We prove that it is impossible to distill more entanglement from a single copy of a two-mode bipartite entangled Gaussian state via LOCC Gaussian operations. More generally, we show that any hypothetical distillation protocol for Gaussian states involving only Gaussian operations would be a deterministic protocol. Finally, we argue that the protocol considered by Eisert et al. [quant-ph/0204052] is the optimum Gaussian distillation protocol for two copies of entangled Gaussian 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 s…
Magic can be distributed non-locally in many-body entangled states, such as the low energy states of condensed matter systems. Using the Bravyi-Kitaev magic state distillation protocol, we find that non-local magic is distillable and can improve the distillation outcome. We analyze a few explicit examples and show that spin squeezing can be used to convert non-distillable states into distillable ones. Our analysis also suggests that the conventional product input states assumed by magic distillation protocols are extremely atypical among general states with distillable magic. It further justif…
We construct multipartite entangled states with underlying W-type structure satisfying positive partial transpose (PPT) condition under any (N-1)|1 partition. Then we show how to distill N-partite secure key form the states using two different methods: direct application of local filtering and novel random key distillation scheme in which we adopt the idea form recent results on entanglement distillation. Open problems and possible implications are also discussed.
We propose an entanglement distillation and purification scheme for symmetric two-mode entangled Gaussian states that allows to asymptotically extract a pure entangled Gaussian state from any input entangled symmetric Gaussian state. The proposed scheme is a modified and extended version of the entanglement distillation protocol originally developed by [Browne et al., Phys. Rev. A 67, 062320 (2003)]. A key feature of the present protocol is that it utilizes a two-copy de-Gaussification procedure that involves a Mach-Zehnder interferometer with single-mode non-Gaussian filters inserted in its t…
We propose a scheme for distillation of free bipartite entanglement from bipartite bound-entangled states. The crucial element of our scheme is an ancillary system that is coupled to the initial bound-entangled state via appropriate weak measurements. We show that in this protocol free entanglement can be always generated with nonzero probability by using a single copy of the bound-entangled state. We also derive a lower bound on the entanglement cost of the protocol, and conclude that, on average, applying weaker measurements results in relatively higher values of free entanglement as well as…
In this paper, we propose two general entanglement distillation protocols, which can concentrate the non-maximally entangled pure W class state. The general protocols are mainly based on the unitary transformation on the auxiliary particle and one of the three entangled particles, and in the second protocol, the entanglement distillation includes two meanings, namely, extracting the concentrated tripartite entangled W state and obtaining the maximally entangled bipartite state from the garbage state, which gives no contribution to the distillation of non-maximally entangled pure W class state.…
We present a way for the entanglement distillation of genuine mixed state. Different from the conventional mixed state in entanglement purification protocol, each components of the mixed state in our protocol is a less-entangled state, while it is always a maximally entangled state. With the help of the weak cross-Kerr nonlinearity, this entanglement distillation protocol does not require the sophisticated single-photon detectors. Moreover, the distilled high quality entangled state can be retained to perform the further distillation. These properties make it more convenient in practical appli…
The distribution of entangled states between distant parties in an optical network is crucial for the successful implementation of various quantum communication protocols such as quantum cryptography, teleportation and dense coding [1-3]. However, owing to the unavoidable loss in any real optical channel, the distribution of loss-intolerant entangled states is inevitably inflicted by decoherence, which causes a degradation of the transmitted entanglement. To combat the decoherence, entanglement distillation, which is the process of extracting a small set of highly entangled states from a large…
Entanglement distillation refers to the task of transforming a collection of weakly entangled pairs into fewer highly entangled ones. It is a core ingredient in quantum repeater protocols, needed to transmit entanglement over arbitrary distances in order to realise quantum key distribution schemes. Usually, it is assumed that the initial entangled pairs are i.i.d. distributed and uncorrelated with each other, an assumption that might not be reasonable at all in any entanglement generation process involving memory channels. Here, we introduce a framework that captures entanglement distillation …
We discuss a specific entanglement distillation scheme under the constraint of finite samples of entangled qubit pairs. It is shown that an iterative process can be explicitly formulated. The average fidelity of this process can be enhanced by introducing conditional storing of entangled qubit pairs in each step of the iteration. We investigate the corresponding limitations on the size and the initial fidelity of the sample.
We present a practical entanglement distillation protocol (EDP) with linear optics for a genuine mixed state. Each components of the genuine mixed state is a pure less-entangled state. After successfully performing this EDP, we can obtain a high quality entangled mixed state. Our EDP can work for both ideal entanglement sources and current available spontaneous parametric down-conversion (SPDC) sources, which makes it feasible in current experimental technology. Moreover, by using the SPDC source, we can obtain a higher fidelity. This protocol can also be used to distill the multi-partite enta…
Entanglement distillation is a procedure for extracting from one or more pairs of entangled qubits a smaller number of pairs with a higher degree of entanglement that is essential for many applications in quantum information science. Schmidt projection is a unique distillation method that can be applied to more than two pairs of entangled qubits at the same time with high efficiency. We implement the Schmidt projection protocol by applying single-photon two-qubit quantum logic to a partially hyperentangled four-qubit input state. The basic elements of the procedure are characterized and we con…
Entanglement distillation is a fundamental building block in long-distance quantum communication. Though known to be useless on their own for distilling Gaussian entangled states, local Gaussian operations may still help to improve non-Gaussian entanglement distillation schemes. Here we show that by applying local squeezing operations, both the performance and the efficiency of existing distillation protocols can be enhanced. We derive the optimal enhancement through local Gaussian unitaries, which can be obtained even in the most natural scenario when Gaussian mixed entangled states are share…
We put forward a method for optimized distillation of partly entangled pairs of qubits into a smaller number of more entangled pairs by recurrent local unitary operations and projections. Optimized distillation is achieved by minimization of a cost function with up to 30 real parameters, which is chosen to be sensitive to the fidelity and the projection probability at each step. We show that in many cases this approach can significantly improve the distillation efficiency in comparison to the present methods.
In this paper, we first propose a general entanglement distillation protocol for three-particle W class state, which can concentrate the state (non-maximally entangled W state). The general protocol is mainly based on the unitary transformations on the auxiliary particles and the entangled particles, and a feasible physical scheme is suggested based on the cavity QED techniques. The protocol and scheme can be extended to the entanglement concentration of N-particle case.
Methods for distilling maximally entangled tripartite (GHZ) states from arbitrary entangled tripartite pure states are described. These techniques work for virtually any input state. Each technique has two stages which we call primary and secondary distillation. Primary distillation produces a GHZ state with some probability, so that when applied to an ensemble of systems, a certain percentage is discarded. Secondary distillation produces further GHZs from the discarded systems. These protocols are developed with the help of an approach to quantum information theory based on absolutely selecti…
The potential impact of future quantum networks hinges on high-quality quantum entanglement shared between network nodes. Unavoidable real-world imperfections necessitate means to improve remote entanglement by local quantum operations. Here we realize entanglement distillation on a quantum network primitive of distant electron-nuclear two-qubit nodes. We demonstrate the heralded generation of two copies of a remote entangled state through single-photon-mediated entangling of the electrons and robust storage in the nuclear spins. After applying local two-qubit gates, single-shot measurements h…
Atomic ensembles are effective memory nodes for quantum communication network due to the long coherence time and the collective enhancement effect for the nonlinear interaction between an ensemble and a photon. Here we investigate the possibility of achieving the entanglement distillation for nonlocal atomic ensembles by the input-output process of a single photon as a result of cavity quantum electrodynamics. We give an optimal entanglement concentration protocol (ECP) for two-atomic-ensemble systems in a partially entangled pure state with known parameters and an efficient ECP for the system…
A recently proposed purification method, in which the Zeno-like measurements of a subsystem can bring about a distillation of another subsystem in interaction with the former, is utilized to yield entangled states between distant systems. It is shown that the measurements of a two-level system locally interacting with other two spatially separated not coupled subsystems, can distill entangled states from the latter irrespectively of the initial states of the two subsystems.
The distribution of entangled states of light over long distances is a major challenge in the field of quantum information. Optical losses, phase diffusion and mixing with thermal states lead to decoherence and destroy the non-classical states after some finite transmission-line length. Quantum repeater protocols, which combine quantum memory, entanglement distillation and entanglement swapping, were proposed to overcome this problem. Here we report on the experimental demonstration of entanglement distillation in the continuous-variable regime. Entangled states were first disturbed by random …
Entanglement distillation, an essential quantum information processing task, refers to the conversion from multiple copies of noisy entangled states to a smaller number of highly entangled states. In this work, we study the non-asymptotic fundamental limits for entanglement distillation. We investigate the optimal tradeoff between the distillation rate, the number of prepared states, and the error tolerance. First, we derive the one-shot distillable entanglement under completely positive partial transpose preserving operations as a semidefinite program and demonstrate an exact characterization…
The quantum states corresponding to a secret key are characterized using the so-called private states, where the key part consisting of a secret key is shielded by the additional systems. Based on the construction, it was shown that a secret key can be distilled from bound entangled states. In this work, I consider the shielded two-qubit states in a key-distillation scenario and derive the conditions under which a secret key can be distilled using the recurrence protocol or the two-way classical distillation, advantage distillation together with one-way postprocessing. From the security condit…
We present a detailed theoretical analysis for the distillation of one copy of a mixed two-mode continuous-variable entangled state using beamsplitters and coherent photon-detection techniques, including conventional on-off detectors and photon number resolving detectors. The initial Gaussian mixed-entangled states are generated by transmitting a two-mode squeezed state through a lossy bosonic channel, corresponding to the primary source of errors in current approaches to optical quantum communication. We provide explicit formulas to calculate the entanglement in terms of logarithmic negativit…
Quantum entanglement plays crucial roles in quantum information processing. Quantum entangled states have become the key ingredient in the rapidly expanding field of quantum information science. Although the nonclassical nature of entanglement has been recognized for many years, considerable efforts have been taken to understand and characterize its properties recently. In this review, we introduce some recent results in the theory of quantum entanglement. In particular separability criteria based on the Bloch representation, covariance matrix, normal form and entanglement witness; lower bound…
It is known that from entangled states that have positive partial transpose it is not possible to distill maximally entangled states by local operations and classical communication (LOCC). A long-standing open question is whether maximally entangled states can be distilled from every state with a non-positive partial transpose. In this paper we study a possible approach to the question consisting of enlarging the class of operations allowed. Namely, instead of LOCC operations we consider k-extendible operations, defined as maps whose Choi-Jamiolkowski state is k-extendible. We find that this c…
Entanglement distillation aims at preparing highly entangled states out of a supply of weakly entangled pairs, using local devices and classical communication only. In this note we discuss the experimentally feasible schemes for optical continuous-variable entanglement distillation that have been presented in [D.E. Browne, J. Eisert, S. Scheel, and M.B. Plenio, Phys. Rev. A 67, 062320 (2003)] and [J. Eisert, D.E. Browne, S. Scheel, and M.B. Plenio, Annals of Physics (NY) 311, 431 (2004)]. We emphasize their versatility in particular with regards to the detection process and discuss the merits …
Efficiency is a key issue in any real implementation of a cryptographic protocol since the physical resources are not unlimited. We will first show that Quantum Key Distribution is possible with an "Entanglement based" scheme with NPPT symmetric Gaussian states in spite of the fact that these systems cannot be distilled with Gaussian operations (they are all bound entangled). In this work we analyze the secrecy properties of Gaussian states under Gaussian operations. Although such operations are useless for quantum distillation, we prove that it is possible to distill efficiently a secret key …
We show that bipartite and tripartite entangled states cannot be used as catalysis states to enable local transformations inbetween inequivalent classes of three-particle entangled states which are non-interchangeable under local transformations. We find the optimal protocol for conversion of a certain family of the W-states and an EPR-pair into the GHZ-state.
We describe various results related to the random distillation of multiparty entangled states - that is, conversion of such states into entangled states shared between fewer parties, where those parties are not predetermined. In previous work [Phys. Rev. Lett. 98, 260501 (2007)] we showed that certain output states (namely Einstein-Podolsky-Rosen (EPR) pairs) could be reliably acquired from a prescribed initial multipartite state (namely the W state) via random distillation that could not be reliably created between predetermined parties. Here we provide a more rigorous definition of what cons…
We construct new entanglement distillation protocols by interpolating between the recurrence and hashing protocols. This leads to asymptotic two-way distillation protocols, resulting in an improvement of the distillation rate for all mixed Bell diagonal entangled states, even for the ones with very high fidelity. We also present a method how entanglement-assisted distillation protocols can be converted into non-entanglement-assisted protocols with the same yield.
Based on the proceding Letter [Int. J. Theor. Phys. 48, 1539 (2009)], we expand the relation between wavelet transformation and Husimi distribution function to the entangled case. We find that the optical complex wavelet transformation can be used to study the entangled Husimi distribution function in phase space theory of quantum optics. We prove that the entangled Husimi distribution function of a two-mode quantum state |phi> is just the modulus square of the complex wavelet transform of exp{-(|eta|^2)/2} with phi(eta)being the mother wavelet up to a Gaussian function.
Blind quantum computation (BQC) provides an efficient method for the client who does not have enough sophisticated technology and knowledge to perform universal quantum computation. The single-server BQC protocol requires the client to have some minimum quantum ability, while the double-server BQC protocol makes the client's device completely classical, resorting to the pure and clean Bell-state shared by two servers. In this paper, we provide a deterministic entanglement distillation protocol in a practical noisy environment for the double-server BQC protocol. This protocol can obtain the pur…
As a natural extension of Fan's paper (arXiv: 0903.1769vl [quant-ph]) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation we find new two-fold complex integration transformation about the Wigner operator (in its entangled form) in phase space quantum mechanics and its inverse transformation. In this way, some operator ordering problems can be solved and the contents of phase space quantum mechanics can be enriched.
The goal of entanglement distillation is to turn a large number of weakly entangled states into a smaller number of highly entangled ones. Practical entanglement distillation schemes offer a tradeoff between the fidelity to the target state, and the probability of successful distillation. Exploiting such tradeoffs is of interest in the design of quantum repeater protocols. Here, we present a number of methods to assess and optimize entanglement distillation schemes. We start by giving a numerical method to compute upper bounds on the maximum achievable fidelity for a desired probability of suc…
We show that all pure entangled states of two $d$-dimensional quantum systems (i.e., two qudits) can be generated from an initial separable state via a universal Yang--Baxter matrix if one is assisted by local unitary transformations.
Entanglement distillation transforms weakly entangled noisy states into highly entangled states, a primitive to be used in quantum repeater schemes and other protocols designed for quantum communication and key distribution. In this work, we present a comprehensive framework for continuous-variable entanglement distillation schemes that convert noisy non-Gaussian states into Gaussian ones in many iterations of the protocol. Instances of these protocols include (a) the recursive-Gaussifier protocol, (b) the temporally-reordered recursive-Gaussifier protocol, and (c) the pumping-Gaussifier proto…
The transformations of $W$-type entangled states by using local operations assisted with classical communication are investigated. For this purpose, a parametrization of the $W$-type states which remains invariant under local unitary transformations is proposed and a complete characterization of the local operations carried out by a single party is given. These are used for deriving the necessary and sufficient conditions for deterministic transformations. A convenient upper bound for the maximum probability of distillation of arbitrary target states is also found.
The multipartite entangled state $|p,ξ_2,ξ_2,...,ξ_n>$ for the total momentum and relative coordinates of n particles is constructed. The corresponding quantum mechanical operator with respect to the classical transformation $p\to e^{λ_1} p$, $ξ_i\to e^{λ_i} ξ_i$, $i=1,...,n$, in the state $|p,ξ_2,ξ_2,...,ξ_n>$ is investigated.
A general form of a two-qubit system is obtained under the effect of Lorentz transformation. We investigate extensively some important classes in the context of quantum information. It is shown Lorentz transformation causes a decay of entanglement and consequently information loses. On the other hand, it generates entangled states between systems prepared initially in a separable states. The partial entangled states are more robust under Lorentz transformation than maximally entangled states. Therefore the rate of information lose is larger for maximum entangled states compared with that for p…
In this paper, we proposed a physical scheme to concentrate the non-maximally entangled atomic pure states via cavity QED by using atomic collision in a far-off-resonant cavity. The most distinctive advantage of our scheme is that there is no excitation of cavity mode during the distillation procedure. Therefore the requirement on the quality of cavity is greatly loosened.
Sources of entangled photon pairs using two parametric down-converters are capable of generating interchangeable entanglement in two different degrees of freedom. The connection between these two degrees of freedom allows the control of the entanglement properties of one, by acting on the other degree of freedom. We demonstrate experimentally, the quantum distillation of the position entanglement using polarization analyzers.
We study the secrecy properties of Gaussian states under Gaussian operations. Although such operations are useless for quantum distillation, we prove that it is possible to distill a secret key secure against any attack from sufficiently entangled Gaussian states with non-positive partial transposition. Moreover, all such states allow for key distillation, when Eve is assumed to perform finite-size coherent attacks before the reconciliation process.
Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We …
Local quantum operations relating multiqubit flip (0-1) and exchange symmetric (FES) states, with the maximum possible probability of success, have been determined by assuming that the states are converted via one-shot FES transformations. It has been shown that certain entangled states are more robust than others, in the sense that the optimum probability of converting these robust states to the states lying in the close neighborhood of separable ones vanish under local FES operations.
Transformations involving only local operations assisted with classical communication are investigated for multipartite entangled pure states having tensor rank 2. All necessary and sufficient conditions for the possibility of deterministically converting truly multipartite, rank-2 states into each other are given. Furthermore, a chain of local operations that successfully achieves the transformation has been identified for all allowed transformations. The identified chains have two nice features: (1) each party needs to carry out at most one local operation and (2) all of these local operatio…
Similar in spirit to the preceding work [Opt. Commun. 282 (2009) 3734] where the relation between optical Fresnel transformation and quantum tomography is revealed, we study this kind of relationship in the two-mode entangled case. We show that under the two-mode Fresnel transformation the bipartite entangled state density |eta><eta| becomes density operator F_2|eta><eta|F_2 ^{dag}=|eta>_{r,s}<eta|, which is just the Radon transform of the two-mode Wigner operator (sigma,gama) in entangled form, where F_2 is an two-mode Fresnel operator in quantum optics, and s,r are the complex-value expressi…
Suppose that we have two entangled states $\ket {ϕ_1}$, $\ket{ψ_1}$ that cannot be converted to any of other two states $\ket{ϕ_2}$, $\ket{ψ_2}$ by local operations and classical communication. We analyze the possibility of locally transforming a superposition of $\ket{ϕ_1}$ and $\ket{ψ_1}$ into a superposition of $\ket{ϕ_2}$ and $\ket{ψ_2}$. By using the Nielsen's theorem we find the necessary and sufficient conditions for this conversion to be performed.
We present an optimal scheme to realize the transformations between single copies of two bipartite entangled states without classical communication between the sharing parties. The scheme achieves the upper bound for the success probabilities [PRA 63, 022301 (2001), PRL 83, 1455 (1999)] of generating maximally entangled states if applied to entanglement concentration. Such strategy also dispenses with the interaction with an ancilla system in the implementation. We also show that classical communications are indispensable in realizing the deterministic transformations of a single bipartite ent…
