as the squeezing of the transmitted microwave field. We show that strong squeezing of the microwave field and robust optomechanical entanglement can be achieved in the presence of moderate therma…
We determine the complete set of generalized spin squeezing inequalities. These are entanglement criteria that can be used for the experimental detection of entanglement in a system of spin-1/2 particles in which the spins cannot be individually addressed. They can also be used to show the presence of bound entanglement in the thermal states of several spin models.
We show that spin squeezing implies pairwise entanglement for arbitrary symmetric multiqubit states. If the squeezing parameter is less than or equal to 1, we demonstrate a quantitative relation between the squeezing parameter and the concurrence for the even and odd states. We prove that the even states generated from the initial state with all qubits being spin down, via the one-axis twisting Hamiltonian, are spin squeezed if and only if they are pairwise entangled. For the states generated via the one-axis twisting Hamiltonian with an external transverse field for any number of qubits great…
In analogy with the spin-squeezing inequality of Wang and Sanders [Physical Review A 68, 012101 (2003)], we find inequalities describing macroscopic polarization correlations that are obeyed by all classical fields, and whose violation implies entanglement of the photons that make up the optical beam. We consider a realistic and exactly-solvable experimental scenario employing polarization- squeezed light from an optical parametric oscillator (OPO) and find that any two photons separated by less than the OPO coherence time are polarization entangled. The polarization entanglement is robust aga…
We investigate steady-state thermal transport and photon statistics in a nonequilibrium hybrid quantum system, in which a qubit is longitudinally and quadratically coupled to an optical resonator. Our calculations are conducted with the method of the quantum dressed master equation combined with full counting statistics. The effect of negative differential thermal conductance is unravelled at finite temperature bias, which stems from the suppression of cyclic heat transitions and large mismatch of two squeezed photon modes at weak and strong qubit-resonator hybridizations, respectively. The gi…
We establish a connection between quantum inequalities (known from quantum field theory on curved spacetimes) and the degree of squeezing in quantum-optical experiments. We prove an inequality which binds the reduction of the electric-field fluctuations to their duration. The bigger the level of fluctuations-suppression the shorter its duration. As an example of the application of this inequality in the case of squeezed light whose phase is controlled with 1% accuracy we derive a limit of -15dB on the allowed degree of squeezing.
We propose to increase the fidelity of two-qubit resonator-induced phase gates in circuit QED by the use of narrowband single-mode squeezed drive. We show that there exists an optimal squeezing angle and strength that erases qubit 'which-path' information leaking out of the cavity and thereby minimizes qubit dephasing during these gates. Our analytical results for the gate fidelity are in excellent agreement with numerical simulations of a cascaded master equation that takes into account the dynamics of the source of squeezed radiation. With realistic parameters, we find that it is possible to…
We determine the complete set of generalized spin squeezing inequalities, given in terms of the collective angular momentum components, for particles with an arbitrary spin. They can be used for the experimental detection of entanglement in an ensemble in which the particles cannot be individually addressed. We also present a large set of criteria involving collective observables different from the angular momentum coordinates. We show that some of the inequalities can be used to detect k-particle entanglement and bound entanglement.
Entangling gates are an essential component of quantum computers. However, generating high-fidelity gates, in a scalable manner, remains a major challenge in all quantum information processing platforms. Accordingly, improving the fidelity and robustness of these gates has been a research focus in recent years. In trapped ions quantum computers, entangling gates are performed by driving the normal modes of motion of the ion chain, generating a spin-dependent force. Even though there has been significant progress in increasing the robustness and modularity of these gates, they are still sensiti…
We study spin squeezing under non-Markovian channels, and consider an ensemble of $N$ independent spin-1/2 particles with exchange symmetry. Each spin interacts with its own bath, and the baths are independent and identical. For this kind of open system, the spin squeezing under decoherence can be investigated from the dynamics of the local expectations, and the multi-qubit dynamics can be reduced into the two-qubit one. The reduced dynamics is obtained by the hierarchy equation method, which is a exact without rotating-wave and Born-Markov approximation. The numerical results show that the sp…
We propose a feedback control mechanism for the squeezing of the phononic mode of a mechanical oscillator. We show how, under appropriate working conditions, a simple adiabatic approach is able to induce mechanical squeezing. We then go beyond the limitations of such a working point and demonstrate the stationary squeezing induced by using repeated measurements and re-initialisation of the state of a two-level system ancilla coupled to the oscillator. Our non-adaptive feedback loop offers interesting possibilities for quantum state engineering and steering in open-system scenarios.
A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-1/2 particles to entanglement conditions for spin-j particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our …
We describe a simple protocol for the single-step generation of $N$-body entangling interactions between trapped atomic ion qubits. We show that qubit state-dependent squeezing operations and displacement forces on the collective atomic motion can generate full $N$-body interactions. Similar to the Mølmer-Sørensen two-body Ising interaction at the core of most trapped ion quantum computers and simulators, the proposed operation is relatively insensitive to the state of motion. We show how this $N$-body gate operation allows the single-step implementation of a family of $N$-bit gate operations …
We introduce and analyze a dispersive qubit readout scheme where two-mode squeezing is generated directly in the measurement cavities. The resulting suppression of noise enables fast, high- fidelity readout of naturally weakly coupled qubits, and the possibility to protect strongly coupled qubits from decoherence by weakening their coupling. Unlike other approaches exploiting squeezing, our setup avoids the difficult task of transporting and injecting with high fidelity an externally-generated squeezed state. Our setup is also surprisingly robust against unwanted non-QND backaction effects, as…
Spin squeezing inequalities (SSI) represent a major tool to probe quantum entanglement among a collection of few-level atoms, and are based on collective spin measurements and their fluctuations. Yet, for atomic ensembles of spin-$j$ atoms and ultracold spinor gases, many experiments can image the populations in all Zeeman sublevels $s=-j, -j+1, \dots, j$, potentially revealing finer features of quantum entanglement not captured by SSI. Here we present a systematic approach which exploits Zeeman-sublevel population measurements in order to construct novel entanglement criteria, and illustrate …
It is shown that even if the linear entropy of mixed two-qubit state is not smaller then 0.457, Bell - CHSH inequalities can be violated. This contradicts the result obtained in the paper of E. Santos [1].
Quantum entanglement reflects itself through non-local correlations among the subsystems of a quantum system. This thesis focuses on constructing a complete set of local invariants characterizing symmetric two qubit systems and analyzing the pairwise entanglement properties like collective spin squeezing - exhibited by multiqubits - in terms of two qubit entanglement invariants. Further, the collective pairwise entanglement properties of symmetric multiqubit states are shown to be captured entirely by the off diagonal block of the covariance matrix.
Pairwise entanglement properties of a symmetric multi-qubit system are analyzed through a complete set of two-qubit local invariants. Collective features of entanglement, such as spin squeezing, are expressed in terms of invariants and a classifcation scheme for pairwise entanglement is proposed. The invariant criteria given here are shown to be related to the recently proposed (Phys. Rev. Lett. 95, 120502 (2005)) generalized spin squeezing inequalities for pairwise entanglement in symmetric multi-qubit states.
We establish a unified view of polygamy of multi-qubit entanglement. We first introduce a two-parameter generalization of entanglement of assistance namely unified entanglement of assistance for bipartite quantum states, and provide an analytic lowerbound in two-qubit systems. We show a broad class of polygamy inequalities of multi-qubit entanglement in terms of unified entanglement of assistance that encapsulates all known multi-qubit polygamy inequalities as special cases. We further show that this class of polygamy inequalities can be improved into tighter inequalities for three-qubit syste…
We show that a necessary and sufficient condition for a set of $n$ one-qubit mixed states to be the reduced states of a pure $n$-qubit state is that their smaller eigenvalues should satisfy polygon inequalities: no one of them can exceed the sum of the others.
A systematic approach is presented to construct non-homogeneous two- and three-qubit Bell-type inequalities. When projector-like terms are subtracted from homogeneous two-qubit CHSH polynomial, non-homogeneous inequalities are attained and the maximal quantum mechanical violation asymptotically equals a constant with the subtracted terms becoming sufficiently large. In the case of three-qubit system, it is found that most significant three-qubit inequalities presented in literature can be recovered in our framework. We aslo discuss the behavior of such inequalities in the loophole-free Bell te…
In this paper, we show that multi-mode qubit states produced via nonlinear optical state truncation driven by classical external pumpings exhibit squeezing condition. We restrict our discussions to the two and three-mode cases.
The squeezing effect arises in the interacting qubit-oscillator system is studied with the presence of a parametric oscillator in the Rabi model. Based on the generalized rotating wave approximation which works well in the wide range of coupling strength as well as detuning, the analytically derived approximate energy spectrum is compared with the numerically determined spectrum of the Hamiltonian. For the initial state of the bipartite system, the dynamical evolution of the reduced density matrix corresponding to the oscillator is obtained by partial tracing over the qubit degree of freedom. …
A set of Bell inequalities classifying the quantum entanglement of four-qubit states is presented. These inequalities involve only two measurement settings per observer and can characterize fully separable, bi-separable and tri-separable quantum states. In addition, a quadratic inequality of the Bell operators for four-qubit systems is derived.
We compare the polynomial invariants for four qubits introduced by Luque and Thibon, PRA {\bf 67}, 042303 (2003), with optimized Bell inequalities and a combination of two qubit concurrences. It is shown for various parameter dependent states from different SLOCC classes that it is possible to measure a genuine 4-qubit entanglement with these polynomials.
Sufficient conditions for (the non-violation of) the Bell-CHSH inequalities in a mixed state of a two-qubit system are: 1) The linear entropy of the state is not smaller than 0.5, 2) The sum of the conditional linear entropies is non-negative, 3) The von Neumann entropy is not smaller than 0.833, 4) The sum of the conditional von Neumann entropies is not smaller than 0.280.
We present a set of Bell inequalities that gives rise to a finer classification of the entanglement for tripartite systems. These inequalities distinguish three possible bi-separable entanglements for three-qubit states. The three Bell operators we employed constitute an external sphere of the separable cube.
A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. Using the qubit portrait method the Bell inequalities for two qubits and two qutrits are discussed in framework of probability representation of quantum mechanics. Semigroup of stochastic matrices is associated with tomographic probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as ansatz to provide a necessary condition f…
Bell inequalities for number measurements are derived via the observation that the bits of the number indexing a number state are proper qubits. Violations of these inequalities are obtained from the output state of the nondegenerate optical parametric amplifier.
We present a family of Bell inequalities involving only two measurement settings of each party for N>2 qubits. Our inequalities include all the standard ones with fewer than N qubits and thus gives a natural generalization. It is shown that all the Greenberger-Horne-Zeilinger states violate the inequalities maximally, with an amount that grows exponentially as 2^{{(N-2)}/2}. The inequalities are also violated by some states that do satisfy all the standard Bell inequalities. Remarkably, our results yield in an efficient and simple way an implementation of nonlocality tests of many qubits favor…
The celebrated Majorana representation is exploited to investigate spin squeezing in different classes of pure symmetric states of N qubits with two distinct spinors, namely the Dicke-class of states. On obtaining a general expression for spin squeezing parameter, the variation of squeezing for different configurations is studied in detail.It is shown that the states in the Dicke-class, characterized by two-distinct non-orthogonal spinors, exhibit squeezing.
