The experimental determination of entanglement is a major goal in the quantum information field. In general the knowledge of the state is required in order to quantify its entanglement. Here we express a lower bound to the robustness of entanglement of a state based only on the measurement of the energy observable and on the calculation of a separability energy. This allows the estimation of entanglement dismissing the knowledge of the state in question.
We show that each entanglement witness detecting given bipartite entangled state provides an estimation of its concurrence. We illustrate our result with several well known examples of entanglement witnesses and compare the corresponding estimation of concurrence with other estimations provided by the trace norm of partial transposition and realignment.
The parameterized entanglement monotone, the $q$-concurrence, is also a reasonable parameterized entanglement measure. By exploring the properties of the $q$-concurrence with respect to the positive partial transposition and realignment of density matrices, we present tight lower bounds of the $q$-concurrence for arbitrary $q\geqslant 2$. Detailed examples are given to show that the bounds are better than the previous ones.
We present a method to estimate entanglement measures in experiments. We show how a lower bound on a generic entanglement measure can be derived from the measured expectation values of any finite collection of entanglement witnesses. Hence witness measurements are given a quantitative meaning without the need of further experimental data. We apply our results to a recent multi-photon experiment [M. Bourennane et al., Phys. Rev. Lett. 92, 087902 (2004)], giving bounds on the entanglement of formation and the geometric measure of entanglement in this experiment.
We give an introduction to Gaussian states and operations. A discussion of the entanglement properties of bipartite Gaussian states in terms of its covariance matrix follows. It is explained how entanglement can be witnessed using feasible measurements, e.g. homodyne measurements. We find that the outcome of such a measurements cannot only witness but also quantify entanglement.
Quantum state teleportation is a protocol where a shared entangled state is used as a quantum channel to transmit quantum information between distinct locations. Here we consider the task of estimating entanglement in teleportation experiments. We show that the data accessible in a teleportation experiment allows to put a lower bound on some entanglement measures, such as entanglement negativity and robustness. Furthermore, we show cases in which the lower bounds are tight. The introduced lower bounds can also be interpreted as quantifiers of the nonclassicality of a teleportation experiment. …
An entanglement concentration scheme for unknown atomic entanglement states is proposed via entanglement swapping in cavity QED. Because the interaction used here is a large-detuned one between two driven atoms and a quantized cavity mode, the effects of the cavity decay and thermal field have been eliminated. These advantages can warrant the experimental feasibility of the current scheme.
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 …
The problem of the estimation of multiple phases (or of commuting unitaries) is considered. This is a sub-model of the estimation of a completely unknown unitary operation where it has been shown in recent works that there are considerable improvements by using entangled input states and entangled measurements. Here it is shown that when estimating commuting unitaries, there is practically no advantage in using entangled input states or entangled measurements.
We explore procedures to detect entanglement of unknown mixed states, which can be experimentally viable. The heart of the method is a hierarchy of simple feasibility problems, which provides sufficient conditions to entanglement. Our numerical investigations indicate that the entanglement is detected with a cost which is much lower than full state tomography. The procedure is applicable to both free and bound entanglement, and involves only single copy measurements.
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.
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…
In quantum information theory, the reliable and effective detection of entanglement is of paramount importance. However, given an unknown state, assessing its entanglement is a challenging task. To attack this problem, we investigate the use of random local measurements, from which entanglement witnesses are then constructed via semidefinite programming methods. We propose a scheme of successively increasing the number of measurements until the presence of entanglement can be unambiguously concluded, and investigate its performance in various examples.
Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement …
Quantum entanglement plays a crucial role in quantum information processing tasks and quantum mechanics, hence quantifying unknown entanglement is a fundamental task. However, this is also challenging, as entanglement cannot be measured by any observables directly. In this paper, we train neural networks to quantify unknown entanglement, where the input features of neural networks are the outcome statistics data produced by locally measuring target quantum states, and the training labels are well-chosen quantities. For bipartite quantum states, this quantity is coherent information, which is a…
An entanglement purification scheme for arbitrary unknown(mixed and pure non-maximally) entangled ionic states is proposed by using linear optical elements. The main advantage of the scheme is that not only two-ion maximally entangled pairs but also four-ion maximally entangled pairs can be extracted from the less entangled pairs. The scheme is within current technology.
We investigate the lower bound obtained from experimental data of a quantum state $ρ$, as proposed independently by Gühne et al. and Eisert et al. for mixed states of three qubits. The measure we consider is the convex-roof extended three-tangle. Our findings highlight an intimate relation to lower bounds obtained recently from so-called characteristic curves of a given entanglement measure. We apply the bounds to estimate the three-tangle present in recently performed experiments aimed at producing a three-qubit GHZ state. A non-vanishing lower bound is obtained if the GHZ-fidelity of the pro…
We propose a scheme for both distilling and quantifying entanglement, applicable to individual copies of an arbitrary unknown two-qubit state. It is realized in a usual two-qubit interferometry with local filtering. Proper filtering operation for the maximal distillation of the state is achieved, by erasing single-qubit interference, and then the concurrence of the state is determined directly from the visibilities of two-qubit interference. We compare the scheme with full state tomography.
We show how to find families of infima and suprema for the entanglement negativity using only a few, low-order moments of the partially transposed density matrix $ρ^{T_2}.$ These moments can be measured using the multi-copy quantum circuits previously given by the author, which define a set of multi-copy expectation values and thus can be used with the replica trick. As such, these bounds are suitable for use with Quantum Monte Carlo methods, and the lower order versions of the estimates may be experimentally accessible for some systems. Using more moments for higher-order versions of these me…
Given n qubits prepared according to the same unknown density operator, we propose a nondestructive measuring method which approximately yields the eigenstates. It is shown that, for any plane which passes through the center point of the Bloch sphere, there exists corresponding projective measurement. By performing these measurements, we can scan the whole Bloch sphere like radar to search for the orientation of and determine the eigenstates. We show the convergency of the measurements. This result actually reveals a mathematical structure of the n-fold Hilbert space .
The high-precision interferometric measurement of an unknown phase is the basis for metrology in many areas of science and technology. Quantum entanglement provides an increase in sensitivity, but present techniques have only surpassed the limits of classical interferometry for the measurement of small variations about a known phase. Here we introduce a technique that combines entangled states with an adaptive algorithm to precisely estimate a completely unspecified phase, obtaining more information per photon that is possible classically. We use the technique to make the first ab initio entan…
The logarithmic derivative (or, quantum score) of a positive definite density matrix appearing in the quantum Fisher information is discussed, and its exact expression is presented. Then, the problem of estimating the parameters in a class of the Werner-type N-qudit states is studied in the context of the quantum Cramér-Rao inequality. The largest value of the lower bound to the error of estimate by the quantum Fisher information is shown to coincide with the separability point only in the case of two qubits. It is found, on the other hand, that such largest values give rise to the universal f…
Whether entanglement in a state can be detected, distilled, and quantified without full state reconstruction is a fundamental open problem. We demonstrate a new scheme encompassing these three tasks for arbitrary two-qubit entanglement, by constructing the optimal entanglement witness for polarization-entangled mixed-state photon pairs without full state reconstruction. With better efficiency than quantum state tomography, the entanglement is maximally distilled by newly developed tunable polarization filters, and quantified by the expectation value of the witness, which equals the concurrence…
We investigate the practicality of the method proposed by Maciel et al. [Phys. Rev. A. 80, 032325(2009)] for detecting the entanglement of two spatial qutrits (3-dimensional quantum systems), which are encoded in the discrete transverse momentum of single photons transmitted through a multi-slit aperture. The method is based on the acquisition of partial information of the quantum state through projective measurements, and a data processing analysis done with semi-definite programs. This analysis relies on generating gradually an optimal entanglement witness operator, and numerical investigati…
We present the generalized version of Wang's protocol[A.M.Wang, Phys.Rev.A 74,032317 (2006)] for the remote implementation(sometimes referred to as quantum remote control) of partially unknown quantum operations. The protocol only requires no more than half of the entanglements used in Bidirectional Quantum State Teleportation. We also propose a protocol for another form of quantum remote control. It can remotely implement a unitary operation which is a combination of the projective representations of a group. Moreover, we prove that the Schmidt rank of the entanglements cannot not be less tha…
A quantum implementation of high-order power method for estimating geometric entanglement of pure states
Equal1 Labs（Equal1实验室）；Department of Mathematics, University of Trento（特伦托大学数学系）；School of Electrical & Electronic Engineering, University College Dublin（都柏林大学学院电气与电子工程学院）
We characterise non-selective global projective measurements capable of increasing quantum entanglement between two particles. We show that non-selective global projective measurements are capable of increasing entanglement between two particles, in particular, entanglement of any pure non-maximally entangled state can be improved in this way (but not of any mixed state) and we provide detailed analysis for two qubits. It is then shown that Markovian open system dynamics can only approximate such measurements, but this approximation converges exponentially fast as illustrated using Araki-Zurek…
Computing localizable entanglement for noisy many-particle quantum states is difficult due to the optimization over all possible sets of local projection measurements. Therefore, it is crucial to develop lower bounds, which can provide useful information about the behaviour of localizable entanglement, and which can be determined by measuring a limited number of operators, or by performing least number of measurements on the state, preferably without performing a full state tomography. In this paper, we adopt two different yet related approaches to obtain a witness-based, and a measurement-bas…
Suppose we have an unknown multipartite quantum state, how can we experimentally find out whether it is genuine multipartite entangled or not? Recall that even for a bipartite quantum state whose density matrix is known, it is already NP-Hard to determine whether it is entangled or not. Therefore, it is hard to efficiently solve the above problem generally. However, since genuine multipartite entanglement is such a fundamental concept that plays a crucial role in many-body physics and quantum information processing tasks, finding realistic approaches to certify genuine multipartite entanglemen…
While the detection of entanglement has been proved already to be quite a difficult task, experimental quantification of entanglement is even more challenging. In this work, we derive an analytical lower bound for the concurrence of a bipartite mix quantum state in arbitrary dimension. The lower bound is experimentally implementable in a feasible way, which enabling quantification of entanglement in a broad variety of cases
We propose a more general method for detecting a set of entanglement measures, i.e. negativities, in an \emph{arbitrary} tripartite quantum state by local operations and classical communication. To accomplish the detection task using this method, three observers, Alice, Bob and Charlie, do not need to perform the partial transposition maps by the structural physical approximation; instead, they are only required to collectively measure some functions via three local networks supplemented by a classical communication. With these functions, they are able to determine the set of negativities rela…
We present two efficient schemes which can simultaneously accomplish hyperentanglement concentration and purification for two-photon four-qubit systems in an unknown partially hyperentangled mixed states. The first can correct errors in the polarization entanglement and extract maximal hyperentanglement in polarization and spatial mode with additional partial frequency entanglement. The second uses additional maximal frequency entanglement to purify and concentrate hyperentanglement in polarization and spatial mode deterministically. Both of the two schemes are only based on existing optical d…
We propose practical schemes for concentrating entanglement of a pair of unknown partially entangled Bell states and three-photon W states with cross-Kerr nonlinearity. In the schemes, utilizing local operations and classical communication, two separated parties can obtain one maximally entangled photon pair from two previously shared partially entangled photon pairs, and three separated parties can obtain one maximally entangled three-photon W state and a maximally entangled cluster state from two identical partially entangled three-photon W state with a certain success probability. Finally, …
A concern has been expressed that ``the Jaynes principle can produce fake entanglement'' [R. Horodecki et al., Phys. Rev. A {\bf 59}, 1799 (1999)]. In this paper we discuss the general problem of distilling maximally entangled states from $N$ copies of a bipartite quantum system about which only partial information is known, for instance in the form of a given expectation value. We point out that there is indeed a problem with applying the Jaynes principle of maximum entropy to more than one copy of a system, but the nature of this problem is classical and was discussed extensively by Jaynes. …
Entanglement concentration from many copies of unknown pure states is discussed, and we propose the protocol which not only achieves entropy rate, but also produces the perfect maximally entangled state. Our protocol is induced naturally from symmetry of $n$-tensored pure state, and is optimal for all the protocols which concentrates entanglement from unknown pure states, in the sense of failure probability. In the proof of optimality, the statistical estimation theory plays a key role, for concentrated entanglement gives a natural estimate of the entropy of entanglement.
We present a systematic entanglement concentration protocol (ECP) for an arbitrary unknown less-entangled three-photon W state, resorting to the optical property of the quantum-dot spins inside one-sided optical microcavities. In our ECP, the parties obtain not only some three-photon systems in the partially entangled with two unknown parameters when one of the parties picks up the robust odd-parity instance with the parity-check gate (PCG) on his two photons, but also some entangled two-photon systems by keeping the even-parity instance in the first step. By exploiting the above three-photon …
We present optimal measuring strategies for the estimation of the entanglement of unknown two-qubit pure states and of the degree of mixing of unknown single-qubit mixed states, of which N identical copies are available. The most general measuring strategies are considered in both situations, to conclude in the first case that a local, although collective, measurement suffices to estimate entanglement, a non-local property, optimally.
We propose a scheme of continuous-variable reversible telecloning, which broadcast the information of an unknown state without loss from a sender to several spatially separated receivers exploiting multipartite entanglement as quantum channels. In this scheme, quantum information of an unknown state is distributed into $M$ optimal clones and $M-1$ anticlones using $2M$% -partite entanglement. For the perfect quantum information distribution that is optimal cloning, $2M$-partite entanglement is required to be a maximum two-party entanglement. Comparing with the quantum telecloning proposed by L…
Transfer of entanglement and information is studied for quantum teleportation of an unknown entangled state through noisy quantum channels. We find that the quantum entanglement of the unknown state can be lost during the teleportation even when the channel is quantum correlated. We introduce a fundamental parameter of correlation information which dissipates linearly during the teleportation through the noisy channel. Analyzing the transfer of correlation information, we show that the purity of the initial state is important in determining the entanglement of the replica state.
An initial unknown quantum state can be determined with a single measurement apparatus by letting it interact with an auxiliary, "Ancilla", system as proposed by Allahverdyan, Balian and Nieuwenhuizen [Phys. Rev. Lett. 92, 120402 (2004)]. In the case of two qubits, this procedure allows to reconstruct the initial state of the qubit of interest S by measuring three commuting observables and therefore by means of a single apparatus, for the total system S + A at a later time. The determinant of the matrix of the linear transformation connecting the measurements of three commuting observables at …
We study the entanglement of formation for arbitrary dimensional bipartite mixed unknown states. Experimentally measurable lower and upper bounds for entanglement of formation are derived.
A bipartite state which is secretly chosen from a finite set of known entangled pure states cannot be immediately useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we introduce a new way to locally manipulate the original quantum system: either identify the state successfully or distill some pure entanglement. Remarkably, if many copies are available, we show that any finite set of entangled pure states, whatever orthogonal or not, can be locally distinguished in this way, which further implies that pure entang…
Quantum teleportation circumvents the uncertainty principle using dual channels: a quantum one consisting of previously-shared entanglement, and a classical one, together allowing the disembodied transport of an unknown quantum state over distance. It has recently been shown that a classical bit can be counterfactually communicated between two parties in empty space, "Alice" and "Bob". Here, by using our "dual" version of the chained quantum Zeno effect to achieve a counterfactual CNOT gate, we propose the first protocol for transporting an unknown qubit counterfactually, that is without any p…
We revisit the problem of mutually unbiased measurements in the context of estimating the unknown state of a $d$-level quantum system, first studied by W. K. Wootters and B. D. fields[7] in 1989 and later investigated by S. Bandyopadhyay et al [3] in 2001 and A. O. Pittenger and M. H. Rubin [6] in 2003. Our approach is based directly on the Weyl operators in the $L^2$-space over a finite field when $d=p^r$ is the power of a prime. When $d$ is not a prime power we sacrifice a bit of optimality and construct a recovery operator for reconstructing the unknown state from the probabilities of eleme…
We examine the problem of estimating the expectation values of two observables when we have a finite number of copies of an unknown qubit state. Specifically we examine whether it is better to measure each of the observables separately on different copies or to perform a joint measurement of the observables on each copy. We find that joint measurements can sometimes provide an advantage over separate measurements, but only if we make estimates of an observable based solely on the results of measurements of that observable. If we instead use both sets of results to estimate each observable then…
Like entanglement, quantum discord quantifies the quantum correlations. Unlike entanglement, whose detection is extremely difficult, the quantum discord of an arbitrary bipartite state allows itself to be detected perfectly by a single observable, the quantum discord witness. In particular, we report a single observable whose expectation value provides a necessary and sufficient condition for the vanishing quantum discord of an arbitrary bipartite unknown state with four copies. A quantum circuit is designed to measure the quantum discord witness by using only local qubit-measurements.
We construct a single observable measurement of which mean value on four copies of an {\it unknown} two-qubit state is sufficient for unambiguous decision whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function which characterizes to some extent the entanglement quantitatively. This function is an entanglement monotone under so--called local pure operations and classical communication (pLOCC) which preserve local dimensions. Moreover it provides tight up…
