One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the $α$th power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, $2\otimes2\otimes3$ systems and $2\otimes2\otimes2^{n}$ systems for $α\geq4\ln2$. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for $0\leqβ\leq2$. Given that the logarithmic negativity and LCREN are not convex these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, $2\otimes2\otimes3$ systems and $2\otimes2\otimes2^{n}$ systems in terms of the $α$th power of logarithmic negativity and LCREN for $α\geq4\ln2$. We also show that the $β$th power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for $0\leqβ\leq2$.
In this paper, we investigate the generalised monogamy inequalities of convex-roof extended negativity (CREN) in multi-level systems. The generalised monogamy inequalities provide the upper and lower bounds of bipartite entanglement, which are obtained by using CREN and the CREN of assistance (CRENOA). Furthermore, we show that the CREN of multi-qubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum and show that the generalised mon…
In this paper, we present some generalized monogamy inequalities based on negativity and convex-roof extended negativity (CREN). These monogamy relations are satisfied by the negativity of $N$-qubit quantum systems $ABC_1\cdots C_{N-2}$, under the partitions $AB|C_1\cdots C_{N-2}$ and $ABC_1|C_2\cdots C_{N-2}$. Furthermore, the $W$-class states are used to test these generalized monogamy inequalities.
Monogamy relations characterize the distributions of entanglement in multipartite systems. We investigate the monogamy relations satisfied by the concurrence of assistance and the negativity of assistance for multiqubit generalized $W$-class states. Analytical monogamy inequalities are presented for both concurrence of assistance and negativity of assistance, which are shown to be tighter than the existing ones. Detailed examples have been presented.
We propose replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MoE). We show that all proven MoE relations using concurrence can be rephrased in terms of CREN. Furthermore we show that higher-dimensional (qudit) extensions of MoE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MoE in terms of concurrence. We further test the CREN version of MoE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MoE for qu…
We propose the square of convex-roof extended negativity(SCREN) as a powerful candidate to characterize strong monogamy of multi-party quantum entanglement. We first provide a strong monogamy inequality of multi-party entanglement using SCREN, and show that the tangle-based multi-qubit strong monogamy inequality can be rephrased by SCREN. We further show that SCREN strong monogamy inequality is still true for the counterexamples that violate tangle-based strong monogamy inequality in higher-dimensional quantum systems rather than qubits. We also analytically show that SCREN strong monogamy ine…
Entanglement negativity is a measure of mixed-state entanglement increasingly used to investigate and characterize emerging quantum many-body phenomena, including quantum criticality and topological order. We present two results for the entanglement negativity: a disentangling theorem, which allows the use of this entanglement measure as a means to detect whether a wave-function of three subsystems $A$, $B$, and $C$ factorizes into a product state for parts $AB_1$ and $B_2C$; and a monogamy relation, which states that if $A$ is very entangled with $B$, then $A$ cannot be simultaneaously very e…
The notion of non-classical correlations is a powerful contrivance for explaining phenomena exhibited in quantum systems. It is well known, however, that quantum systems are not free to explore arbitrary correlations---the church of the smaller Hilbert space only accepts monogamous congregants. We demonstrate how to characterize the limits of what is quantum mechanically possible with a computable measure, entanglement negativity. We show that negativity only saturates the standard linear monogamy inequality in trivial cases implied by its monotonicity under LOCC, and derive a necessary and su…
In this paper, we study the $α$th power monogamy properties related to the entanglement measure in bipartite states. The monogamy relations related to the $α$th power of negativity and the Convex- Roof Extended Negativity are obtained for N-qubit states. We also give a tighter bound of hierarchical monogamy inequality for the entanglement of formation. We find that the GHZ state and W state can be used to distinguish the $α$th power the concurrence for $0<α<2$. Furthermore, we compare concurrence with negativity in terms of monogamy property and investigate the difference between them.
We develop theories of entanglement distribution and of entanglement dynamics for qudit systems, which incorporate previous qubit formulations. Using convex-roof extended negativity, we generalize previous qubit results for entanglement distribution with isotropic states and for entanglement dynamics with the depolarizing channel, and we establish a relation between these two types of entanglement networks.
We propose a new entanglement measure to quantify three qubits entanglement in terms of negativity. A monogamy inequality analogous to Coffman-Kundu-Wootters (CKW) inequality is established. This consequently leads to a definition of residual entanglement, which is referred to as three-$π$ in order to distinguish from three-tangle. The three-$π$ is proved to be a natural entanglement measure. By contrast to the three-tangle, it is shown that the three-$π$ always gives greater than zero values for $W$ and GHZ classes, implying there always exists three-way entanglement for them, and three-tangl…
We present a set of inequalities based on mean values of quantum mechanical observables nonlinear entanglement witnesses for bipartite quantum systems. These inequalities give rise to sufficient and necessary conditions for separability of all bipartite pure states and even some mixed states. In terms of these mean values of quantum mechanical observables a measurable lower bound of the convex-roof extension of the negativity is derived.
We extend the concept of the negativity, a good measure of entanglement for bipartite pure states, to mixed states by means of the convex-roof extension. We show that the measure does not increase under local quantum operations and classical communication, and derive explicit formulae for the entanglement measure of isotropic states and Werner states, applying the formalism presented by Vollbrecht and Werner [Phys. Rev. A {\bf 64}, 062307 (2001)].
We investigate monogamy relations related to quantum entanglement for $n-$qubit quantum systems. General monogamy inequalities are presented to the $β$th $(β\in(0,2))$ power of concurrence, negativity and the convex-roof extended negativity, as well as the $β$th $(β\in(0,\sqrt{2}))$ power of entanglement of formation. These monogamy relations are complementary to the existing ones with different regions of parameter $β$. In additions, new monogamy relations are also derived which include the existing ones as special cases.
The monogamy relation for quantum correlations is not satisfied by all measures for all multiparty quantum states. We prove that an arbitrary quantum state which is nonmonogamous for negativity will become monogamous if a finite number of copies of the same state is provided. We refer this as activation of nonmonogamous states. We also show that multiple copies of a state satisfy monogamy for negativity if it does so for a single copy. The results are true for all quantum states of an arbitrary number of parties. Moreover, we find that two different three-qubit pure states which individually v…
We provide a characterization of multiqubit entanglement monogamy and polygamy constraints in terms of negativity. Using the square of convex-roof extended negativity (SCREN) and the Hamming weight of the binary vector related to the distribution of subsystems proposed in Kim (Phys Rev A 97: 012334, 2018), we provide a new class of monogamy inequalities of multiqubit entanglement based on the $α$th power of SCREN for $α\geq1$ and polygamy inequalities for $0\leqα\leq1$ in terms of squared convex-roof extended negativity of assistance (SCRENoA). For the case $α<0$, we give the corresponding pol…
Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in literatures are not complete. We establish here a strict framework for defining multipartite entanglement measure (MEM): apart from the postulates of bipartite measure, a genuine MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM and a tightly complete monogamy relation for the genuine MEM. Consequently, we propose MEMs which…
For multipartite entangled states, entanglement monogamy is an important property. We investigate the monogamy relations for multiqubit generalized W-class states. We present new analytical monogamy inequalities satisfied by the $x$-th power of the dual of convex-roof extended negativity, namely CRENOA, for $x\geq2$ and $x\leq0$. As for The squared Rényi-$α$ entanglement (SR$α$E) with $α$ in the region $[(\sqrt 7 - 1)/2,(\sqrt {13} - 1)/2]$, we show the upper bound of SR$α$E.
Here we investigate the monogamy relations for the generalized W class. Monogamy inequalities for the generalized W class in terms of the $β$-th ($β\in (0,2)$) power of the concurrence, the concurrence of assistance and the negativity are presented. At last, under some restricted conditions of the generalized W class, we present stronger monogamy inequalities.
Monogamy relations characterize the distributions of entanglement in multipartite systems. We investigate monogamy relations for multiqubit generalized W-class states. We present new analytical monogamy inequalities for the concurrence of assistance, which are shown to be tighter than the existing ones. Furthermore, analytical monogamy inequalities are obtained for the negativity of assistance.
We investigate the monogamy relations related to the concurrence, the entanglement of formation, convex-roof extended negativity, Tsallis-q entanglement and R'enyi-α entanglement, the polygamy relations related to the entanglement of formation, Tsallis-q entanglement and R'enyi-α entanglement. Monogamy and polygamy inequalities are obtained for arbitrary multipartite qubit systems, which are proved to be tighter than the existing ones. Detailed examples are presented.
Monogamy relations characterize the distributions of entanglement in multipartite systems. We investigate monogamy relations related to the concurrence C, the entanglement of formation E, negativity Nc and Tsallis-q entanglement Tq. Monogamy relations for the $α$th power of entanglement have been derived, which are tighter than the existing entanglement monogamy relations for some classes of quantum states. Detailed examples are presented.
Exploring an analytical expression for the convex roof of the pure state squared concurrence for rank 2 mixed states the entanglement of a system of three particles under decoherence is studied, using the monogamy inequality for mixed states and the residual entanglement obtained from it. The monogamy inequality is investigated both for the concurrence and the negativity in the case of local independent phase damping channel acting on generalized GHZ states of three particles and the local independent amplitude damping channel acting on generalized W state of three particles. It is shown that …
We introduce two tangle-based four-party entanglement measures $t_1$ and $t_2$, and two negativity-based measures $n_1$ and $n_2$, which are derived from the monogamy relations. These measures are computed for three four-qubit maximally entangled and W states explicitly. We also compute these measures for the rank-$2$ mixture $ρ_4 = p \ket{\mbox{GHZ}_4} \bra{\mbox{GHZ}_4} + (1 - p) \ket{\mbox{W}_4} \bra{\mbox{W}_4}$ by finding the corresponding optimal decompositions. It turns out that $t_1 (ρ_4)$ is trivial and the corresponding optimal decomposition is equal to the spectral decomposition. Pr…
We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a special case. Given $n$ matrices $A_i$, $i=1,\ldots,n$, of the same size, let $Z_1$ and $Z_2$ be the block matrices $Z_1:=(A_jA_i^*)_{i,j=1}^n$ and $Z_2:=(A_j^*A_i)_{i,j=1}^n$. Then the conjectured inequality is \[ \left(||Z_1||_1-\trace Z_1\right)^2 + \left(||Z_2||_1-\trace Z_2\right)^2 \le \left(\sum_{i\neq j} ||A_i||_2 ||A_j||_2\right)^2. \] We prove this inequality for the already challenging case $n=2$ with $A_1$ equal to the identity matrix.
We provide an evidence for the validity of strong monogamy inequality of multi-party quantum entanglement using square of convex-roof extended negativity(SCREN). We first consider a large class of multi-qudit mixed state that are in a partially coherent superposition of a generalized W-class state and the vacuum, and provide some useful properties about this class of states. We show that monogamy inequality of multi-qudit entanglement in terms of SCREN holds for this class of states. We further show that SCREN strong monogamy inequality of multi-qudit entanglement also holds for this class of …
We investigate the general monogamy and polygamy relations satisfied by quantum correlation measures. We show that there exist two real numbers $α$ and $β$ such that for any quantum correlation measure $Q$, $Q^x$ is monogamous if $x\geq α$ and polygamous if $0\leq x\leq β$ for a given multipartite state $ρ$. For $β<x<α$, we show that the monogamy relation can be superactivated by finite $m$ copies $ρ^{\otimes m}$ of $ρ$ for nonadditive correlation measures. As a detailed example, we use the negativity as the quantum correlation measure to illustrate such superactivation of monogamy properties.…
We present an interesting monogamy equation for $(2 \otimes 2 \otimes n)$-dimensional pure states, by which a quantity is found to characterize the tripartite entanglement with the GHZ type and W typeentanglements as a whole. In particular, we, for the first time, reveals that for any quantum state of a pair of qubits, the difference between the two remarkable entanglement measures, concurrence and negativity, characterizes the W type entanglement of tripartite pure states with the two-qubit state as reduced density.
In this paper, we present some monogamy relations of multiqubit quantum entanglement in terms of the βth power of concurrence, entanglement of formation and convex-roof extended negativity. These monogamy relations are proved to be tighter than the existing ones, together with detailed examples showing the tightness.
We show that any measure of entanglement that on pure bipartite states is given by a strictly concave function of the reduced density matrix is monogamous on pure tripartite states. This includes the important class of bipartite measures of entanglement that reduce to the (von Neumann) entropy of entanglement. Moreover, we show that the convex roof extension of such measures (e.g., entanglement of formation) are monogamous also on \emph{mixed} tripartite states. To prove our results, we use the definition of monogamy without inequalities, recently put forward[Gour and Guo, Quantum \textbf{2}, …
Monogamy and Polygamy are important properties of entanglement, which characterize the entanglement distribution of multipartite systems. We study general monogamy and polygamy relations based on the $α$th $(0\leqα\leq γ)$ power of entanglement measures and the $β$th $(β\geq δ)$ power of assisted entanglement measures, respectively. We illustrate that these monogamy and polygamy relations are tighter than the inequalities in the article [Quantum Inf Process 19, 101], so that the entanglement distribution can be more precisely described for entanglement states that satisfy stronger constraints.…
Monogamy of quantum correlations is a vibrant area of research because of its potential applications in several areas in quantum information ranging from quantum cryptography to co-operative phenomena in many-body physics. In this paper, we investigate conditions under which monogamy is preserved for functions of quantum correlation measures. We prove that a monogamous measure remains monogamous on raising its power, and a non-monogamous measure remains non-monogamous on lowering its power. We also prove that monogamy of a convex quantum correlation measure for arbitrary multipartite pure quan…
The monogamy of quantum entanglement captures the property of limitation in the distribution of entanglement. Various monogamy relations exist for different entanglement measures that are important in quantum information processing. Our goal in this work is to propose a general monogamy inequality for all entanglement measures on entangled qubit systems. The present result provide a unified model for various entanglement measures including the concurrence, the negativity, the entanglement of formation, Tsallis-q entropy, Renyi-q entropy, and Unified-(q,s) entropy. We then proposed tightened mo…
We investigate the generalized monogamy and polygamy relations $N$-qubit systems. We give a general upper bound of the $α$th ($0\leqα\leq2$) power of concurrence for $N$-qubit states. The monogamy relations satisfied by the $α$th ($0\leqα\leq2$) power of concurrence are presented for $N$-qubit pure states under the partition $AB$ and $C_1 . . . C_{N-2}$, as well as under the partition $ABC_1$ and $C_2\cdots C_{N-2}$. These inequalities give rise to the restrictions on entanglement distribution and the trade off of entanglement among the subsystems. Similar results are also derived for negativi…
We study the monogamy and polygamy inequalities of quantum correlations in arbitrary dimensional multipartite quantum systems. We first derive the monogamy inequality of the $α$th ($0\leqα\leq\frac{r}{2}, r\geq2$) power of concurrence for any $2\otimes2\otimes2^{n-2}$ tripartite states and generalize it to the $n$-qubit quantum states. In addition to concurrence, we show that the monogamy relations are satisfied by other quantum correlation measures such as entanglement of formation. Moreover, the polygamy inequality of the $β$th ($β\leq0$) power of concurrence and the $β$th ($β\geq s, 0\leq s…
The genuine multiparticle negativity is a measure of genuine multiparticle entanglement which can be computed numerically. We present several results how this entanglement measure can be characterized analytically. First, we show that with an appropriate normalization this measure can be seen as coming from a mixed convex roof construction. Based on this, we determine its value for $n$-qubit GHZ-diagonal states and four-qubit cluster-diagonal states.
Monogamy and polygamy relations characterize the distributions of entanglement in multipartite systems. We provide classes of monogamy and polygamy inequalities of multiqubit entanglement in terms of concurrence, entanglement of formation, negativity, Tsallis-$q$ entanglement and Rényi-$α$ entanglement, respectively. We show that these inequalities are tighter than the existing ones for some classes of quantum states.
We investigate the evolution of entanglement in the Fenna-Matthew-Olson (FMO) complex based on simulations using the scaled hierarchy equation of motion (HEOM) approach. We examine the role of multipartite entanglement in the FMO complex by direct computation of the convex roof optimization for a number of measures, including some that have not been previously evaluated. We also consider the role of monogamy of entanglement in these simulations. We utilize the fact that the monogamy bounds are saturated in the single exciton subspace. This enables us to compute more measures of entanglement ex…
We study the distribution of quantum correlations characterized by monogamy relations in multipartite systems. By using the Hamming weight of the binary vectors associated with the subsystems, we establish a class of monogamy inequalities for multiqubit entanglement based on the $α$th ($α\geq 2$) power of concurrence, and a class of polygamy inequalities for multiqubit entanglement in terms of the $β$th ($0\leq β\leq2$) power of concurrence and concurrence of assistance. Moveover, we give the monogamy and polygamy inequalities for general quantum correlations. Application of these results to q…
We report a nuclear magnetic resonance experiment, which simulates the quantum transverse Ising spin system in a triangular configuration and further show that the monogamy of quantum correlations can be used to distinguish between the frustrated and non-frustrated regimes in the ground state of this system. Adiabatic state preparation methods are used to prepare the ground states of the spin system. We employ two different multipartite quantum correlation measures to analyze the experimental ground state of the system in both the frustrated and non-frustrated regimes. In particular, we use mu…
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 $ρ$. 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 charact…
The quantification of quantum entanglement is a central issue in quantum information theory. Recently, Gao \emph{et al}. ( \href{http://dx.doi.org/10.1103/PhysRevLett.112.180501}{Phys. Rev. Lett. \textbf{112}, 180501 (2014)}) pointed out that the maximum of entanglement measure of the permutational invariant part of $ρ$ ought to be a lower bound on entanglement measure of the original state $ρ$, and proposed that this argument can be used as an additional requirement for (multipartite) entanglement measures. Whether any individual proposed entanglement measure satisfies the requirement still h…
We show that for all n-mode Gaussian states of continuous variable systems, the entanglement shared among n parties exhibits the fundamental monogamy property. The monogamy inequality is proven by introducing the Gaussian tangle, an entanglement monotone under Gaussian local operations and classical communication, which is defined in terms of the squared negativity in complete analogy with the case of n-qubit systems. Our results elucidate the structure of quantum correlations in many-body harmonic lattice systems.
Monogamy relations place restrictions on the shareability of quantum corellations in multipartite states. Being an intrinsic quantum feature, monogamy property throws light on {\emph{residual}} entanglement, an entanglement which is not accounted for by the pairwise entanglement in the state. Expressed in terms of suitable pairwise entanglement measures such as concurrence, the monogamy inequality leads to the evaluation of {\emph{tangle}}, a measure of residual entanglement. In this work, we explore monogamy relations in pure symmetric multiqubit states constituted by two distinct spinors, th…
Squashed entanglement and entanglement of purification are quantum mechanical correlation measures and defined as certain minimisations of entropic quantities. We present the first non-trivial calculations of both quantities. Our results lead to the conclusion that both measures can drop by an arbitrary amount when only a single qubit of a local system is lost. This property is known as "locking" and has previously been observed for other correlation measures, such as the accessible information, entanglement cost and the logarithmic negativity. In the case of squashed entanglement, the results…
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they a…
Using the convex-roof extended negativity and the negativity of assistance as quantifications of bipartite entanglement, we consider the possible remotely-distributed entanglement. For two pure states $\ketϕ_{AB}$ and $\ketψ_{CD}$ on bipartite systems $AB$ and $CD$, we first show that the possible amount of entanglement remotely distributed on the system $AC$ by joint measurement on the system $BD$ is not less than the product of two amounts of entanglement for the states $\ketϕ_{AB}$ and $\ketψ_{CD}$ in two-qubit and two-qutrit systems. We also provide some sufficient conditions, for which th…
For continuous-variable systems, we introduce a measure of entanglement, the continuous variable tangle ({\em contangle}), with the purpose of quantifying the distributed (shared) entanglement in multimode, multipartite Gaussian states. This is achieved by a proper convex roof extension of the squared logarithmic negativity. We prove that the contangle satisfies the Coffman-Kundu-Wootters monogamy inequality in all three--mode Gaussian states, and in all fully symmetric $N$--mode Gaussian states, for arbitrary $N$. For three--mode pure states we prove that the residual entanglement is a genuin…
Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structu…
We study the loss of entanglement of bipartite state subjected to discarding or measurement of one qubit. Examining the behavior of different entanglement measures, we find that entanglement of formation, entanglement cost, and logarithmic negativity are lockable measures in that it can decrease arbitrarily after measuring one qubit. We prove that any convex and asymptotically non-continuous measure is lockable. As a consequence, all the convex roof measures can be locked. Relative entropy of entanglement is shown to be a non-lockable measure.
We study bipartite entangled states in arbitrary dimensions and obtain different bounds for the entanglement measures in terms of teleportation fidelity. We find that there is a simple relation between negativity and teleportation fidelity for pure states but for mixed states, an upper bound is obtained for negativity in terms of teleportation fidelity using convex-roof extension negativity (CREN). However, with this it is not clear how to distinguish betweeen states useful for teleportation and positive partial transpose (PPT) entangled states. Further, there exists a strong conjecture in the…
We present observable lower bounds for several bipartite entanglement measures including entanglement of formation, geometric measure of entanglement, concurrence, convex-roof extended negativity, and G-concurrence. The lower bounds facilitate estimates of these entanglement measures for arbitrary finite-dimensional bipartite states. Moreover, these lower bounds can be calculated analytically from the expectation value of a single observable. Based on our results, we use several real experimental measurement data to get lower bounds of entanglement measures for these experimentally realized st…
Experimentally quantifying entanglement and coherence are extremely important for quantum resource theory. However, because the quantum state tomography requires exponentially growing measurements with the number of qubits, it is hard to quantify entanglement and coherence based on the full information of the experimentally realized multipartite states. Fortunately, other methods have been found to directly measure the fidelity of experimental states without quantum state tomography. Here we present a fidelity-based method to derive experimentally accessible lower bounds for measures of genuin…
Using measures of entanglement such as negativity and tangles we provide a detailed analysis of entanglement structures in pure states of non-interacting qubits. The motivation for this exercise primarily comes from holographic considerations, where entanglement is inextricably linked with the emergence of geometry. We use the qubit systems as toy models to probe the internal structure, and introduce some useful measures involving entanglement negativity to quantify general features of entanglement. In particular, our analysis focuses on various constraints on the pattern of entanglement which…
