A new simple entanglement measurement method is proposed for the bright EPR beams generated from a non-degenerate optical parametric amplifier operating at deamplification. Due to the output signal and idler modes are frequency degenerate and in phase, the needed phase shift of interference for the measurement of the correlated phase quadratures and anti-correlated amplitude quadratures can be accomplished by a quarter-wave plate and a half wave plate without separating the signal and idler beam. Therefore, phase locking and local oscillators are avoided.
We study the loss of entanglement of bipartite state subjected to discarding or measurement of one qubit. Examining the behavior of different entanglement measures, we find that entanglement of formation, entanglement cost, and logarithmic negativity are lockable measures in that it can decrease arbitrarily after measuring one qubit. We prove that any convex and asymptotically non-continuous measure is lockable. As a consequence, all the convex roof measures can be locked. Relative entropy of entanglement is shown to be a non-lockable measure.
Recently, we have shown how the phase of an electromagnetic field can be determined by measuring the population of either of the two states of a two-level atomic system excited by this field, via the so-called Bloch-Siegert oscillation resulting from the interference between the co- and counter-rotating excitations. Here, we show how a degenerate entanglement, created without transmitting any timing signal, can be used to teleport this phase information. This phase-teleportation process may be applied to achieve frequency-locking of remote oscillators, thereby facilitating the process of synch…
Type II optical parametric oscillators are amongst the highest-quality sources of quantum-correlated light. In particular, when pumped above threshold, such devices generate a pair of bright orthogonally-polarized beams with strong continuous-variable entanglement. However, these sources are of limited practical use, because the entangled beams emerge with different frequencies and a diffusing phase-difference. It has been proven that the use of an internal wave-plate coupling the modes with orthogonal polarization is capable of locking the frequencies of the emerging beams to half the pump fr…
We investigate quantum properties of phase-locked light beams generated in a nondegenerate optical parametric oscillator (NOPO) with an intracavity waveplate. This investigation continuous our previous analysis presented in Phys.Rev.A 69, 05814 (2004), and involves problems of continuous-variable quadrature entanglement in the spectral domain, photon-number correlations as well as the signatures of phase-locking in the Wigner function. We study the role of phase-localizing processes on the quantum correlation effects. The peculiarities of phase-locked NOPO in the self-pulsing instability opera…
Quantum correlations are investigated theoretically in a two-spin system with the dipole-dipole interactions in the NMR multiple-pulse spin-locking experiments. We consider two schemes of the multiple-pulse spin-locking. The first scheme consists of $π/2$-pulses only and the delays between the pulses can differ. The second scheme contains $φ$-pulses ($0<φ<π$) and has equal delays between them. We calculate entanglement for both schemes for an initial separable state. We show that entanglement is absent for the first scheme at equal delays between $π/2$-pulses at arbitraty temperatures. Entangl…
The phenomenon of nonlocality without entanglement(NLWE) arises in discriminating multi-party quantum separable states. Recently, it has been found that the post-measurement information about the prepared subensemble can lock or unlock NLWE in minimum-error discrimination of non-orthogonal separable states. Thus, it is natural to ask whether the availability of the post-measurement information can influence on the occurrence of NLWE even in other state-discrimination strategies. Here, we show that the post-measurement information can be used to lock as well as unlock the occurrence of NLWE in …
Squashed entanglement and entanglement of purification are quantum mechanical correlation measures and defined as certain minimisations of entropic quantities. We present the first non-trivial calculations of both quantities. Our results lead to the conclusion that both measures can drop by an arbitrary amount when only a single qubit of a local system is lost. This property is known as "locking" and has previously been observed for other correlation measures, such as the accessible information, entanglement cost and the logarithmic negativity. In the case of squashed entanglement, the results…
Phase-locking governs the phase noise in classical clocks through effects described in precise mathematical terms. We seek here a quantum counterpart of these effects by working in a finite Hilbert space. We use a coprimality condition to define phase-locked quantum states and the corresponding Pegg-Barnett type phase operator. Cyclotomic symmetries in matrix elements are revealed and related to Ramanujan sums in the theory of prime numbers. The employed mathematical procedures also emphasize the isomorphism between algebraic number theory and the theory of quantum entanglement
We study the Wigner function of phase-locked nondegenerate optical parametric oscillator and find the signatures of both phase-locking and self-pulsing phenomena in phase space. We also analyze the problem of continuous-variable entanglement in the self-pulsing instability regime.
The paper concerns spontaneous asymptotic phase-locking and synchronization in two-qubit systems undergoing continuous Markovian evolution described by Lindbladian dynamics with normal Lindblad operators. Using analytic methods, all phase-locking-enforcing mechanisms within the given framework are obtained and classified. Detailed structures of their respective attractor spaces are provided and used to explore their properties from various perspectives. Amid phase-locking processes those additionally enforcing identical stationary parts of both qubits are identified, including as a special cas…
Optical approaches to quantum computation require the creation of multi-mode photonic quantum states in a controlled fashion. Here we experimentally demonstrate phase locking of two all-optical quantum memories, based on a concatenated cavity system with phase reference beams, for the time-controlled release of two-mode entangled single-photon states. The release time for each mode can be independently determined. The generated states are characterized by two-mode optical homodyne tomography. Entanglement and nonclassicality are preserved for release-time differences up to 400 ns, confirmed by…
We experimentally demonstrated that the quantum correlations of amplitude and phase quadratures between signal and idler beams produced from a non-degenerate optical parametric amplifier (NOPA) can be significantly improved by using a mode cleaner in the pump field and reducing the phase fluctuations in phase locking systems. Based on the two technical improvements the quantum entanglement measured with a two-mode homodyne detector is enhanced from ~ 4 dB to ~ 6 dB below the quantum noise limit using the same NOPA and nonlinear crystal.
Recently, we have shown theoretically [1] as well as experimentally [2] how the phase of an electromagnetic field can be determined by measuring the population of either of the two states of a two-level atomic system excited by this field, via the so-called Bloch-Siegert oscillation resulting from the interference between the co- and counter-rotating excitations. Here, we show how a degenerate entanglement, created without transmitting any timing signal, can be used to teleport this phase information. This phase-teleportation process may be applied to achieve wavelength teleportation, which in…
We present the design and experimental proof of principle of a low threshold optical parametric oscillator (OPO) that continuously oscillates over a large bandwidth allowed by phase matching. The large oscillation bandwidth is achieved with a selective two-photon loss that suppresses the inherent mode competition, which tends to narrow the bandwidth in conventional OPOs. Our design performs pairwise mode-locking of many frequency pairs, in direct equivalence to passive mode-locking of ultrashort pulsed lasers. The ability to obtain high powers of continuous \textit{and} broadband down-converte…
We develop theoretical and numerical tools for the quantification of entanglement in systems with continuous degrees of freedom. Continuous variable entanglement swapping is introduced and based on this idea we develop methods of entanglement purification for continuous variable systems. The success of these entanglement purification methods is then assessed using these tools.
We give a protocol and criteria for demonstrating unconditional teleportation of continuous-variable entanglement (i.e., entanglement swapping). The initial entangled states are produced with squeezed light and linear optics. We show that any nonzero entanglement (any nonzero squeezing) in both of two entanglement sources is sufficient for entanglement swapping to occur. In fact, realization of continuous-variable entanglement swapping is possible using only {\it two} single-mode squeezed states.
Entanglement detection and estimation are fundamental in quantum information science. Compared with discrete-variable states, for which lots of efficient entanglement detection criteria and lower bounds of entanglement measures have been proposed, the continuous-variable entanglement is much less understood. Here we shall present a family of entanglement witnesses based on continuous-variable local orthogonal observables (CVLOOs) to detect and estimate entanglement of Gaussian and non-Gaussian states, especially for bound entangled states. By choosing an optimal set of CVLOOs our entanglement …
Entanglement is a crucial resource in quantum information theory. We investigate the use of different forms of entangled states in continuous variable quantum teleportation, specifically the use of a finite-basis entanglement resource. We also consider the continuous variable teleportation of finite-basis states, such as qubits, and present results that point to the possibility of an efficient conditional scheme for continuous variable teleportation of such states with near-unit fidelity using finite-basis entanglement.
This article discusses entanglement between two subsystems, one with discrete degrees of freedom and the other with continuous degrees of freedom. The overlap integral between continuous variable wave functions emerges as an important parameter to characterize this kind entanglement. ``Beam-like'' entanglement and ``shape-like'' entanglement are contrasted. One example of this kind of entanglement is between between the spin degrees of freedom and the momentum degrees of freedom for a non-relativistic particle. This intraparticle entanglement is Galilean invariant.
We theoretically show that bright six-partite continuous-variable entanglement can be generated using cascaded four-wave mixing effects of third-order nonlinearity atomic systems above threshold. The six-partite continuous-variable entanglement among the six cavity fields with different frequencies is analyzed by applying optimized inseparability criteria proposed by Van Loock and Furusawa.
Starting from arbitrary Hilbert spaces, we reduce the problem to verify entanglement of any bipartite quantum state to finite dimensional subspaces. Hence, entanglement is a finite dimensional property. A generalization for multipartite quantum states is also given.
We present a model to realize quantum feedback control of continuous variable entanglement. It consists of two interacting bosonic modes subject to amplitude damping and achieving entangled Gaussian steady state. The possibility to greatly improve the degree of entanglement by means of Markovian (direct) feedback is then shown.
We suggest an experimentally realizable scheme to test entanglement of a mixed Gaussian continuous variable state. We find that the entanglement condition is simplified for the family of Gaussian states which are relevant to experimental realization. The entanglement condition is then shown to be directly related to joint homodyne measurements. We show how robust the proposed test of entanglement is against imperfect detection efficiency.
We explore the possibility of entanglement detection in continuous variable systems by entanglement witnesses based on covariance matrices, constructible from random homodyne measurements. We propose new linear constraints characterizing the entanglement witnesses based on second moments, and use them in a semidefinite program providing the optimal entanglement test for given random measurements. We test the method on the class of squeezed vacuum states and study the efficiency of entanglement detection in general unknown covariance matrices.
We introduce a multi-mode squeezing coefficient to characterize entanglement in $N$-partite continuous-variable systems. The coefficient relates to the squeezing of collective observables in the $2N$-dimensional phase space and can be readily extracted from the covariance matrix. Simple extensions further permit to reveal entanglement within specific partitions of a multipartite system. Applications with nonlinear observables allow for the detection of non-Gaussian entanglement.
We investigate how entanglement can be transferred between continuous variable and qubit systems. We find that a two-mode squeezed vacuum state and a continuous variable Werner state can be converted to the product states of infinitive number of two-qubit states while keeping the entanglement. The reverse process is also possible.
We propose the definition of the geometric measure of entanglement for continuous variable states. On the basis of this definition we examine entanglement of the graph states obtained as a result of action of a unitary operator on the ground state of a system of $N$ noninteracting harmonic oscillators. We find that the entanglement of a harmonic oscillator with other ones is defined by the value of its vertex degree.
We provide a method to construct entanglement criteria for arbitrary multipartite systems of discrete or continuous variables and hybrid combinations of both. While any set of local operators generates a sufficient condition for entanglement of arbitrary quantum states, a suitable set leads to a necessary and sufficient criterion for pure states. The criteria are readily implementable with existing technology and reveal entanglement that remains undetected by the respective state-of-the-art methods for discrete and continuous variables.
We present several entanglement conditions in order to detect bound entangled states in continuous variable systems. Specifically, Werner and Wolf [Phys. Rev. Lett. 86, 3658 (2001)] and Horodecki and Lewenstein [Phys. Rev. Lett. 85, 2657 (2000)] have proposed examples of bound entangled Gaussian state and bound entangled non-Gaussian state, respectively, of which entanglement can be detected by using our entanglement conditions.
The cavity-optomechanical radiation pressure interaction provides the means to create entanglement between a mechanical oscillator and an electromagnetic field interacting with it. Here we show how we can utilize this entanglement within the framework of time-continuous quantum control, in order to engineer the quantum state of the mechanical system. Specifically, we analyze how to prepare a low-entropy mechanical state by (measurement-based) feedback cooling operated in the blue detuned regime, the creation of bipartite mechanical entanglement via time-continuous entanglement swapping, and pr…
We study a proposal of generating macroscopic continuous-variable entanglement with two coupled waveguides respectively carrying optical damping and optical gain. Moreover, a squeezing element is added into one or both waveguides. We show that quantum noise effect existing in the process is essential to the degree of the generated entanglement. It will totally eliminate the entanglement in the setup of adding the squeezing element into the waveguide filled with optical damping material, but will not completely damp the entanglement to zero in the other configurations of having the squeezing el…
We propose a method to improve the performance of two entanglement-based continuous-variable quantum key distribution protocols using noiseless linear amplifiers. The two entanglement-based schemes consist of an entanglement distribution protocol with an untrusted source and an entanglement swapping protocol with an untrusted relay. Simulation results show that the noiseless linear amplifiers can improve the performance of these two protocols, in terms of maximal transmission distances, when we consider small amounts of entanglement, as typical in realistic setups.
We demonstrate the principle of one-sided device-independent continuous variable (CV) quantum information. In situations of no trust, we show by enactment how the use of standard CV entanglement criteria can mislead Charlie into thinking Alice and Bob share entanglement, when the data is actually generated classically using a Local Hidden Variable theory based on the Wigner function. We distinguish between criteria that demonstrate CV entanglement, and criteria that demonstrate the CV Einstein-Podolsky-Rosen (EPR) steering paradox. We show that the latter, but not the former, are necessarily o…
We report the first experiment of continuous-variable measurement-device-independent quantum key distribution that enables secret key generation at a symbol rate of 5 MBaud without frequency and optical phase locking. This is achieved by using a new relay structure based on a polarization-based 90-degree optical hybrid and a well-designed DSP pipeline.
We report the experimental transformation of quadrature entanglement between two optical beams into continuous variable polarization entanglement. We extend the inseparability criterion proposed by Duan, et al. [Duan00] to polarization states and use it to quantify the entanglement between the three Stokes operators of the beams. We propose an extension to this scheme utilizing two quadrature entangled pairs for which all three Stokes operators between a pair of beams are entangled.
We discuss how continous-variable quantum states such as coherent states and two-mode squeezed states can be encoded in phase-reference independent ways.
We observed continuous-variable entanglement between the bright beams emitted above threshold by an ultrastable optical parametric oscillator, classically phase-locked at a frequency difference of 161.8273240(5) MHz. The amplitude-difference squeezing is -3 dB and the phase-sum one is -1.35 dB. Besides proving entanglement in a new physical system, the phase-locked OPO, such unprecedented frequency-difference stability paves the way for transferring entanglement between different optical frequencies and densely implementing continuous-variable quantum information in the frequency domain.
Inspired by the discrete-variable pairwise entanglement, in this work, we in theory analyze the continuous-variable pairwise entanglement between microwave modes based on a hybrid optoelectromechanical system, where the multi-pair microwave superconducting circuits simutaneously interact with each other via a mechanical resonator, which forms a Fabry-Pérot cavity along with a standing mirror. With experimentally reachable parameter settings, wanted entanglement can be acheived when the pair number up to 10, and more is also available, which has the potential to be useful in quantum technologie…
We derive several entanglement criteria for bipartite continuous variable quantum systems based on the Shannon entropy. These criteria are more sensitive than those involving only second-order moments, and are equivalent to well-known variance product tests in the case of Gaussian states. Furthermore, they involve only a pair of quadrature measurements, and will thus should prove extremely useful the experimental identification of entanglement.
In this work we analyse and compare the continuous variable tripartite entanglement available from the use of two concurrent or cascaded $χ^{(2)}$ nonlinearities. We examine both idealised travelling-wave models and more experimentally realistic intracavity models, showing that tripartite entangled outputs are readily producible. These may be a useful resource for such applications as quantum cryptography and teleportation.
We introduce the definition of generic bound entanglement for the case of continuous variables. We provide some examples of bound entangled states for that case, and discuss their physical sense in the context of quantum optics. We rise the question of whether the entanglement of these states is generic. As a byproduct we obtain a new many parameter family of bound entangled states with positive partial transpose. We also point out that the ``entanglement witnesses'' and positive maps revealing the corresponding bound entanglement can be easily constructed.
We investigate the possibility of characterizing two-party entanglement by measuring correlations of Stokes operators in polarized bright light beams. We adapt a general separability criterion to such operators. We then show that entanglement purification can only be singled out for a particular protocol.
The entanglement produced by a bilinear Hamiltonian in continuous variables has been thoroughly studied and widely used. In contrast, the physics of entanglement resulting from nonlinear interaction described by partially degenerate high-order Hamiltonians remains unclear. Here, we derive a hierarchy of sufficient and necessary conditions for the positive-partial-transposition separability of bipartite nonlinear quantum states. The proposed criteria detect the nonpositive-partial-transposition inseparability of higher-order moments of states, which provides a systematic framework for the chara…
We propose a scheme to generate robust stationary continuous-variable entanglement in optomechanical systems, based on geometrical nonlinearity that occurs for large mechanical displacements. Such nonlinearity is often used to correct the dynamics of the systems in the strong coupling regime. It appears that geometrical nonlinearity enhances the entanglement and shifts its maximum towards high detuning values. Using the experimental parameters, we find that such a scheme generates a very robust entanglement against thermal decoherence even at room temperature. Our results show that geometrical…
We introduce a new protocol in continuous variable entanglement distillation that requires only linear temporal and constant physical or spatial resources.Compared with the existing protocols, our protocol uses atom-photon entanglement to substitute for atom-atom entanglement in existing protol, which will not only save the number of quantum memories, but also has the advantages in operation efficiency and temporal saving. The distillation can be realized by a fixed model of iterations, in which only three quantum memories and photon detectors are required.
We derive a family of entanglement criteria for continuous variable systems based on the Rényi entropy of complementary distributions. We show that these entanglement witnesses can be more sensitive than those based on second-order moments, as well as previous tests involving the Shannon entropy [Phys. Rev. Lett. \textbf{103}, 160505 (2009)]. We extend our results to include the case of discrete sampling, and develop another set of entanglement tests using the discrete Tsallis entropy. We provide several numerical results which show that our criteria can be used to identify entanglement in a n…
We propose two probabilistic entanglement concentration schemes for a single copy of two-mode squeezed vacuum state. The first scheme is based on the off-resonant interaction of a Rydberg atom with the cavity field while the second setup involves the cross Kerr interaction, auxiliary mode prepared in a strong coherent state and a homodyne detection. We show that the continuous-variable entanglement concentration allows us to improve the fidelity of teleportation of coherent states.
We describe an entanglement purification protocol to generate maximally entangled states with high efficiencies from two-mode squeezed states or from mixed Gaussian continuous entangled states. The protocol relies on a local quantum non-demolition measurement of the total excitation number of several continuous variable entangled pairs. We propose an optical scheme to do this kind of measurement using cavity enhanced cross--Kerr interactions.
We derive necessary conditions in terms of the variances of position and momentum linear combinations for all kinds of separability of a multi-party multi-mode continuous-variable state. Their violations can be sufficient for genuine multipartite entanglement, provided the combinations contain both conjugate variables of all modes. Hence a complete state determination, for example by detecting the entire correlation matrix of a Gaussian state, is not needed.
Photon subtraction and addition are experimental means of generating non-Gaussian states from Gaussian states. Coherent subtraction or addition is a combination of photon subtractions or additions. The resultant states are quite general non-Gaussian states. The states can be photon number entangled states with arbitrary coefficients. We derive the entanglement conditions for several classes of coherent subtraction or coherent addition bipartite continuous variable states. One of the entanglement conditions is necessary and sufficient.
We derive a hierarchy of separability criteria for multi-mode continuous variable systems. They permit to study in a unified way the k-partite entanglement of broad classes of Gaussian and non- Gaussian states. With specific examples we demonstrate the strength of the criteria, and, we discuss their assessment based on data obtained from Gaussian measurements.
We present experimental demonstration of quadrature and polarization entanglement generated via the interaction between a coherent linearly polarized field and cold atoms in a high finesse optical cavity. The non linear atom-field interaction produces two squeezed modes with orthogonal polarizations which are used to generate a pair of non separable beams, the entanglement of which is demonstrated by checking the inseparability criterion for continuous variables recently derived by Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)] and calculating the entanglement of formation [Giedke et al., Phys…
We address the non-Markovian entanglement dynamics for bimodal continuous variable quantum systems interacting with two independent structured reservoirs. We derive an analytical expression for the entanglement of formation without performing the Markov and the secular approximations. We observe a variety of qualitative features such as entanglement sudden death, dynamical generation, and protection for two types of Lorentzian spectral densities, assuming the two modes initially excited in a twin-beam state. Our quantitative analysis shows that these cases with different reservoir spectrum, th…
