J. Sperling, W. Vogel
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.
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The verification of quantum entanglement under the influence of realistic noise and decoherence is crucial for the development of quantum technologies. Unfortunately, a full entanglement characterization is generally not possible with most entanglement criteria such as entanglement witnesses or the partial transposition criterion. In particular, so called bound entanglement cannot be certified via the partial transposition criterion. Here we present the full entanglement verification of dephased qubit and qutrit Werner states via entanglement quasiprobabilities. Remarkably, we are able to reve…
The Hong-Ou-Mandel interference dip is caused by an entangled state, a delocalized bi-photon state. We propose a method of detecting this entanglement by utilizing inverse Hong-Ou-Mandel interference, while taking into account vacuum and multi-photon contaminations, phase noise, and other imperfections. The method uses just linear optics and photodetectors, and for single-mode photodetectors we find a lower bound on the amount of entanglement.
High-order entanglement embedded in multipartite multilevel quantum systems (qudits) with many degrees of freedom (DOFs) plays an important role in quantum foundation and quantum engineering. Verifying high-order entanglement without the restriction of system complexity is a critical need in any experiments on general entanglement. Here, we introduce a scheme to efficiently detect genuine high-order entanglement, such as states close to genuine qudit Bell, Greenberger-Horne-Zeilinger, and cluster states as well as multilevel multi-DOF hyperentanglement. All of them can be identified with two l…
We construct a method for verifying mode entanglement of N-mode W states. The ideal W state contains exactly one excitation symmetrically shared between N modes, but our method takes the existence of higher numbers of excitations into account, as well as the vacuum state and other deviations from the ideal state. Moreover, our method distinguishes between full N-party entanglement and states with M-party entanglement with M<N, including mixtures of the latter. We specialize to the case N=4 for illustrative purposes. In the optical case, where excitations are photons, our method can be implemen…
Quantum entangled states have shown distinguished features beyond any classical state. Many methods like quantum state tomography have been presented to verify entanglement. In this work, we aim to identify unknown entanglements with partial information of the state space by developing a nonlinear entanglement witness. The witness consists of a generalized Greenberger-Horne-Zeilinger-like paradox expressed by Pauli observables, and a nonlinear inequality expressed by density matrix elements. First, we verify unknown bipartite entanglements and study the robustness of entanglement witnesses aga…
We prove experimentally the predicted existence of a three-qubit quantum state with genuine multipartite entanglement which can be certified solely from its separable two-qubit reduced density matrices. The qubits are encoded into different degrees of freedom of a pair of correlated photons and the state is prepared by letting the photons to propagate through a linear optical circuit. The presence of genuine multipartite entanglement is verified by finding numerically a fully decomposable entanglement witness acting nontrivially only on the reductions of the global state. Our result confirms v…
Three different methods have been discussed to verify continuous variable entanglement of intense light beams. We demonstrate all three methods using the same set--up to facilitate the comparison. The non--linearity used to generate entanglement is the Kerr--effect in optical fibres. Due to the brightness of the entangled pulses, standard homodyne detection is not an appropriate tool for the verification. However, we show that by using large asymmetric interferometers on each beam individually, two non-commuting variables can be accessed and the presence of entanglement verified via joint meas…
Recent experiments to test Bell's inequality using entangled photons and ions aimed at tests of basic quantum mechanical principles. Interesting results have been obtained and many loopholes could be closed. In this paper we want to point out that tests of Bell's inequality also play an important role in verifying atom entanglement schemes. We describe as an example a scheme to prepare arbitrary entangled states of N two-level atoms using a leaky optical cavity and a scheme to entangle atoms inside a photonic crystal. During the state preparation no photons are emitted and observing a violatio…
The ability to generate and verify multipartite entanglement is an important benchmark for near-term quantum devices devices. We develop a scalable entanglement metric based on multiple quantum coherences, and demonstrate experimentally on a 20-qubit superconducting device - the IBM Q System One. We report a state fidelity of 0.5165$\pm$0.0036 for an 18-qubit GHZ state, indicating multipartite entanglement across all 18 qubits. Our entanglement metric is robust to noise and only requires measuring the population in the ground state; it can be readily applied to other quantum devices to verify …
Verifying Energy-Time Entanglement with Irregularly Sampled Correlations
针对不规则滤波器采样下的能量-时间纠缠验证问题，提出由测量数据构建不高估真实关联的连续变量概率密度，实现无需态或设备假设的完整认证。
By combining a squeezed propagating microwave field and an unsqueezed vacuum field on a hybrid (microwave beam-splitter), we generate entanglement between the two output modes. We verify that we have generated entangled states by making independent and efficient single-quadrature measurements of the two output modes. We observe the entanglement witness $E_\mathrm{W}=-0.263^{+0.001}_{-0.036}$ and the negativity $N=0.0824^{+0.01}_{-0.0004}$ with measurement efficiencies at least $26\pm{0.1}\%$ and $41\pm{0.2}\%$ for channel~1 and 2 respectively. These measurements show that the output two-mode s…
Entanglement is at the heart of most quantum information tasks, and therefore considerable effort has been made to find methods of deciding the entanglement content of a given bipartite quantum state. Here, we prove a fundamental limitation to deciding if an unknown state is entangled or not: we show that any quantum measurement which can answer this question necessarily gives enough information to identify the state completely. Therefore, only prior information regarding the state can make entanglement detection less expensive than full state tomography in terms of the demanded quantum resour…
Entanglement verification and measurement is essential for experimental tests of quantum mechanics and also for quantum communication and information science. Standard methods of verifying entanglement in a bipartite mixed state require detection of both particles and involve coincidence measurement. We present a method that enables us to verify and measure entanglement in a two-photon mixed state without detecting one of the photons, i.e., without performing any coincidence measurement or postselection. We consider two identical sources, each of which can generate the same two-photon mixed st…
Witnessing the Effective Entanglement in the COW Protocol
本文提出了一种双参数族的纠缠见证者，用于验证COW协议中的有效纠缠，并通过实验数据展示了其检测能力。
We apply the approach of verifying entanglement, which is based on the sole knowledge of the dimension of the underlying physical system to the entanglement based version of the BB84 quantum key distribution protocol. We show that the familiar one-way key rate formula holds already if one assumes the assumption that one of the parties is measuring a qubit and no further assumptions about the measurement are needed.
Measurements provide a novel mechanism for generating the entanglement resource necessary for performing scalable quantum computation. Recently, we proposed a method for performing parity measurements in a coupled quantum dot system. In this paper we generalise this scheme and perform a comprehensive analytic and numerical study of environmental factors. We calculate the effects of possible error sources including non-ideal photon detectors, ineffective spin-selective excitation and dot distinguishability (both spatial and spectral). Furthermore, we present an experimental approach for verifyi…
Verifying the security of a continuous variable quantum communication protocol via quantum metrology
本研究建立了量子计量学与安全通信的联系，提出三方感知协议并映射为安全访问与通信协议，证明多方协同工作能获得高安全性资产并实现比单独工作更高的互信息。
Entanglement is a unique nature of quantum theory and has tremendous potential for application. Nevertheless, the complexity of quantum entanglement grows exponentially with an increase in the number of entangled particles. Here we introduce a quantum state concentration scheme which decomposes the multipartite entangled state into a set of bipartite and tripartite entangled states. It is shown that the complexity of the entanglement induced by the large number of particles is transformed into the high dimensions of bipartite and tripartite entangled states for pure quantum systems. The result…
Entanglement, one of the most intriguing aspects of quantum mechanics, marks itself into different features of quantum states. For this reason different criteria can be used for verifying entanglement. In this paper we review some of the entanglement criteria casted for continuous variable states and link them to peculiar aspects of the original debate on the famous EPR paradox. Moreover, we give a handy expression for valuating Bell-type non-locality on Gaussian states. We also present the experimental measurement of a particular realization of the Bell operator over continuous variable entan…
Continuous weak measurement allows localizing open quantum systems in state space, and tracing out their quantum trajectory as they evolve in time. Efficient quantum measurement schemes have previously enabled recording quantum trajectories of microwave photon and qubit states. We apply these concepts to a macroscopic mechanical resonator, and follow the quantum trajectory of its motional state conditioned on a continuous optical measurement record. Starting with a thermal mixture, we eventually obtain coherent states of 78% purity--comparable to a displaced thermal state of occupation 0.14. W…
Local unitary equivalence of arbitrary-dimensional multipartite quantum states
本文基于广义 Bloch 表示的系数矩阵构造任意维度两体及多体量子态的局域酉不变量，并将其用于验证局域酉等价和估计并发度。
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.
Single-photon entangled states, i.e. states describing two optical paths sharing a single photon, constitute the simplest form of entanglement. Yet they provide a valuable resource in quantum information science. Specifically, they lie at the heart of quantum networks, as they can be used for quantum teleportation, swapped and purified with linear optics. The main drawback of such entanglement is the difficulty in measuring it. Here, we present and experimentally test an entanglement witness allowing one not only to say whether a given state is path-entangled but also that entanglement lies in…
This paper has been withdrawn.
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 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.
Identifying genuine entanglement of lossy noisy very large scale continuous variable Greenberger-Horne-Zeilinger state
本文提出基于不确定性关系和符号矩阵技术的通用高效框架，用于检测高损耗噪声环境下非常大规模连续变量多体纠缠状态的真实纠缠。
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…
Scalable multipartite entanglement criteria for continuous variables
针对连续变量系统，提出基于匹配纠缠见证与积分不确定性关系的通用多方纠缠检测判据，利用协方差矩阵可快速检测各类纠缠。
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.
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