We present a scheme for multipartite entanglement purification of quantum systems in a Greenberger-Horne-Zeilinger state with quantum nondemolition detectors (QNDs). This scheme does not require the controlled-not gates which cannot be implemented perfectly with linear optical elements at present, but QNDs based on cross-Kerr nonlinearities. It works with two steps, i.e., the bit-flipping error correction and the phase-flipping error correction. These two steps can be iterated perfectly with parity checks and simple single-photon measurements. This scheme does not require the parties to possess sophisticated single photon detectors. These features maybe make this scheme more efficient and feasible than others in practical applications.
In tasks, where multipartite entanglement plays a central role, state purification is, due to inevitable noise, a crucial part of the procedure. We consider a scenario exploiting the multipartite entanglement in a straightforward multipartite purification algorithm and compare it to bipartite purification procedures combined with state teleportation. While complete purification requires an infinite amount of input states in both cases, we show that for an imperfect output fidelity the multipartite procedure exhibits a major advantage in terms of input states used.
We present a multipartite entanglement purification scheme in a Greenberger-Horne-Zeilinger state for electrons based on their spins and their charges. This scheme works for purification with two steps, i.e., bit-flipping error correction and phase-error flip error correction. By repeating these two steps, the parties in quantum communication can get some high-fidelity multipartite entangled electronic systems.
We describe an entanglement purification protocol (EPP) for multipartite three-level atomic entangled pair using photonic Faraday rotation. In this EPP, the multipartite atomic entangled state can be purified with the help of some single photons. This protocol resorts to the photonic Faraday rotation to realize the function of the controlled-Not (CNOT) gate. The purified multipartite atomic entangled state can be retained and to be repeated to reach a higher fidelity.
A method for producing an upper bound for all multipartite purification protocols is devised, based on knowing the optimal protocol for purifying bipartite states. When applied to a range of noise models, both local and correlated, the optimality of certain protocols can be demonstrated for a variety of graph and valence bond states.
It is shown that ponderomotive force can be used to purify entangled states. The protocol is based on the possibility to exploit such force for a local quantum nondemolition measurement of the total excitation number of continuos variable entangled pairs.
The entanglement resource required for quantum information processing comes in a variety of forms, from Bell states to multipartite GHZ states or cluster states. Purifying these resources after their imperfect generation is an indispensable step towards using them in quantum architectures. While this challenge, both in the case of Bell pairs and more general multipartite entangled states, is mostly overcome in the presence of perfect local quantum hardware with unconstrained qubit register sizes, devising optimal purification strategies for finite-size realistic noisy hardware has remained elu…
We propose an entanglement purification protocol for high-dimensional multipartite systems. In the protocol we can select a subensemble in a pure generalized Greenberger-Horne-Zeilinger (GHZ) state. This post-selection can be made by detecting the noise which contaminated the initial pure ensemble when the systems past through a noisy environment. For the detection of noise we investigate correlation properties of GHZ states and analyze their possible errors due to a noisy environment. We show that the presented protocol is more efficient than a simple generalization of the purification protoc…
Multipartite entanglement purification is revisited by using the Local operations and classical communications(LOCCs). We demonstrate our idea by considering the tripartite case, i.e. the purification of tripartite entanglement. We express the general tripartite entangled states in a special representation of total spin operators $J_{123}^{2}$ and $% J_{12}^{2}$ of tripartite system with eigenvalues 15/4 and 2 respectively. This basis is a \textit{genuine basis} because it consists of of all the genuine entangled states of tripartite system. Our protocol is a recurrence one, and only two copie…
Multipartite entanglement plays an important role in controlled quantum teleportation, quantum secret sharing, quantum metrology and some other important quantum information branches. However, the maximally multipartite entangled state will degrade into the mixed state because of the noise. We present an efficient multipartite entanglement purification protocol (EPP) which can distill the high quality entangled states from low quality entangled states for N-photon systems in a Greenberger-Horne-Zeilinger (GHZ) state in only linear optics. After performing the protocol, the spatial-mode entangl…
We present a protocol in which two or more parties can share multipartite entanglement over noisy quantum channels. The protocol is based on the entanglement purification presented by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)] and the quantum teleportation via an isotropic state. We show that a nearly perfect purification implies a nearly perfect sharing of multipartite entanglement between two parties so that the protocol can assure a faithful sharing of multipartite entanglement with Shor and Preskill's proof on the entanglement purification.
Record Loss Sets a Rare-Trajectory Limit on Quantum Purification
School of Instrumentation and Optoelectronic Engineering, Beihang University（北京航空航天大学仪器科学与光电工程学院）；Institute of Large-Scale Scientific Facility, Beihang University（北京航空航天大学大型科学装置研究院）；Hangzhou Institute of Extremely-Weak Magnetic Field Major National Science and Technology Infrastructure（杭州极弱磁场重大科技基础设施研究院）；Hangzhou Innovation Institute, Beihang University（北京航空航天大学杭州创新研究院）
We present a high-efficiency multipartite entanglement purification protocol (MEPP) for electron-spin systems in a Greenberger-Horne-Zeilinger state based on their spins and their charges. Our MEPP contains two parts. The first part is our normal MEPP with which the parties can obtain a high-fidelity N-electron ensemble directly, similar to the MEPP with controlled-not gates. The second one is our recycling MEPP with entanglement link from N'-electron subsystems (2 < N' < N). It is interesting to show that the N'-electron subsystems can be obtained efficiently by measuring the electrons with p…
We present an efficient multipartite entanglement purification protocol (MEPP) for N-photon systems in a Greenberger-Horne-Zeilinger state with parity-check detectors. It contains two parts. One is the conventional MEPP with which the parties can obtain a high-fidelity N-photon ensemble directly, similar to the MEPP with controlled-not gates. The other is our recycling MEPP in which the entanglement link is used to produce some $N$-photon entangled systems from entangled N'-photon subsystems (2 \leq N'<N) coming from the instances which are just discarded in all existing conventional MEPPs. Th…
We introduce the distribution of a secret multipartite entangled state in a real-world scenario as a quantum primitive. We show that in the presence of noisy quantum channels (and noisy control operations) any state chosen from the set of two-colorable graph states (CSS codewords) can be created with high fidelity while it remains unknown to all parties. This is accomplished by either blind multipartite entanglement purification, which we introduce in this paper, or by multipartite entanglement purification of enlarged states, which offers advantages over an alternative scheme based on standar…
We give a review on entanglement purification for bipartite and multipartite quantum states, with the main focus on theoretical work carried out by our group in the last couple of years. We discuss entanglement purification in the context of quantum communication, where we emphasize its close relation to quantum error correction. Various bipartite and multipartite entanglement purification protocols are discussed, and their performance under idealized and realistic conditions is studied. Several applications of entanglement purification in quantum communication and computation are presented, w…
We introduce a new information-theoretic measure of multipartite quantum/classical correlations $Δ_P$, by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT corresponden…
Locally maximally entangleable states (LMESs) constitute a large set of multipartite states, containing for instance all stabilizer states. LMESs are uniquely characterized by (2n-1) phases, where n denotes the number of qubits. We consider here those LMES whose phases are either 0 or π and present a multipartite entanglement purification protocol for arbitrary such states. In contrast to all previously known recurrence protocols this protocol uses a novel ingredient, which is required due to the quantum correlations contained in the various LMESs. We compare this scheme to previously known en…
In this work we generalize the entanglement of purification and its conjectured holographic dual to conditional and multipartite versions of the same, where the optimization defining the entanglement of purification is now optimized in either a constrained way or over multiple parties. We separately derive new constraints on both the conditional entanglement of purification and its conjectured holographic dual object that match, further reinforcing the likelihood of this conjecture. We also show that the multipartite objects we define, despite obeying several of the same inequalities, are not …
We present a theoretical proposal for a physical implementation of entanglement concentration and purification protocols for two-mode squeezed microwave photons in circuit quantum electrodynamics (QED). First, we give the description of the cross-Kerr effect induced between two resonators in circuit QED. Then we use the cross-Kerr media to design the effective quantum nondemolition (QND) measurement on microwave-photon number. By using the QND measurement, the parties in quantum communication can accomplish the entanglement concentration and purification of nonlocal two-mode squeezed microwave…
We study the preparation and distribution of high-fidelity multi-party entangled states via noisy channels and operations. In the particular case of GHZ and cluster states, we study different strategies using bipartite or multipartite purification protocols. The most efficient strategy depends on the target fidelity one wishes to achieve and on the quality of transmission channel and local operations. We show the existence of a crossing point beyond which the strategy making use of the purification of the state as a whole is more efficient than a strategy in which pairs are purified before the…
Hyperentanglement is a promising resource in quantum information processing, especially for increasing the channel capacity of long-distance quantum communication. Hyperentanglement purification is an important method to obtain high-fidelity nonlocal hyperentangled states from mixed hyperentangled states in a long-distance quantum communication process with noisy channels. Here, we present a two-step hyperentanglement purification protocol for nonlocal mixed hyperentangled states with polarization bit-flip errors and spatial-mode phase-flip errors, resorting to polarization-spatial phase-check…
This paper will address the question of the distillation of entanglement from a finite number of multi-partite mixed states. It is shown that if one can distill a pure entangled state from n copies of a mixed state $σ_{ABC...}$ there must be at least a subspace in whole Hilbert space of the all copies such that the projection of $σ_{ABC...}^{\otimes n}$ onto the subspace is a pure entangled state. We also show that the purification of entanglement or distillation of entanglement can be carried out by local joint projective measurements with the help of classical communication and local general…
Any bipartite entanglement witness $W$ can be written as $W=c_σI-σ$, where $σ$ is a quantum state, $I$ is the identity matrix, and $c_σ$ is a non-negative number. We present a general method to extend the given entanglement witness to multipartite cases via purification, partial purification, and direct tensor of the quantum state $σ$. Our methods extend $σ$ but leave the parameter $c_σ$ untouched. This is very valuable since the parameter is generally not easy to compute.
In [J. C. Howell and J. A. Yeazell, Phys. Rev. A 62, 012102 (2000)], a proposal is made to generate entangled macroscopically distinguishable states of two spatially separated traveling optical modes. We model the decoherence due to light scattering during the propagation along an optical transmission line and propose a setup allowing an entanglement purification from a number of preparations which are partially decohered due to transmission. A purification is achieved even without any manual intervention. We consider a nondemolition configuration to measure the purity of the state as contrast…
We present a way for entanglement purification based on two parametric down-conversion (PDC) sources with cross-Kerr nonlinearities. It is comprised of two processes. The first one is a primary entanglement purification protocol for PDC sources with nondestructive quantum nondemolition (QND) detectors by transferring the spatial entanglement of photon pairs to their polarization. In this time, the QND detectors act as the role of controlled-not (CNot) gates. Also they can distinguish the photon number of the spatial modes, which provides a good way for the next process to purify the entangleme…
Recently, two-photon six-qubit hyperentangled states were produced in experiment and they can improve the channel capacity of quantum communication largely. Here we present a scheme for the hyperentanglement purification of nonlocal two-photon systems in three degrees of freedom (DOFs), including the polarization, the first-longitudinal-momentum, and the second longitudinal momentum DOFs. Our hyperentanglement purification protocol (hyper-EPP) is constructed with two steps resorting to parity-check quantum nondemolition measurement on the three DOFs and SWAP gates, respectively. With these two…
We investigate measurement-based entanglement purification protocols (EPP) in the presence of local noise and imperfections. We derive a universal, protocol-independent threshold for the required quality of the local resource states, where we show that local noise per particle of up to 24% is tolerable. This corresponds to an increase of the noise threshold by almost an order of magnitude, based on the joint measurement-based implementation of sequential rounds of few-particle EPP. We generalize our results to multipartite EPP, where we encounter similarly high error thresholds.
We propose two schemes for concentration of hyperentanglement of nonlocal multipartite states which are simultaneously entangled in the polarization and spatial modes. One scheme uses an auxiliary singlephoton state prepared according to the parameters of the less-entangled states. The other scheme uses two less-entangled states with unknown parameters to distill the maximal hyperentanglement. The procrustean concentration is realized by two parity check measurements in both the two degrees of freedom. Nondestructive quantum nondemolition detectors based on cross-Kerr nonlinearity are used to …
As the hyperentanglement of photon systems presents lots of unique opportunities in high-capacity quantum networking, the hyperentanglement purification protocol (hyper-EPP) becomes a vital project work and the quality of its accomplishment attracts much attention recently. Here we present the first theoretical scheme of faithful hyper-EPP for nonlocal two-photon systems in two degrees of freedom (DOFs) by constructing several fidelity-robust quantum circuits for hyper-encoded photons. With this faithful hyper-EPP, the bit-flip errors in both the polarization and spatial-mode DOFs can be effic…
Efficiency-Induced Freezing in Quantum-State Purification
School of Instrumentation and Optoelectronic Engineering, Beihang University（北京航空航天大学仪器科学与光电工程学院）；Institute of Large-Scale Scientific Facility, Beihang University（北京航空航天大学大型科学仪器中心）；Hangzhou Institute of Extremely-Weak Magnetic Field Major National Science and Technology Infrastructure（杭州极弱磁场重大科技基础设施研究所）；Hangzhou Innovation Institute, Beihang University（北京航空航天大学杭州创新研究院）
Microwave photons have become very important qubits in quantum communication as the first quantum satellite has been launched successfully. Therefore, it is a necessary and meaningful task for ensuring the high security and efficiency of microwave-based quantum communication in practice. Here, we present an original polarization entanglement purification protocol for nonlocal microwave photons based on the cross-Kerr effect in circuit quantum electrodynamics (QED). Our protocol can solve the problem that the purity of maximally entangled states used for constructing quantum channels will decre…
We analyze the description of quantum many-body mixed states using matrix product states and operators. We consider two such descriptions: (i) as a matrix product density operator of bond dimension D, and (ii) as a purification that is written as a matrix product state of bond dimension D'. We show that these descriptions are inequivalent in the sense that D' cannot be upper bounded by D only. Then we provide two constructive methods to obtain (ii) out of (i). The sum of squares (sos) polynomial method scales exponentially in the number of different eigenvalues, and its approximate version is …
We show that pure states of multipartite quantum systems are multiseparable (i.e. give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the quantification of multipartite pure-state entanglement are discussed. Further, as an application of the techniques used here, we show that any purification of a bipartite PPT bound entangled state is tri-inseparable, i.e. has none of its three bipartite partial traces separable.
Based on quantum complementary relations (QCRs) and a purification scenario, we analyze a class of N-qubit mixed states that are entangled but do not have two-, and genuine three-, four-, ..., N-qubit entanglements. It is shown that entanglement (one-tangle or negativity) in these mixed states is closely related to the QCR entanglement of their purified states. In particular, it is elaborated that when the mixed state does not have multipartite tangles (two- and higher tangles), its entanglement is actually a kind of genuine multipartite QCR entanglement between the system and its environment.
We present a method for multipartite entanglement purification of any stabilizer state shared by several parties. In our protocol each party measures the stabilizer operators of a quantum error-correcting code on his or her qubits. The parties exchange their measurement results, detect or correct errors, and decode the desired purified state. We give sufficient conditions on the stabilizer codes that may be used in this procedure and find that Steane's seven-qubit code is the smallest error-correcting code sufficient to purify any stabilizer state. An error-detecting code that encodes two qubi…
We explore ways to quantify multipartite correlations, in quantum information and in holography. We focus on optimized correlation measures, linear combinations of entropies minimized over all possible purifications of a state that satisfy monotonicity conditions. These contain far more information about correlations than entanglement entropy alone. We present a procedure to derive such quantities, and construct a menagerie of symmetric optimized correlation measures on three parties. These include tripartite generalizations of the entanglement of purification, the squashed entanglement, and t…
We present two economical one-step error-correction protocols for multipartite polarization-entangled systems in a Greenberger-Horne-Zeilinger state. One uses spatial entanglement to correct errors in the polarization entanglement of an N-photon system, resorting to linear optical elements. The other uses frequency entanglement to correct errors in the polarization entanglement of an N-photon system. The parties in quantum communication can obtain a maximally entangled state from each N-photon system transmitted with one step in these two protocols, and both of their success probabilities are …
We introduce a repeater scheme to efficiently distribute multipartite entangled states in a quantum network with optimal scaling. The scheme allows to generate graph states such as 2D and 3D cluster states of growing size or GHZ states over arbitrary distances, with a constant overhead per node/channel that is independent of the distance. The approach is genuine multipartite, and is based on the measurement-based implementation of multipartite hashing, an entanglement purification protocol that operates on a large ensemble together with local merging/connection of elementary building blocks. W…
We study the quantification of genuine multipartite entanglement (GME) for general multipartite states. A set of inequalities satisfied by the entanglement of $N$-partite pure states is derived by exploiting the restrictions on entanglement distributions, showing that the bipartite entanglement between each part and its remaining ones cannot exceed the sum of the other partners with their remaining ones. Then a series of triangles, named concurrence triangles, are established corresponding to these inequalities. Proper genuine multipartite entanglement measures are thus constructed by using th…
We study bipartite quantum discord as a manifestation of a multipartite entanglement structure in the tripartite purified system. In particular, we find that bipartite quantum discord necessarily manifests itself in the presence of both bipartite and tripartite entanglement in the purification. This allows one to understand the asymmetry of quantum discord (between D[A,B] and D[B,A]) in terms of entanglement monogamy. As instructive special cases, we study discord for qubits and Gaussian states in detail. As a result of this we shed new light on a counterintuitive property of Gaussian states: …
Informational Mpemba Effect for Fast State Purification in Non-Hermitian System
Institute of Atomic and Molecular Sciences, Academia Sinica（中央研究院原子与分子科学研究所）；Department of Physics, National Central University（国立中央大学物理系）；Department of Physics, National Taiwan Normal University（国立台湾师范大学物理系）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
Optimising multipartite entanglement manipulation in quantum networks
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Delft University of Technology（代尔夫特理工大学）；QuTech（奎泰克）
We introduce a new multipartite communication scheme, with the aim to enable the senders to remotely and obliviously provide the receivers with an arbitrary amount of multipartite entanglement. The scheme is similar to Remote State Preparation (RSP). However, we show that even though the receivers are restricted to local unitary operations, the required resources for remote entanglement preparation are less than for RSP. In order to do so we introduce a novel canonical form of arbitrary multipartite pure states describing an arbitrary number of qubits. Moreover, we show that if the receivers a…
We discuss two methods that, through a combination of cyclically gluing copies of a given $n$-party boundary state in AdS/CFT and a canonical purification, creates a bulk geometry that contains a boundary homologous minimal surface with area equal to 2 or 4 times the $n$-party entanglement wedge cross-section, depending on the parity of the party number and choice of method. The areas of the minimal surfaces are each dual to entanglement entropies that we define to be candidates for the $n$-party reflected entropy. In the context of AdS$_3$/CFT$_2$, we provide a boundary interpretation of our …
The concepts of quantum correlation complexity and quantum communication complexity were recently proposed to quantify the minimum amount of resources needed in generating bipartite classical or quantum states in the single-shot setting. The former is the minimum size of the initially shared state $σ$ on which local operations by the two parties (without communication) can generate the target state $ρ$, and the latter is the minimum amount of communication needed when initially sharing nothing. In this paper, we generalize these two concepts to multipartite cases, for both exact and approximat…
Quantum computing is among the most far-reaching technologies of the 21st century, tackling challenges at the cutting edge of physics. This new paradigm in computer science harnesses quantum entanglement, one striking non-intuitive feature of quantum mechanics and a cornerstone of quantum information, to provide computation with a quantum speed-up over the best-known classical algorithms and to enable encrypted data communication against eavesdropping. The bulk of this article is focused on providing a deep and abiding understanding of non-relativistic matrix mechanics by demonstrating the fun…
The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states $σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}$, where $|χ_{1}\rangle=α|0\rangle+β|1\rangle$, $|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle$. We have shown that the state $σ_{AB}$ represent a classical state when $n$ is odd while it represent a non-classical state when $n$ is even. The purification of the state $σ_{AB}$ is studied and found that the purification is possible if the spectral decomposit…
We consider the problem of generating multipartite entangled states in a quantum network upon request. We follow a top-down approach, where the required entanglement is initially present in the network in form of network states shared between network devices, and then manipulated in such a way that the desired target state is generated. This minimizes generation times, and allows for network structures that are in principle independent of physical links. We present a modular and flexible architecture, where a multi-layer network consists of devices of varying complexity, including quantum netw…
