Preparation of distilled and purified continuous-variable entangled states

How to concentrate non-maximally entangled states for quantum communication is a fundamental problem in quantum information. In this paper, we will apply generalized measurements to entanglement concentration of known non-maximally entangled pure states in arbitrary dimensional system. How to design the generalized measurements for the unambiguous discrimination of linearly independent non-orthogonal states is crucial for the concentration of the known non-maximally entangled states. The result shows that, any known non-maximally entangled pure state (for arbitrary dimensional system) can be transformed to the maximally entangled state only by introducing a qubit as ancilla and a joint unitary transformation operation on one of the entangled particles and the ancilla. In addition, because the less entangled state of each fail round will be re-concentrated too, the entanglement waste during the concentration process will be greatly reduced.

We propose an optimal entanglement concentration protocol (ECP) for nonlocal \(N\)-electron systems in a partially entangled Greenberger–Horne–Zeilinger (GHZ) pure state, resorting to charge detection and the projection measurement on an additional electron. For each nonlocal \(N\)-electron system in a partially entangled GHZ state, one party in quantum communication, say Alice first entangles it with an additional electron, and then, she projects the additional electron into an orthogonal basis for dividing the \(N\)-electron systems into two groups. In the first group, the \(N\) parties obtain a subset of \(N\)-electron systems in a maximally entangled state directly. In the second group, they obtain some less-entangled \(N\)-electron systems which are the resource for the entanglement concentration in the next round. By iterating the entanglement concentration process several times, the present ECP has the maximal success probability, the theoretical limit of an ECP as it just equals to the entanglement of the partially entangled state, far higher than others. Moreover, this ECP for an \(N\)-electron GHZ-type state requires only one additional electron, not two or more, and it does not resort to a collective unitary evolution, far different from others, which may decrease the difficulty for its implementation in experiment. When it is used for an \(N\)-electron W-type state, \(N-1\) additional electrons are required only.

We propose an optimal nonlocal entanglement concentration protocol (ECP) for\nmulti-photon systems in a partially entangled pure state, resorting to the\nprojection measurement on an additional photon. One party in quantum\ncommunication first performs a parity-check measurement on her photon in an\nN-photon system and an additional photon, and then she projects the additional\nphoton into an orthogonal Hilbert space for dividing the original $N$-photon\nsystems into two groups. In the first group, the N parties will obtain a subset\nof $N$-photon systems in a maximally entangled state. In the second group, they\nwill obtain some less-entangled N-photon systems which are the resource for the\nentanglement concentration in the next round. By iterating the entanglement\nconcentration process several times, the present ECP has the maximal success\nprobability which is just equivalent to the entanglement of the partially\nentangled state. That is, this ECP is an optimal one.\n

We address the degradation of continuous variable (CV) entanglement distillation in the presence of thermal environment and nonunity detecting efficiency. We explore the logarithmic negativity as a criteria and compare the output entanglement between the schemes of single-photon subtraction (PS) and two-PS. The probability of success is also investigated. We show that compared with detector efficiency, thermal noise plays a more dominant role in Gaussian entanglement distillation. Our results shed more light on the noise mechanism that influences realistic CV entanglement distillation and is of experimental relevance in developing a more efficient entanglement distillation scheme.

We consider one-way quantum state merging and entanglement distillation under compound and arbitrarily varying source models. Regarding quantum compound sources, where the source is memoryless, but the source state an unknown member of a certain set of density matrices, we continue investigations begun in the work of Bjelaković et al. [“Universal quantum state merging,” J. Math. Phys. 54, 032204 (2013)] and determine the classical as well as entanglement cost of state merging. We further investigate quantum state merging and entanglement distillation protocols for arbitrarily varying quantum sources (AVQS). In the AVQS model, the source state is assumed to vary in an arbitrary manner for each source output due to environmental fluctuations or adversarial manipulation. We determine the one-way entanglement distillation capacity for AVQS, where we invoke the famous robustification and elimination techniques introduced by Ahlswede. Regarding quantum state merging for AVQS we show by example that the robustification and elimination based approach generally leads to suboptimal entanglement as well as classical communication rates.

Different mathematics forms of entangled state can judge prepared many particles is if entangled. Entangled state is a key role of quantum information technology. The more entangled states are generated, the better performances are presented. In order to make it more convenient to generate entangled state or judge the state prepared is entangled, we investigate three-qubit GHZ state and expand recurrence relation based on an expand recurrence relation of multi-qubit state. Furthermore, new forms of maximally five and seven-qubit entangled states are reconstructed via three-qubit GHZ state. These new forms stand for universal law of n-qubit and [Formula: see text]-qubit can have direct application in quantum entangle state generation and quantum communications.

New approach to the fidelity of quantum teleportation based on the entangled state generated by beam splitter

Quantum entanglement has a central role in many areas of physics. To grasp the essence of this phenomenon, it is fundamental to understand how different manifestations of entanglement relate to each other. In 1999, Peres conjectured that Bell nonlocality is equivalent to distillability of entanglement. The intuition of Peres was that the non-classicality of an entangled state, as witnessed via Bell inequality violation, implies that pure entanglement can be distilled from this state, hence making it useful for quantum information protocols. Subsequently, the Peres conjecture was shown to hold true in several specific cases, and became a central open question in quantum information theory. Here we disprove the Peres conjecture by showing that an undistillable bipartite entangled state--a bound entangled state--can violate a Bell inequality. Hence Bell nonlocality implies neither entanglement distillability, nor non-positivity under partial transposition. This clarifies the relation between three fundamental aspects of entanglement.

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 bounds, subadditivity property of concurrence and tangle, fully entangled fraction related to the optimal fidelity of quantum teleportation, and entanglement distillation will be discussed in detail.

We propose and analyze a multifunctional setup consisting of high-finesse optical cavities, beam splitters, and phase shifters. The basic scheme projects arbitrary photonic two-mode input states onto the subspace spanned by the product of Fock states $|n\ensuremath{\rangle}|n\ensuremath{\rangle}$ with $n=0,1,2,\dots{}.$ This protocol does not only provide the possibility to conditionally generate highly entangled photon number states as resource for quantum information protocols but also allows one to test and hence purify this type of quantum states in a communication scenario, which is of great practical importance. The scheme is especially attractive as a generalization to many modes allows for distribution and purification of entanglement in networks. In an alternative working mode, the setup allows for quantum nondemolition number resolved photodetection in the optical domain.

Field-orthogonal temporal mode analysis of optical fields was recently developed to form a new framework of quantum information science. But so far, the exact profiles of the temporal modes are not known, which makes it difficult to achieve mode selection and de-multiplexing. A novel feedback-iteration method which, combined with the stimulated emission method, can give rise to the exact forms of the temporal mode structure of pulse-pumped spontaneous parametric processes both for high gain parametric process, which gives rise to quantum entanglement in continuous variables, and for the low gain case, which produces a two-photon entangled state for discrete variables.For the temporal mode analysis in high gain situations, the common treatment of parametric interaction Hamiltonian does not consider the issue of time ordering problem of interaction Hamiltonian and thus leads to the inaccurate conclusion that the mode structure and the temporal mode functions do not change as the gain increases. We use an approach that is usually employed for treating nonlinear interferometers and avoids the time ordering issue. This allows us to derive an evolution equation in differential-integral form. Numerical solutions for high gain situations indicate a gain-dependent mode structure that has its mode distributions changed and mode functions broadened as the gain increases. This will enable us to have a complete picture of the mode structure of parametric processes and produce high quality quantum sources for a variety of applications of quantum technology.To verify the feedback-iteration method which measures temporal mode structure directly, we measure the joint spectral density of photon pairs produced with the spontaneous parametric down-conversion process of a pulse-pumped PPKTP crystal. The measurement method is based on a stimulated emission process which significantly improves the measurement time and accuracy compared with old spectrally resolved photon coincidence measurement. With the measured joint spectral density, the amplitude of the temporal modes can be obtained with the mathematical tool of singular value decomposition and compared with those measured directly with the feedback-iteration method.Because the parametric amplifier is in essence a linear four-port device, it couples and linearly mixes two inputs before amplifying and sending them to two output ports. We show that for quadrature phase amplitudes, a parametric amplifier can replace beam splitters to play the role of mixer. We apply this idea to a continuous-variable quantum state teleportation scheme in which a parametric amplifier replaces a beam splitter in the Bell measurement. We show that this scheme is loss-tolerant in the Bell measurement process and thus demonstrate the advantage of parametric amplifiers over beam splitter in the applications in quantum measurement.

Entanglement is essential for quantum information processing, but is limited by noise. We address this by developing high-yield entanglement distillation protocols with several advancements. (1) We extend the 2-to-1 recurrence entanglement distillation protocol to higher-rate n-to-(n−1) protocols that can correct any single-qubit errors. These protocols are evaluated through numerical simulations focusing on fidelity and yield. We also outline a method to adapt any classical error-correcting code for entanglement distillation, where the code can correct both bit-flip and phase-flip errors by incorporating Hadamard gates. (2) We propose a constant-depth decoder for stabilizer codes that transforms logical states into physical ones using single-qubit measurements. This decoder is applied to entanglement distillation protocols, reducing circuit depth and enabling protocols derived from high-performance quantum error-correcting codes. We demonstrate this by evaluating the circuit complexity for entanglement distillation protocols based on surface codes and quantum convolutional codes. (3) Our stabilizer entanglement distillation techniques advance quantum computing. We propose a fault-tolerant protocol for constant-depth encoding and decoding of arbitrary states in surface codes, with potential extensions to more general quantum low-density parity-check codes. This protocol is feasible with state-of-the-art reconfigurable atom arrays and surpasses the limits of conventional logarithmic depth encoders. Overall, our study integrates stabilizer formalism, measurement-based quantum computing, and entanglement distillation, advancing both quantum communication and computing.

Multi-copy nested entanglement purification for quantum repeaters

Einstein-Podolsky-Rosen (EPR)-entangled state light field at a telecommunication wavelength of 1.5 μm is an important quantum source for realizing the continuous variable quantum information processing and some quantum protocols over optical fiber channel. When the EPR-entangled state light field is distributed over the optical fiber channel, the disentanglement is always present because the the EPR entangled state interacts with the fiber channel. It affects the performance of quantum information processing. In this paper, we theoretically calculate the positive partial transposition (PPT) of the entangled state distributed over the optical fiber channel in the single-channel and dual-channel distribution scheme, respectively. Three types of initial entangled light field are considered and analyzed, they being an initial EPR entangled state, an EPR entangled state with asymmetric quadratures, and an EPR entangled state with asymmetric modes. Furthermore, the influence of the extra noise in the optical fiber on the transmission distance of EPR entangled state over the optical fiber channel is investigated. In the single-channel scheme or dual-channel scheme, the extra noise in the optical fiber channel leads the entangled state light field to be disentangled, and the transmission distance of EPR entangled state over the optical fiber channel to decrease rapidly with the increase of the extra noise. For maintaining the robustness of EPR entangled states in lossy optical fiber channels, the dual-channel scheme has more stringent requirements for the correlation quadrature symmetry and purity of the initial entangled state than the single-channel scheme. In the single fiber noise channel scheme, the maximum transmission distance and the robustness of the EPR entangled states with asymmetric modes are not sensitive to the asymmetry between modes. The change of asymmetry between modes does not lead to being disentangled. The maximum transmission distance does not change either. However, the decrease of asymmetry between modes results in the disentanglement in the double fiber noise channels’ scheme. The maximum transmission distance is reduced and the sudden death occurs to the entanglement. The present results will lay a foundation for continuous variables quantum information processing based on optical fiber, such as realizing continuous variables quantum communication over optical fiber and constructing metropolitan quantum network over optical fiber.

The distribution of maximal entanglement stands as a pivotal technique in long-distance practical quantum communication. Specifically, achieving high-fidelity entanglement distribution leans upon efficacious entanglement purification and entanglement concentration. In this paper, we materialize error-rejecting nonlocal entanglement purification protocol (EPP) and entanglement concentration protocol (ECP) for solid-state emitter systems assisted by the scattering property of photon-emitter in one-dimensional waveguide. The parity-check gates of both the EPP and ECP can herald the faulty events with the help of the detectors that click, so they are accomplished, in principle, with unity fidelities, which are conducive to implement two protocols. Moreover, additional emitters are unnecessary except a single photon.