Localizable Entanglement Research Articles

Controlling gain with loss: Bounds on localizable entanglement in multiqubit systems

We investigate the relation between the amount of entanglement localized on a chosen subsystem of a multiqubit system via local measurements on the rest of the system, and the bipartite entanglement that is lost during this measurement process. We study a number of paradigmatic pure states, including the generalized Greenberger--Horne--Zeilinger (GHZ), the generalized $W$ (gW), Dicke, and the generalized Dicke states. For the generalized GHZ and $W$ states, we analytically derive bounds on localizable entanglement in terms of the entanglement present in the system prior to the measurement. Also, for the Dicke and the generalized Dicke states, we demonstrate that with increasing system size, localizable entanglement tends to be equal to the bipartite entanglement present in the system over a specific partition before measurement. We extend the investigation numerically in the case of arbitrary multiqubit pure states. We also analytically determine the modification of these results, including the proposed bounds, in situations where these pure states are subjected to single-qubit phase-flip noise on all qubits. Additionally, we study one-dimensional paradigmatic quantum spin models, namely, the transverse-field XY model and the XXZ model in an external field, and numerically demonstrate a cubic dependence of the localized entanglement on the lost entanglement. We show that this relation is robust even in the presence of disorder in the strength of the external field.

Dipole-Dipole Effect to Limits of Entanglement in Multipartite Spin Chain: A Monte Carlo Study

The entanglement of the ferromagnetically ordered isotropic spin-1/2 chain is discussed. The analytically deriving concurrence of a two-qubit state allows focusing on the effect of dipolar interaction (D). Low fields enable tuning creation/extinction of entangled states, particularly at low temperatures. There is a joint effect of the applied field and dipolar interaction which can’t be disregarded. We perform Quantum Monte Carlo simulations on quantifying localizable entanglement (LE) in terms of upper/lower bounds. Findings reveal that D and B_z are decisive parameters on the production of entanglement including creation and extinction. A non-monotonic behavior has occurred under high fields at the critical temperature. However, strong D provides the stability of LE values concerning distance herewith conserving the unity at low temperatures under zero field. Rival regions are observed for the distant nearest neighbors, particularly odd ones.

Distinguishing phases via non-Markovian dynamics of entanglement in topological quantum codes under parallel magnetic field

We investigate the static and the dynamical behavior of localizable entanglement and its lower bounds on nontrivial loops of topological quantum codes with parallel magnetic field. Exploiting the connection between the stabilizer states and graph states in the absence of the parallel field and external noise, we identify a specific measurement basis, referred to as the canonical measurement basis, that optimizes localizable entanglement when measurement is restricted to single-qubit Pauli measurements only, thereby providing a lower bound. We also propose an approximation of the lower bound that can be computed for larger systems according to the computational resource in hand. Additionally, we compute a lower bound of the localizable entanglement that can be computed by determining the expectation value of an appropriately designed witness operator. We study the behavior of these lower bounds in the vicinity of the topological to nontopological quantum phase transition of the system, and perform a finite-size scaling analysis. We also investigate the dynamical features of these lower bounds when the system is subjected to Markovian or non-Markovian single-qubit dephasing noise. We find that in the case of the non-Markovian dephasing noise, at large time, the canonical measurement-based lower bound oscillates with a larger amplitude when the initial state of the system undergoing dephasing dynamics is chosen from the nontopological phase, compared to the same for an initial state from the topological phase. These features can be utilized to distinguish the topological phase of the system from the nontopological phase in the presence of dephasing noise.

Hierarchies of localizable entanglement due to spatial distribution of local noise

Complete characterization of a noisy multipartite quantum state in terms of entanglement requires full knowledge of how the entanglement content in the state is affected by the spatial distribution of noise in the state. Specifically, we find that if the measurement basis in the protocol of computing localizable entanglement and the basis of the Kraus operator representing the local noisy channel do not commute, the information regarding the noise is retained in the system even after the qubit is traced out after measurement. Using this result and the basic properties of entanglement under noise, we present a set of hierarchies that localizable entanglement over a specific subsystem in a multiqubit state can obey when local noise acts on the subparts or on all the qubits of the whole system. In particular, we propose two types of hierarchies---one tailored according to the number of noisy unmeasured qubits, and the other one that depends additionally on the cardinality of the set of noisy measured qubits, leading to the classification of quantum states. We report the percentage of states satisfying the proposed hierarchies in the case of random three- and four-qubit systems and show, using both analytical methods and numerical simulations, that in almost all the cases, anticipated hierarchies tend to hold with the variation of the strength of noise.

Realizing a deterministic source of multipartite-entangled photonic qubits

Sources of entangled electromagnetic radiation are a cornerstone in quantum information processing and offer unique opportunities for the study of quantum many-body physics in a controlled experimental setting. Generation of multi-mode entangled states of radiation with a large entanglement length, that is neither probabilistic nor restricted to generate specific types of states, remains challenging. Here, we demonstrate the fully deterministic generation of purely photonic entangled states such as the cluster, GHZ, and W state by sequentially emitting microwave photons from a controlled auxiliary system into a waveguide. We tomographically reconstruct the entire quantum many-body state for up to N = 4 photonic modes and infer the quantum state for even larger N from process tomography. We estimate that localizable entanglement persists over a distance of approximately ten photonic qubits.

Scalable characterization of localizable entanglement in noisy topological quantum codes

Topological quantum error correcting codes have emerged as leading candidates towards the goal of achieving large-scale fault-tolerant quantum computers. However, quantifying entanglement in these systems of large size in the presence of noise is a challenging task. In this paper, we provide two different prescriptions to characterize noisy stabilizer states, including the surface and the color codes, in terms of localizable entanglement over a subset of qubits. In one approach, we exploit appropriately constructed entanglement witness operators to estimate a witness-based lower bound of localizable entanglement, which is directly accessible in experiments. In the other recipe, we use graph states that are local unitary equivalent to the stabilizer state to determine a computable measurement-based lower bound of localizable entanglement. If used experimentally, this translates to a lower bound of localizable entanglement obtained from single-qubit measurements in specific bases to be performed on the qubits outside the subsystem of interest. Towards computing these lower bounds, we discuss in detail the methodology of obtaining a local unitary equivalent graph state from a stabilizer state, which includes a new and scalable geometric recipe as well as an algebraic method that applies to general stabilizer states of arbitrary size. Moreover, as a crucial step of the latter recipe, we develop a scalable graph-transformation algorithm that creates a link between two specific nodes in a graph using a sequence of local complementation operations. We develop open-source Python packages for these transformations, and illustrate the methodology by applying it to a noisy topological color code, and study how the witness and measurement-based lower bounds of localizable entanglement varies with the distance between the chosen qubits.

Uniform decoherence effect on localizable entanglement in random multiqubit pure states

We investigate the patterns in distributions of localizable entanglement over a pair of qubits for random multi-qubit pure states. We observe that the mean of localizable entanglement increases gradually with increasing the number of qubits of random pure states while the standard deviation of the distribution decreases. The effects on the distributions, when the random pure multi-qubit states are subjected to local as well as global noisy channels, are also investigated. Unlike the noiseless scenario, the average value of the localizable entanglement remains almost constant with the increase in the number of parties for a fixed value of noise parameter. We also find out that the maximum strength of noise under which entanglement survives can be independent of the localizable entanglement content of the initial random pure states.

Estimating localizable entanglement from witnesses

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 the 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-based lower bounds for localizable entanglement. The former is determined by the minimal amount of entanglement that can be present in a subsystem of the multipartite quantum state, which is consistent with the expectation value of an entanglement witness. Determining this bound does not require any information about the state beyond the expectation value of the witness operator, which renders this approach highly practical in experiments. The latter bound of localizable entanglement is computed by restricting the local projection measurements over the qubits outside the subsystem of interest to a suitably chosen basis. We discuss the behaviour of both lower bounds against local physical noise on the qubits, and discuss their dependence on noise strength and system size. We also analytically determine the measurement-based lower bound in the case of graph states under local uncorrelated Pauli noise.

Multipartite entanglement and quantum Fisher information in conformal field theories

Bipartite entanglement entropy of a segment with the length $l$ in $1+1$ dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\ln l+\gamma$, where $c$ is the central charge of the CFT and $\gamma$ is a cut-off dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have {\it{more}} bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy but also have more multipartite entanglement content. In particular, we argue that since systems with smaller {\it{smallest scaling dimension}} have bigger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in $1+1$ dimensions. Furthermore, we also argue that the QFI decays under renormalization group (RG) flow between two unitary CFTs. Finally, we also comment on the non-conformal but scale invariant systems.

Symmetry-protected topological order at nonzero temperature

We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global on-site symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such on-site symmetries are not thermally stable. Second, we prove that the 3D cluster state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a global generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high error tolerance of this 3D cluster state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry.

Proposal for demonstration of long-range cluster state entanglement in the presence of photon loss

Photonic cluster states are a crucial resource for optical quantum computing. Recently a quantum dot single photon source has been demonstrated to produce strings of single photons in a small linear cluster state. Sources of this kind could produce much larger cluster states, but high photon loss rates make it impossible to characterize the entanglement generated by quantum state tomography. We present a benchmarking method for such sources that can be used to demonstrate useful long-range entanglement with currently available collection/detection efficiencies below 1%. The measurement of the polarization state of single photons in different bases can provide an estimate for the three-qubit correlation function ⟨ZXZ⟩. This value constrains correlations spanning more than three qubits, which in turn provide a lower bound for the localizable entanglement between any two qubits in the large state produced by the source. Finite localizable entanglement can be established by demonstrating ⟨ZXZ⟩>23. This result enables photonic experiments demonstrating computationally useful entanglement with currently available technology.

Multipartite entanglement accumulation in quantum states: Localizable generalized geometric measure

Multiparty quantum states are useful for a variety of quantum information and computation protocols. We define a multiparty entanglement measure based on local measurements on a multiparty quantum state, and an entanglement measure averaged on the post-measurement ensemble. Using the generalized geometric measure as the measure of multipartite entanglement for the ensemble, we demonstrate, in the case of several well-known classes of multipartite pure states, that the localized multipartite entanglement can exceed the entanglement present in the original state. We also show that measurement over multiple parties may be beneficial in enhancing localizable multipartite entanglement. We point out that localizable generalized geometric measure faithfully signals quantum critical phenomena in well-known quantum spin models even when considerable finite-size effect is present in the system.

Post-measurement bipartite entanglement entropy in conformal field theories

We derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in $1+1$ dimensional conformal field theories with periodic and open boundary conditions. After defining the set up we will check numerically the validity of our results in the case of Klein-Gordon field theory (coupled harmonic oscillators) and spin-$1/2$ XX chain in a magnetic field. The agreement between analytical results and the numerical calculations is very good. We also find a lower bound for localizable entanglement in coupled harmonic oscillators.

Extracting entangled qubits from Majorana fermions in quantum dot chains through the measurement of parity.

We propose a scheme for extracting entangled charge qubits from quantum-dot chains that support zero-energy edge modes. The edge mode is composed of Majorana fermions localized at the ends of each chain. The qubit, logically encoded in double quantum dots, can be manipulated through tunneling and pairing interactions between them. The detailed form of the entangled state depends on both the parity measurement (an even or odd number) of the boundary-site electrons in each chain and the teleportation between the chains. The parity measurement is realized through the dispersive coupling of coherent-state microwave photons to the boundary sites, while the teleportation is performed via Bell measurements. Our scheme illustrates localizable entanglement in a fermionic system, which serves feasibly as a quantum repeater under realistic experimental conditions, as it allows for finite temperature effect and is robust against disorders, decoherence and quasi-particle poisoning.

Entangled Three Qutrit Coherent States and Localizable Entanglement

Choosing I-concurrence as the measure of bipartite entanglement and using von Neumann projective local measurements, localizable entanglement (LE) in a three-qutrit system is studied. A superposition of the qutrit-coherentstate of this system is considered ant its LE is obtained and analyzed as a function of the relevant parameters. It is observed that one may achieve the maximal entanglement or no entanglement at all, depending on the choice of the parameters involved.

Fermionized photons in the ground state of one-dimensional coupled cavities

The density matrix renormalization group algorithm is used to characterize the ground states of one-dimensional coupled cavities in the regime of low photon densities. Numerical results for photon and spin excitation densities, one- and two-body correlation functions, superfluid and condensate fractions, as well as the entanglement entropy and localizable entanglement are obtained for the Jaynes-Cummings-Hubbard (JCH) model and are compared with those for the Bose-Hubbard (BH) model where applicable. The results indicate that a Tonks-Girardeau phase, in which the photons are strongly fermionized, appears between the Mott-insulating and superfluid phases as a function of the intercavity coupling. In fact, the superfluid density is found to be zero in a wide region outside the Mott-insulator phase boundary. The presence of two different species of excitation (spin and photon) in the JCH model gives rise to properties with no analog in the BH model, such as the (quasi)condensation of spin excitations and the spontaneous generation of entanglement between the atoms confined to each cavity.

Universal Quantum Computation with Little Entanglement

We show that universal quantum computation can be achieved in the standard pure-state circuit model while the entanglement entropy of every bipartition is small in each step of the computation. The entanglement entropy required for large-scale quantum computation even tends to zero. Moreover we show that the same conclusion applies to many entanglement measures commonly used in the literature. This includes e.g., the geometric measure, localizable entanglement, multipartite concurrence, squashed entanglement, witness-based measures, and more generally any entanglement measure which is continuous in a certain natural sense. These results demonstrate that many entanglement measures are unsuitable tools to assess the power of quantum computers.

Matrix product states with long-range localizable entanglement

We derive a criterion to determine when a translationally invariant matrix product state (MPS) has long-range localizable entanglement, where that quantity remains finite in the thermodynamic limit. We give examples fulfilling this criterion and eventually use it to obtain all such MPS with bond dimension 2 and 3.

Three-step approach to Fokker-Planck equation for an interacting open waveguide system and localizable entanglement dynamics

We consider a system of three coupled single-mode waveguides each locally interacting with its own Gaussian environment and present a general solution for this coupled system initially in any Gaussian state using the symplectic operations. We investigate the dynamics of two-mode localizable entanglement contained in the evolved state when the system is initially in three-mode bisymmetric Gaussian state in contact with the independent decoherence. We show that such an entanglement exhibits a damped oscillation in a regime of weak waveguide-waveguide coupling and small mean photon numbers of the bath. Remarkably, we find that the entanglement can reappear after the long-time death and arrives at a steady-state oscillation, whose maximum depends strongly on both the squeezing of the bath and the coupling strength between these waveguides. Finally, we generalize the approach to a common squeezed environment case.

Dynamics of localizable entanglement in a qutrit chain with Dzyaloshinskii–Moriya interaction

Dynamics of localizable entanglement in a qutrit chain, in the presence of the Dzyaloshinskii---Moriya (DM) interaction is studied. Three distinct initial states, namely, superposition of the ground and the first excited state (SGE), a GHZ state and a superposition of qutrit coherent states (SQCS) are considered in this investigation. While the ground and the first excited state exhibit the maximum of entanglement, the latter is diminished for any superposition of the states. In both SGE and GHZ cases, localizable entanglement (LE) oscillates and its period is a decreasing function of the ratio of the strength of DM interaction and the spin coupling constant (DS ratio), but its maximum value is independent of the latter. In SQCS case, LE also oscillates in time at small values of DS ratio; its average is reduced as the strength of the DM interaction increases and gains its maximum average and the highest peaks at a specific value of the coherence parameter.