Entanglement Negativity Research Articles

Dynamics of charge-resolved entanglement after a local quench

Quantum entanglement and its main quantitative measures, the entanglement entropy and entanglement negativity, play a central role in many body physics. An interesting twist arises when the system considered has symmetries leading to conserved quantities: Recent studies introduced a way to define, represent in field theory, calculate for 1+1D conformal systems, and measure, the contribution of individual charge sectors to the entanglement measures between different parts of a system in its ground state. In this paper, we apply these ideas to the time evolution of the charge-resolved contributions to the entanglement entropy and negativity after a local quantum quench. We employ conformal field theory techniques and find that the known dependence of the total entanglement on time after a quench, $S_A \sim \log(t)$, results from $\sim\sqrt{\log(t)}$ significant charge sectors, each of which contributes $\sim\sqrt{\log(t)}$ to the entropy. We compare our calculation to numerical results obtained by the time-dependent density matrix renormalization group algorithm and exact solution in the noninteracting limit, finding good agreement between all these methods.

The entanglement of atoms with two-photon transitions in the presence of ac Stark shift of energy levels

In this paper, we investigated the dynamics of atomic entanglement in a quantum system consisting of two identical two-level atoms (qubits) resonantly interacting with a mode of a thermal cavity field through degenerate two-photon transitions, in the presence of a Stark energy level shift. An analytical expression is obtained for the atom entanglement (negativity) parameter for separable and entangled initial states of atoms. The influence of the Stark shift on the degree of atom-atom entanglement is considered. It is established that the Stark shift leads to a significant increase in the degree of atom entanglement in the case of separable initial states of atoms and to stabilization of atomic entanglement in the case of entangled initial states of atoms.

Entanglement negativity in Galilean conformal field theories

We obtain the entanglement negativity for various bipartite zero and finite temperature pure and mixed state configurations in a class of $(1+1)$-dimensional Galilean conformal field theories. In this context we establish a construction for computing the entanglement negativity for such bipartite states involving a suitable replica technique. Our construction exactly reproduces certain universal features observed for entanglement negativity of corresponding states in relativistic $(1+1)$-dimensional conformal field theories.

Covariant holographic entanglement negativity for disjoint intervals in AdS_3/CFT_2

We advance a construction for the covariant holographic entanglement negativity for time dependent mixed states of disjoint intervals in (1+1) dimensional conformal field theories (CFT_{1+1}) dual to bulk non static AdS_3 geometries. Application of our proposal to such mixed states in a CFT_{1+1} dual to bulk non extremal and extremal rotating BTZ black holes exactly reproduces the replica technique results in the large central charge limit. We also investigate the time dependent holographic entanglement negativity for such mixed states in a CFT_{1+1} dual to a bulk Vaidya-AdS_3 geometry in the context of their thermalization involving bulk black hole formation.

Time evolution of entanglement negativity from black hole interiors

We investigate the time evolution of entanglement negativity following a global quench for mixed state configurations of two disjoint and adjacent intervals in a (1 + 1)-dimensional conformal field theory (CFT1+1) dual to the eternal black hole sliced in half by an end of the world brane, through the AdS3/CFT2 correspondence. To this end we obtain the time evolution of the holographic entanglement negativity for such mixed states from a dual bulk eternal black hole geometry and elucidate the relevant geodesic structures. The holographic entanglement negativity for such mixed states, following a global quench is described by half of the results for the eternal black hole. Significantly our results exactly match with the corresponding CFT1+1 computations.

Sharma–Mittal quantum discord

We demonstrate a generalization of quantum discord using a generalized definition of von-Neumann entropy, which is Sharma-Mittal entropy; and the new definition of discord is called Sharma-Mittal quantum discord. Its analytic expressions are worked out for two qubit quantum states as well as Werner, isotropic, and pointer states as special cases. The R{\'e}nyi, Tsallis, and von-Neumann entropy based quantum discords can be expressed as limiting cases for of Sharma-Mittal quantum discord. We also numerically compare all these discords and entanglement negativity.

Holographic entanglement negativity conjecture for adjacent intervals in AdS3/CFT2

We propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in (1+1)-dimensional dual conformal field theories through the AdS/CFT correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture we obtain the entanglement negativity of adjacent intervals in zero and finite temperature (1+1)-dimensional holographic conformal field theories dual to the bulk AdS3 vacuum and the Euclidean BTZ black hole respectively. Our holographic conjecture exactly reproduces the conformal field theory results obtained through the replica technique, in the large central charge limit. We briefly elucidate the corresponding issue for the AdSd+1/CFTd scenario.

Finite-temperature entanglement negativity of free fermions

The entanglement entropy of free fermions with a Fermi surface is known to obey a logarithmic scaling and violate the area law in all dimensions. Here, we would like to see how temperature affects the logarithmic scaling behavior. To this end, we compute the entanglement negativity of free fermions using the fermionic partial transpose developed in our earlier paper (Shapourian et al 2017 Phys. Rev. B 95 165101). In one dimension, we analytically derive the leading order term in the finite-temperature entanglement negativity and show how the entanglement negativity indicates a crossover from a quantum entangled state to a classical thermal state, where the entanglement is completely lost. We explain how the one-dimensional result can be generalized to codimension-one Fermi surface of arbitrary shape in higher dimensions. In both one and two dimensions, we check that our analytical results agree with the numerical simulation of free fermions on a lattice.

Methods to estimate entanglement in teleportation experiments

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. Thus, our findings provide a quantitative relation between teleportation and entanglement.

Holographic entanglement negativity for disjoint intervals in AdS_3/CFT_2

We advance a holographic construction for the entanglement negativity of bipartite mixed state configurations of two disjoint intervals in (1+1) dimensional conformal field theories (CFT_{1+1}) through the AdS_3/CFT_2 correspondence. Our construction constitutes the large central charge analysis of the entanglement negativity for mixed states under consideration and involves a specific algebraic sum of bulk space like geodesics anchored on appropriate intervals in the dual CFT_{1+1}. The construction is utilized to compute the holographic entanglement negativity for such mixed states in CFT_{1+1}s dual to bulk pure AdS_3 geometries and BTZ black holes respectively. Our analysis exactly reproduces the universal features of corresponding replica technique results in the large central charge limit which serves as a consistency check.

Singularity in entanglement negativity across finite-temperature phase transitions

Phase transitions at a finite (i.e. non-zero) temperature are typically dominated by classical correlations, in contrast to zero temperature transitions where quantum mechanics plays an essential role. Therefore, it is natural to ask if there are any signatures of a finite temperature phase transition in measures that are sensitive only to quantum correlations. Here we study one such measure, namely, entanglement negativity, across finite temperature phase transitions in several exactly solvable Hamiltonians and find that it is a singular function of temperature across the transition. As an aside, we also calculate the entanglement of formation exactly in a related, interacting model.

Entanglement negativity of fermions: Monotonicity, separability criterion, and classification of few-mode states

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 fermionic partial transpose in various fermionic states, where we explain why the bosonic partial transpose fails in distinguishing separable states of fermions. Finally, we explore a set of entanglement quantities which distinguish different classes of entangled states of a system with two and three fermionic modes. In doing so, we prove that vanishing entanglement negativity is a necessary and sufficient condition for separability of $N\geq 2$ fermionic modes with respect to the bipartition into one mode and the rest. We further conjecture that the entanglement negativity of inseparable states which mix local fermion-number parity is always non-vanishing.

Entropic uncertainty relations in the spin-1 Heisenberg model

The uncertainty relations build an intrinsic lower bound to the measurement precision for arbitrary two incompatible observables and hence deemed as a backbone in quantum theory, strikingly distinguishing from classical physics. In this work, we examine the quantum-memory-assisted entropic uncertainty relations (QMA-EUR) in two-qutrit spin-1 Heisenberg XYZ and XXX chains under homogeneous magnetic fields, respectively. We specifically derive the dynamical evolution of QMA-EUR for various incompatible measurements on Pauli operators (mutually unbiased bases) and SU(3) generators in the Heisenberg XXX and XYZ models when spin A is the object to be measured and B is served as quantum memory during information processing. Notably, it has been found in the case of mutual unbiased-base measurements that, firstly, the larger coupling strength J between A and B can induce the degradation of the measurement’s uncertainty; secondly, the entropic uncertainty is extremely dependent on the coupling strength and the external magnetic field, which would bring the uncertainty on the inflation. In addition, we unveil the dynamical behaviors of the uncertainty measured on SU(3) generators, and it proves that the uncertainty’s dynamics are subtly different from those in the former. Moreover, we explore the relationship between the entropic uncertainty and the system’s entanglement (negativity) and declare that the dynamics of the entropic uncertainty of interest are approximately anti-correlated with that of negativity. At last, the entanglement’s dynamics of the probed system in the XXX and XYZ models are revealed and analyzed by means of negativity, respectively. Hence, our observations might pave the way to understand the dynamical traits of the entropic uncertainty during measurement-based information processing.

Triangle-like inequalities related to coherence and entanglement negativity

Quantum coherence and entanglement are two key features in quantum mechanics and play important roles in quantum information processing and quantum computation. We provide a general triangle-like inequality satisfied by the $l_1$-norm measure of coherence for convex combination of arbitrary $n$ pure states of a quantum state $\rho$. Furthermore, we present triangle-like inequality for the convex-roof extended negativity for any states of rank 2, which gives a positive answer to a conjecture raised in [Phys. Rev. A 96, 062308 (2017)]. Detailed examples are given to illustrate the relations characterized by the triangle-like inequalities.

Estimation of entanglement negativity of a two-qubit quantum system with two measurements

Numerous work had been done to quantify the entanglement of a two-qubit quantum state, but it can be seen that previous works were based on joint measurements on two copies or more than two copies of the quantum state under consideration. In this work, we show that a single copy and two measurements are enough to estimate the entanglement quantifier like entanglement negativity and concurrence. To achieve our goal, we establish a relationship between the entanglement negativity and the minimum eigenvalue of the structural physical approximation of the partial transpose of an arbitrary two-qubit state. The derived relation makes it possible to estimate entanglement negativity experimentally by Hong-Ou-Mandel interferometry with only two detectors. Also, we derive the upper bound of the concurrence of an arbitrary two-qubit state and have shown that the upper bound can be realized in experiment. We will further show that the concurrence of i) an arbitrary pure two-qubit state and ii) a particular class of mixed states, namely, rank-2 quasi-distillable mixed states, can be exactly estimated with two measurements.

Holographic entanglement negativity for conformal field theories with a conserved charge

We study the application of our recent holographic entanglement negativity conjecture for mixed states of adjacent subsystems in conformal field theories with a conserved charge. In this context we obtain the holographic entanglement negativity for zero and finite temperature mixed state configurations in d-dimensional conformal field theories dual to bulk extremal and non extremal charged AdS_{d+1} black holes. Our results conform to quantum information theory expectations and constitute significant consistency checks for our conjecture.

Covariant holographic entanglement negativity

We propose a covariant holographic conjecture for the entanglement negativity of bipartite mixed states in (1+1)-dimensional conformal field theories dual to bulk non static AdS_{3} configurations. Application of our conjecture to (1+1)-dimensional conformal field theories dual to bulk non extremal and extremal rotating BTZ black holes exactly reproduce the corresponding entanglement negativity obtained through the replica technique, in the large central charge limit. We briefly discuss the issue of the generalization of our conjecture to higher dimensions.

Holographic entanglement negativity for adjacent subsystems in AdSd+1/CFTd

We establish our recently proposed holographic entanglement negativity conjecture for mixed states of adjacent subsystems in conformal field theories with concrete higher dimensional examples. In this context we compute the holographic entanglement negativity for mixed states of adjacent subsystems in $d$-dimensional conformal field theories dual to bulk $AdS_{d+1}$ vacuum and $AdS_{d+1}$-Schwarzschild black holes. These representative examples provide strong indication for the universality of our conjecture which affirms significant implications for diverse applications.

Entanglement negativity, holography and black holes

We investigate the application of our recent holographic entanglement negativity conjecture for higher dimensional conformal field theories to specific examples which serve as crucial consistency checks. In this context we compute the holographic entanglement negativity for bipartite pure and finite temperature mixed state configurations in d-dimensional conformal field theories dual to bulk pure AdS_{d+1} geometry and AdS_{d+1}-Schwarzschild black holes respectively. It is observed that the holographic entanglement negativity characterizes the distillable entanglement for the finite temperature mixed states through the elimination of the thermal contributions. Significantly our examples correctly reproduce universal features of the entanglement negativity for corresponding two dimensional conformal field theories, in higher dimensions.

Holographic quantum entanglement negativity

We advance a holographic conjecture for the entanglement negativity of bipartite quantum states in (1 + 1)-dimensional conformal field theories in the AdS3/CF T2 framework. Our conjecture exactly reproduces the replica technique results in the large central charge limit, for both the pure state described by the CF T1+1 vacuum dual to bulk the pure AdS3 geometry and the finite temperature mixed state dual to a Euclidean BTZ black hole respectively. The holographic entanglement negativity characterizes the distillable entanglement and reduces to a specific sum of holographic mutual informations. We briefly allude to a possible higher dimensional generalization of our conjecture in a generic AdSd+1/CF Td scenario.