The computable measure of the mixed-state entanglement, the negativity, is shown to admit a clear geometrical interpretation, when applied to Schmidt-correlated (SC) states: the negativity of a SC state equals a distance of the state from a pertinent separable state. As a consequence, a SC state is separable if and only if its negativity vanishes. Another remarkable consequence is that the negativity of a SC can be estimated "at a glance" on the density matrix. These results are generalized to mixtures of SC states, which emerge in certain quantum-dynamical settings.
In this report we consider the three dimensional subset of the space of states of two qubits that may be written in the so called standard form. For those states we show that different measures of entanglement, specifically concurrence, negativity and the Hilbert-Schmidt distance are proportional to the euclidean distance between the point representing the state in the three dimensional parameter space and the set of separable states.
We show that higher order inter-group covariances involving even number of qubits are necessarily positive semidefinite for N qubit separable states, which are completely symmetric under permutations of the qubits. This identification leads to a family of sufficient conditions of inseparability based on the negativity of 2kth order inter-group covariance matrices of symmetric N-qubit systems. These conditions have a simple structure and detect entanglement in all even partitions of the symmetric multiqubit system. The observables involved are feasible experimental quantities and do not demand …
We study the ordering of two-qubit states with respect to the degree of bipartite entanglement using the Wootters concurrence -- a measure of the entanglement of formation, and the negativity -- a measure of the entanglement cost under the positive-partial-transpose-preserving operations. For two-qubit pure states, the negativity is the same as the concurrence. However, we demonstrate analytically on simple examples of various mixtures of Bell and separable states that the entanglement measures can impose different orderings on the states. We show which states, in general, give the maximally d…
It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity …
Quantum systems generally exhibit different kinds of correlations. In order to compare them on equal footing, one uses the so-called distance-based approach where different types of correlations are captured by the distance to different sets of states. However, these quantifiers are usually hard to compute as their definition involves optimization aiming to find the closest states within the set. On the other hand, negativity is one of the few computable entanglement monotones, but its comparison with other correlations required further justification. Here we place negativity as part of a fami…
We investigate the physical mechanism behind the counterintuitive phenomenon, the distribution of continuous-variable entanglement between two distant modes by sending a third separable auxiliary mode between them. For this purpose, we propose a new more simple and more efficient protocol resulting in distributed entanglement with more than an order of the magnitude higher logarithmic negativity than in the previously proposed protocol. This new protocol shows that the distributed entanglement originates from the entanglement of one mode and the auxiliary mode used for distribution which is fi…
Like a silver thread, quantum entanglement [1] runs through the foundations and breakthrough applications of quantum information theory. It cannot arise from local operations and classical communication (LOCC) and therefore represents a more intimate relationship among physical systems than we may encounter in the classical world. The `nonlocal' character of entanglement manifests itself through a number of counterintuitive phenomena encompassing Einstein-Podolsky-Rosen paradox [2,3], steering [4], Bell nonlocality [5] or negativity of entropy [6,7]. Furthermore, it extends our abilities to pr…
Detecting spacelike vacuum entanglement at all distances and promoting negativity to a necessary and sufficient entanglement measure in many-body regimes
We explore sufficient conditions for inseparability in mixed states with a globally conserved charge, such as a particle number. We argue that even separable states may contain entanglement in fixed charge sectors, as long as the state can not be separated into charge conserving components. As a witness of symmetric inseparability we study the number entanglement (NE), $ΔS_m$, defined as the entropy change due to a subsystem's charge measurement. Whenever $ΔS_m > 0$, there exist inseparable charge sectors, having finite (logarithmic) negativity, even when the full state is either separable or …
We study quantum information aspects of the fermionic entanglement negativity recently introduced in [Phys. Rev. B 95, 165101 (2017)] based on the fermionic partial transpose. In particular, we show that it is an entanglement monotone under the action of local quantum operations and classical communications--which preserves the local fermion-number parity-- and satisfies other common properties expected for an entanglement measure of mixed states. We present fermionic analogs of tripartite entangled states such as W and Greenberger-Horne-Zeilinger states and compare the results of bosonic and …
We extend the relation between absolutely maximally entangled (AME) states and quantum maximum distance separable (QMDS) codes by constructing whole families of QMDS codes from their parent AME states. We introduce a reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. We establish that AME state…
The many-body entanglement between two finite (size-$d$) disjoint vacuum regions of non-interacting lattice scalar field theory in one spatial dimension -- a $(d_A \times d_B)_{\rm mixed}$ Gaussian continuous variable system -- is locally transformed into a tensor-product "core" of $(1_A \times 1_B)_{\rm mixed}$ entangled pairs. Accessible entanglement within these core pairs exhibits an exponential hierarchy, and as such identifies the structure of dominant region modes from which vacuum entanglement could be extracted into a spatially separated pair of quantum detectors. Beyond the core, rem…
We present an unifying approach to the quantification of entanglement based on entanglement witnesses, which includes several already established entanglement measures such as the negativity, the concurrence and the robustness of entanglement. We then introduce an infinite family of new entanglement quantifiers, having as its limits the best separable approximation measure and the generalized robustness. Gaussian states, states with symmetry, states constrained to super-selection rules and states composed of indistinguishable particles are studied under the view of the witnessed entanglement. …
Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. In this article we apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance between a given state and the set of separable states. While Hilbert Schmidt Distance does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide here a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outpu…
We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendibleproduct basis bound entangled states (UPB BE states). The method relies on the fact that an optimalentanglement witness is given by a plane perpendicular to a line between the reference state, entanglementof which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds anapproximation of CSS. In this article, we investigate if this approximation can be good enough toyield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those givenby Ban…
We present a proof-of-principle experiment demonstrating measurement of the collectibility, a nonlinear entanglement witness proposed by Rudnicki et al. [Phys. Rev. Lett. 107, 150502 (2011)]. This entanglement witness works for both mixed and pure two-qubit states. In the later case it can be used to measure entanglement in terms of the negativity. We measured the collectibility for three distinct classes of photonic polarization-encoded two-qubit states, i.e., maximally entangled, separable and maximally mixed states. We demonstrate that the measurement procedure is feasible and robust agains…
Continuous-variable quantum key distribution (CVQKD) is considered to be an alternative to classical cryptography for secure communication. However, its transmission distance is restricted to metropolitan areas, given that it is affected by the channel excess noise and losses. In this paper, we present a scheme for implementing long-distance CVQKD using separable Gaussian states. This tunable QKD protocol requires separable Gaussian states, which are squeezed and displaced, along with the assistance of classical communication and available linear optics compoments. This protocol originates fro…
We study the time evolution of the entanglement negativity after a local quantum quench in (1+1)-dimensional conformal field theories (CFTs), which we introduce by suddenly joining two initially decoupled CFTs at their endpoints. We calculate the negativity evolution for both adjacent intervals and disjoint intervals explicitly. For two adjacent intervals, the entanglement negativity grows logarithmically in time right after the quench. After developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case of two spatially separated intervals, a ligh…
We investigate the logarithmic negativity in strongly-disordered spin chains in the random-singlet phase. We focus on the spin-1/2 random Heisenberg chain and the random XX chain. We find that for two arbitrary intervals the disorder-averaged negativity and the mutual information are proportional to the number of singlets shared between the two intervals. Using the strong-disorder renormalization group (SDRG), we prove that the negativity of two adjacent intervals grows logarithmically with the intervals length. In particular, the scaling behavior is the same as in conformal field theory, but …
We examine the thermal entanglement of XXZ-type Heisenberg chains in the presence of a uniform magnetic field along the z axes, through the evaluation of the negativity associated with bipartitions of the whole system and subsystems. Limit temperatures for non-zero global negativities are shown to depend on the asymmetry $Δ$ but not on the uniform field, and can be much higher than those limiting pairwise entanglement. It is also shown that global bipartite entanglement may exist for T>0 even for $Δ\geq 1$, i.e., when the system is fully aligned (and hence separable) at T=0, and that the bipar…
In this paper we want to investigate the possibility of transferring entanglement to two three-level separable atomic states over large distance using the quantum repeater protocol. In detail, our model consists of eight three-level atoms where only the pairs (1,2), (3,4), (5,6) and (7,8) are prepared in maximally entangled states. Performing suitable interaction between non-entangled three-level atoms (2,3) and (6,7) in two-mode cavities with photon leakage rates in the presence of spontaneous emission leads to producing entanglement between atoms (1,4) and (5,8), separately. Finally, the ent…
We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their "mutual negativity" and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a cla…
We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We will give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. We will furthermore chara…
Following recent work of Beigi and Shor, we investigate PPT states that are "heavily entangled." We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least 1/4. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of 2-ε from the separable states, we need a dimension of 2^{poly(\log(1/ε))}, as opposed to 2^{poly(1/ε)} given by the construction of Beigi and Shor. We do so by exploiting the so called {\…
The exact nature of the many-body localization transition remains an open question. An aspect which has been posited in various studies is the emergence of scale invariance around this point, however the direct observation of this phenomenon is still absent. Here we achieve this by studying the logarithmic negativity and mutual information between disjoint blocks of varying size across the many-body localization transition. The two length scales, block sizes and the distance between them, provide a clear quantitative probe of scale invariance across different length scales. We find that at the…
We focus on characterizing entanglement of high dimensional bipartite states using various statistical correlators for two-qudit mixed states. The salient results obtained are as follows: (a) A scheme for determining the entanglement measure given by Negativity is explored by analytically relating it to the widely used statistical correlators viz. mutual predictability, mutual information and Pearson Correlation coefficient for different types of bipartite arbitrary dimensional mixed states. Importantly, this is demonstrated using only a pair of complementary observables pertaining to the mutu…
Given the density matrix rho of a bipartite quantum state, the quantum separability problem asks whether rho is entangled or separable. In 2003, Gurvits showed that this problem is NP-hard if rho is located within an inverse exponential (with respect to dimension) distance from the border of the set of separable quantum states. In this paper, we extend this NP-hardness to an inverse polynomial distance from the separable set. The result follows from a simple combination of works by Gurvits, Ioannou, and Liu. We apply our result to show (1) an immediate lower bound on the maximum distance betwe…
Entanglement has developed into an essential concept for the characterization of phases and phase transitions in ground states of quantum many-body systems. In this work, we use the logarithmic negativity to study the spatial entanglement structure in the transverse-field Ising chain both in the ground state and at nonzero temperatures. Specifically, we investigate the entanglement between two disjoint blocks as a function of their separation, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond wh…
We determine the conditions for the existence of a pair of degenerate parity breaking separable eigenstates in general arrays of arbitrary spins connected through $XYZ$ couplings of arbitrary range and placed in a transverse field, not necessarily uniform. Sufficient conditions under which they are ground states are also provided. It is then shown that in finite chains, the associated definite parity states, which represent the actual ground state in the immediate vicinity of separability, can exhibit entanglement between any two spins regardless of the coupling range or separation, with the r…
In recent years, various aspects of theoretical models with long range interactions have attracted attention, ranging from out-of-time-ordered correlators to entanglement. In the present paper, entanglement properties of a simple non-local model with long-range interactions in the form of a fractional Laplacian is investigated in both static and a quantum quench scenario. Logarithmic negativity, which is a measure for entanglement in mixed states is calculated numerically. In the static case, it is shown that the presence of long-range interaction ensures that logarithmic negativity decays muc…
In this paper we show an explicit construction of multipartite class of entangled states with the PPT (Positive Partial Transposition) property in every cut. We investigate properties of this class of states focusing on the trace distance from the set of separable states. We provide an explicit sub-class of the multipartite entangled PPT states which are arbitrary far from the set of separable states. We argue, that in the multipartite case the mentioned distance increases with dimension of the local Hilbert space. In our construction is we do not have to use many copies of initial state livin…
Every entangled state can be perturbed, for instance by decoherence, and stay entangled. For a large class of pure entangled states, we show how large the perturbation can be. Our class includes all pure bipartite and all maximally entangled states. For an entangled state, E, the constucted neighborhood of entangled states is the region outside two parallel hyperplanes, which sandwich the set of all separable states. The states for which these neighborhoods are largest are the maximally entangled ones. As the number of particles, or the dimensions of the Hilbert spaces for two of the particles…
We present several criteria for genuine multipartite entanglement from universal uncertainty relations based on majorization theory. Under non-negative Schur-concave functions, the vector-type uncertainty relation generates a family of infinitely many detectors to check genuine multipartite entanglement. We also introduce the concept of $k$-separable circles via geometric distance for probability vectors, which include at most $(k-1)$-separable states. The entanglement witness is also generalized to a universal entanglement witness which is able to detect the $k$-separable states more accurate…
In the standard geometric approach to a measure of entanglement of a pure state, $\sin^2θ$ is used, where $θ$ is the angle between the state to the closest separable state of products of normalized qubit states. We consider here a generalization of this notion to separable states consisting of products of unnormalized states of different dimension. In so doing, the entanglement measure $\sin^2θ$ is found to have an interpretation as the distance between the state to the closest separable state. We also find the components of the closest separable state and its norm have an interpretation in te…
Université de Genève（日内瓦大学）；Institute for Quantum Optics and Quantum Information — IQOQI Vienna, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所——维也纳量子光学与量子信息研究所）；Technische Universität Wien（维也纳工业大学）
Finding the closest separable state to a given target state is a notoriously difficult task, even more difficult than deciding whether a state is entangled or separable. To tackle this task, we parametrize separable states with a neural network and train it to minimize the distance to a given target state, with respect to a differentiable distance, such as the trace distance or Hilbert--Schmidt distance. By examining the output of the algorithm, we obtain an upper bound on the entanglement of the target state, and construct an approximation for its closest separable state. We benchmark the met…
We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.
It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for t…
A geometric understanding of entanglement is proposed based on local measurements. Taking recourse to the general structure of density matrices in the framework of Euclidean geometry, we first illustrate our approach for bipartite Werner states. It is demonstrated that separable states satisfy certain geometric constraints that entangled states do not. A separability criterion for multiparty Werner states of arbitrary dimension is derived. This approach can be used to determine separability across any bipartition of a general density matrix and leads naturally to a computable measure of entang…
We investigate the time evolution of odd entanglement entropy (OEE) and logarithmic negativity (LN) for the thermofield double (TFD) states in free scalar quantum field theories using the covariance matrix approach. To have mixed states, we choose non-complementary subsystems, either adjacent or disjoint intervals on each side of the TFD. We find that the time evolution pattern of OEE is a linear growth followed by saturation. On a circular lattice, for longer times the finite size effect demonstrates itself as oscillatory behavior. In the limit of vanishing mass, for a subsystem containing a …
In this thesis there are two topics: On the entangling capacity, in terms of negativity, of quantum operations, and on the supremum of negativity for symmetric Gaussian states. Positive partial transposition (PPT) states are an important class of states in quantum information. We show a method to calculate bounds for entangling capacity, the amount of entanglement that can be produced by a quantum operation, in terms of negativity, a measure of entanglement. The bounds of entangling capacity are found to be associated with how non-PPT (PPT preserving) an operation is. A length that quantifies …
A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan $P$ function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $τ$, related to the minimal amount of Gaussian noise which changes a nonpositive $P$ function into a positive one; (2) the nonclassical distance $D$, defined as the Bures di…
We show that long-distance quantum correlations probe short-distance physics. Two disjoint regions of the latticized, massless scalar field vacuum are numerically demonstrated to become separable at distances beyond the negativity sphere, which extends to infinity in the continuum limit. The size of this quantum coherent volume is determined by the highest momentum mode supported in the identical regions, each of diameter $d$. More generally, effective field theories (EFTs), describing a system up to a given momentum scale $Λ$, are expected to share this feature -- entanglement between regions…
We investigate the effects of relative motion on entanglement harvesting by considering a pair of Unruh-Dewitt detectors moving at arbitrary, but independent and constant velocities, both linearly interacting with the vacuum scalar field. Working within the weak coupling approximation, we find that the Negativity is a (non-elementary) function of the relative velocity of the detectors, as well as their energy gaps and minimal separation. We find parameter regions where Negativity increases with velocity up to a maximum and then decreases, reaching zero at some sublight velocity. At any given r…
The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated to a hyperbolic space with a Minkowski metric, which is divided in two regions - separablelike and entangledlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distances measures - one related to the purity and another related to amount of entanglement. The second …
Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely realignment and partial transpose. This bipartite quantity, denoted here as $R_{12}$, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three qubit states via their two-qubit…
