Entanglement Of Gaussian States Research Articles

Experimental Demonstration of Quantum Steering Swapping with Gaussian Entangled States

AbstractAs a key element in quantum repeaters, entanglement swapping establishes entanglement between two independent entangled states. Besides entanglement, quantum steering has been identified as an essential quantum resource and has broad applications in quantum communication. Although entanglement swapping has been realized experimentally, quantum steering swapping still remains a challenge since the condition of quantum steering is stronger than that of entanglement. In this article, quantum steering swapping between a three‐mode and a two‐mode Gaussian entangled state in a lossy channel is demonstrated. By choosing the optimum gain in the classical channel that corresponds to the maximum transmission distance, the one‐way and two‐way steerabilities in a new three‐mode Gaussian state located in distant nodes are achieved. The obtained collective steerability from two users to one user has potential application in quantum secret sharing. The results make a crucial step toward applications of quantum steering in asymmetric quantum information processing.

Remote Preparation of Optical Cat States Based on Gaussian Entanglement

AbstractRemote state preparation enables one to prepare and manipulate quantum state non‐locally. As an essential quantum resource, optical cat state is usually prepared locally by subtracting photons from a squeezed vacuum state. For remote quantum information processing, it is essential to prepare and manipulate optical cat states remotely based on Gaussian entanglement, which remains a challenge. Here, experimental preparation of optical cat states based on a remotely distributed two‐mode Gaussian entangled state in a lossy channel is presented. By performing photon subtraction and homodyne projective measurement at Alice's station, an optical cat state is prepared remotely at Bob's station. Furthermore, the prepared cat state is rotated by changing Alice's measurement basis of homodyne detection, which demonstrates the remote manipulation of it. By distributing two modes of the two‐mode Gaussian entangled state in lossy channels, it is demonstrated that the remotely prepared cat state can tolerate much more loss in Alice's channel than that in Bob's channel. It is also shown that cat states with amplitudes larger than 2 can be prepared by increasing the squeezing level and subtracting photon numbers. The results make a crucial step toward remote hybrid quantum information processing involving discrete‐ and continuous‐variable techniques.

Circuit Complexity in Z2 EEFT

Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in Z2 Even Effective Field Theories (Z2 EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such as ϕ4, ϕ6, and ϕ8. To facilitate our computation, we regularize the theory by putting it on a lattice. First, we consider a simple case of two oscillators and later generalize the results to N oscillators. This study was carried out for nearly Gaussian states. In our computation, the reference state is an approximately Gaussian unentangled state, and the corresponding target state, calculated from our theory, is an approximately Gaussian entangled state. We compute the complexity using the geometric approach developed by Nielsen, parameterizing the path-ordered unitary transformation and minimizing the geodesic in the space of unitaries. The contribution of higher-order operators to the circuit complexity in our theory is discussed. We also explore the dependency of complexity on other parameters in our theory for various cases.

Noisy propagation of Gaussian states in optical media with finite bandwidth

We address propagation and entanglement of Gaussian states in optical media characterised by nontrivial spectral densities. In particular, we consider environments with a finite bandwidth J(omega ) = J_0 left[ theta (omega -Omega ) - theta (omega - Omega - delta )right] , and show that in the low temperature regime Tll Omega ^{-1}: (i) secular terms in the master equation may be neglected; (ii) attenuation (damping) is strongly suppressed; (iii) the overall diffusion process may be described as a Gaussian noise channel with variance depending only on the bandwidth. We find several regimes where propagation is not much detrimental and entanglement may be protected form decoherence.

Time-dependent quantum teleportation via a parametric converter

Quantum state teleportation is an important protocol that plays a pivotal role in various quantum information tasks. Here we theoretically investigate quantum state teleportation by exploiting a general two-mode Gaussian entangled state produced by a parametric converter when two single-mode Gaussian states (SMGSs) in terms of the non-classicality and purity are employed as inputs to the parametric converter. In particular, the time-dependent teleportation fidelity is analyzed with respect to the squeezing parameter and phase-space quadratures of the teleported squeezed coherent state. We show that the teleportation fidelity is maximal when ratio of the expectation values of photon number in the two modes of the evolved Gaussian entangled state is equal to 1. Quantum state teleportation in terms of the purity and non-classicality of the general two SMGSs seems to be a good choice for experimental realization of quantum communication and information processing.

A local area quantum teleportation network based on an array of electrically activated graphene waveguides.

We present a scheme to generate a continuous variable (CV) multipartite entangled state using an array of plasmonic graphene waveguides that are activated by nonclassical driving microwave modes. Within this scheme, we can exploit the interaction of two light fields coupled to the same microwave mode in each waveguide to produce any type of multipartite Gaussian entangled state. A teleportation network is illustrated using the resultant CV multipartite entangled state. In particular, the proposed setup enables coherent state teleportation across remotely connected nodes with fidelity above a threshold limit of 2/3, providing secure quantum teleportation networking even in the presence of losses.

Experimental Demonstration of Remotely Creating Wigner Negativity via Quantum Steering.

Non-Gaussian states with Wigner negativity are of particular interest in quantum technology due to their potential applications in quantum computing and quantum metrology. However, how to create such states at a remote location remains a challenge, which is important for efficiently distributing quantum resource between distant nodes in a network. Here, we experimentally prepare an optical non-Gaussian state with negative Wigner function at a remote node via local non-Gaussian operation and shared Gaussian entangled state existing quantum steering. By performing photon subtraction on one mode, Wigner negativity is created in the remote target mode. We show that the Wigner negativity is sensitive to loss on the target mode, but robust to loss on the mode performing photon subtraction. This experiment confirms the connection between the remotely created Wigner negativity and quantum steering. As an application, we present that the generated non-Gaussian state exhibits metrological power in quantum phase estimation.

Experimental demonstration of robustness of Gaussian quantum coherence

Besides quantum entanglement and steering, quantum coherence has also been identified as a useful quantum resource in quantum information. It is important to investigate the evolution of quantum coherence in practical quantum channels. In this paper, we experimentally quantify the quantum coherence of a squeezed state and a Gaussian Einstein–Podolsky–Rosen (EPR) entangled state transmitted in Gaussian thermal noise channel. By reconstructing the covariance matrix of the transmitted states, quantum coherence of these Gaussian states is quantified by calculating the relative entropy. We show that quantum coherence of the squeezed state and the Gaussian EPR entangled state is robust against loss and noise in a quantum channel, which is different from the properties of squeezing and Gaussian entanglement. Our experimental results pave the way for application of Gaussian quantum coherence in lossy and noisy environments.

Dissipative Engineering of Gaussian Entangled States in Harmonic Lattices with a Single-Site Squeezed Reservoir.

We study the dissipative preparation of many-body entangled Gaussian states in bosonic lattice models which could be relevant for quantum technology applications. We assume minimal resources, represented by systems described by particle-conserving quadratic Hamiltonians, with a single localized squeezed reservoir. We show that in this way it is possible to prepare, in the steady state, the wide class of pure states which can be generated by applying a generic passive Gaussian transformation on a set of equally squeezed modes. This includes nontrivial multipartite entangled states such as cluster states suitable for measurement-based quantum computation.

Gouy phase of type-I SPDC-generated biphotons

We consider a double Gaussian approximation to describe the wavefunction of twin photons (also called a biphoton) created in a nonlinear crystal via a type-I spontaneous parametric downconversion (SPDC) process. We find that the wavefunction develops a Gouy phase while it propagates, being dependent of the two-photon correlation through the Rayleigh length. We evaluate the covariance matrix and show that the logarithmic negativity, useful in quantifying entanglement in Gaussian states, although Rayleigh-dependent, does not depend on the propagation distance. In addition, we show that the two-photon entanglement can be connected to the biphoton Gouy phase as these quantities are Rayleigh-length-related. Then, we focus the double Gaussian biphoton wavefunction using a thin lens and calculate a Gouy phase that is in reasonable agreement with the experimental data of D. Kawase et al. published in Ref. [1].

Continuous variable quantum teleportation via entangled Gaussian state generated by a linear beam splitter

We theoretically implement the protocol of continuous variable quantum teleportation wherein Alice and Bob share a state built by a linear beam splitter as general two-mode Gaussian entangled state. Using the non-classicality and purity of the two single-mode Gaussian states employed initially at the input of the beam splitter, the unknown state teleportation protocol is explicitly implemented in terms of squeezing and phase space quadratures. We show that fidelity of the teleported state is maximal when the gain factor g is equal to 1 and gradually decreases when g deviates from 1 on either side. We further find that teleportation fidelity decreases when one of the input states of the beam splitter is replaced by the state of thermal photons.

Methods for Derivation of Density Matrix of Arbitrary Multi-Mode Gaussian States from Its Phase Space Representation**Supported by the National Natural Science Foundation of China under Grant Nos 11574400 and 11204379, the Beijing Institute of Technology Research Fund Program for Young Scholars, and the NSFC-ICTP Proposal under Grant No 11981240356.

We present a method for derivation of the density matrix of an arbitrary multi-mode continuous variable Gaussian entangled state from its phase space representation. An explicit computer algorithm is given to reconstruct the density matrix from Gaussian covariance matrix and quadrature average values. As an example, we apply our method to the derivation of three-mode symmetric continuous variable entangled state. Our method can be used to analyze the entanglement and correlation in continuous variable quantum network with multi-mode quantum entanglement states.

Continuous-variable entangled states of light carrying orbital angular momentum

The orbital angular momentum of light, unlike spin, is an infinite-dimensional discrete variable and may hence offer enhanced performances for encoding, transmitting, and processing quantum information. Hitherto, this degree of freedom of light has been studied mainly in the context of quantum states with definite number of photons. On the other hand, field-quadrature continuous-variable quantum states of light allow implementing many important quantum protocols not accessible with photon-number states. Here, we realize a scheme based on a $q$-plate device for endowing a bipartite continuous-variable Gaussian entangled state with nonzero orbital angular momentum. We then apply a reconfigurable homodyne detector working directly with such nonzero orbital angular momentum modes in order to retrieve experimentally their entire quantum-state covariance matrix, thus providing a full characterization of their quantum fluctuation properties. Our work is a step towards generating multipartite continuous-variable entanglement in a single optical beam.

Monogamy relation in multipartite continuous-variable quantum teleportation

Quantum teleportation (QT) is a fundamentally remarkable communication protocol that also finds many important applications for quantum informatics. Given a quantum entangled resource, it is crucial to know to what extent one can accomplish the QT. This is usually assessed in terms of output fidelity, which can also be regarded as an operational measure of entanglement. In the case of multipartite communication when each communicator possesses a part of $N$-partite entangled state, not all pairs of communicators can achieve a high fidelity due to monogamy property of quantum entanglement. We here investigate how such a monogamy relation arises in multipartite continuous-variable (CV) teleportation particularly using a Gaussian entangled state. We show a strict monogamy relation, i.e. a sender cannot achieve a fidelity higher than optimal cloning limit with more than one receiver. While this seems rather natural owing to the no-cloning theorem, a strict monogamy relation still holds even if the sender is allowed to individually manipulate the reduced state in collaboration with each receiver to improve fidelity. The local operations are further extended to non-Gaussian operations such as photon subtraction and addition, and we demonstrate that the Gaussian cloning bound cannot be beaten by more than one pair of communicators. Furthermore we investigate a quantitative form of monogamy relation in terms of teleportation capability, for which we show that a faithful monogamy inequality does not exist.

New formula for calculating the fidelity of teleportation and its applications

Quantum teleportation plays an important role in quantum information science. In order to obtain the effect of quantum teleportation of a quantum state by using an entangled resource, the fidelity of teleporting the quantum state should be calculated. Braunstein and Kimble[Phys. Rev. Lett. 80 869 (1998)] derived a formula of calculating the fidelity of quantum teleportation for Gaussian entangled resource and any input state to be teleported. Then, the point is how to calculate the quantum teleportation fidelity for any entangled resource. In this paper, werealize this purpose by using the entangled state representation. First, we derive the Weyl expansion of any density operator by using the completeness relation between coherent state and P-representation. Then using the orthogonal property of entangled state representation and the traditional Kimble-Braunstein scheme of quantum teleportation, we further derive the mean density operator of the output state, which means that we establish the relation between the output density operator and the characteristic functions of the input state to be teleported and the entangled resources. The characteristic function of the output state is also derived which is in the concise form relating these two characteristic functions above. Then we further obtain a new formula for calculating the quantum teleportation fidelity for the coherent state input and any two-mode entangled resource. It is shown that the fidelity of teleportation can be easily calculated when the Q-function of the normally ordering form of entangled resource is known. This is a convenient way of obtaining the fidelity of teleportation. As its applications, some Gaussian and non-Gaussian entangled states are examined to teleport the coherent state, whose results are correct.

Conservation relation of nonclassicality and entanglement for Gaussian states in a beam splitter

We study the relation between single-mode nonclassicality and two-mode entanglement in a beam splitter. We show that single-mode nonclassicality (the entanglement potential) of incident light cannot be transformed into two-mode entanglement completely after a single beam splitter. Some of the entanglement potential remains as single-mode nonclassicality in the two entangled output modes. Two-mode entanglement generated in the process can be equivalently quantified as an increase in the minimum uncertainty widths (or decrease in the squeezing) of the output states compared to the input states. We use the nonclassical depth and logarithmic negativity as single-mode nonclassicality and entanglement measures, respectively. We realize that a conservation relation between the two quantities can be adopted for Gaussian states, if one works in terms of uncertainty width. This conservation relation is extended to many sets of beam splitters.

Entanglement and separability in the noncommutative phase-space scenario

Quantumness and separability criteria for continuous variable systems are discussed for the case of a noncommutative (NC) phase-space. In particular, the quantum nature and the entanglement configuration of NC two-mode Gaussian states are examined. Two families of covariance matrices describing standard quantum mechanics (QM) separable states are deformed into a NC QM configuration and then investigated through the positive partial transpose criterium for identifying quantum entanglement. It is shown that the entanglement of Gaussian states may be exclusively induced by switching on the NC deformation. Extensions of some preliminary results are presented.

Entanglement due to noncommutativity in phase space

The entanglement criterion for continuous variable systems and the conditions under which the uncertainty relations are fulfilled are generalized to the case of a noncommutative phase space. The quantum nature and the separability of noncommutative two-mode Gaussian states are examined. It is shown that the entanglement of Gaussian states may be exclusively induced by switching on the noncommutative deformation.

Effect of excess noise on continuous variable entanglement sudden death and Gaussian quantum discord

A symmetric two-mode Gaussian entangled state is used to investigate the effect of excess noise on entanglement sudden death and Gaussian quantum discord with continuous variables. The results show that the excess noise in the channel can lead to entanglement sudden death of a symmetric two-mode Gaussian entangled state, while Gaussian quantum discord never vanishes. As a practical application, the security of a quantum key distribution (QKD) scheme based on a symmetric two-mode Gaussian entangled state against collective Gaussian attacks is analyzed. The calculation results show that the secret key cannot be distilled when entanglement vanishes and only quantum discord exists in such a QKD scheme.

Displacement-enhanced entanglement distillation of single-mode-squeezed entangled states

It has been shown that entanglement distillation of Gaussian entangled states by means of local photon subtraction can be improved by local Gaussian transformations. Here we show that a similar effect can be expected for the distillation of an asymmetric Gaussian entangled state that is produced by a single squeezed beam. We show that for low initial entanglement, our largely simplified protocol generates more entanglement than previous proposed protocols. Furthermore, we show that the distillation scheme also works efficiently on decohered entangled states as well as with a practical photon subtraction setup.