We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.
Coffman, Kundu and Wootters presented the 3-tangle of three qubits in [Phys. Rev. A 61, 052306 (2000)]. Wong and Christensen extended the 3-tangle to even number of qubits, known as $n$-tangle [Phys. Rev. A 63, 044301 (2001)]. In this paper, we propose a generalization of the 3-tangle to any odd $n$-qubit pure states and call it the $n$-tangle of odd $n$ qubits. We show that the $n$-tangle of odd $n$ qubits is invariant under permutations of the qubits, and is an entanglement monotone. The $n$-tangle of odd $n$ qubits can be considered as a natural entanglement measure of any odd $n$-qubit pur…
A necessary condition of the maximally multipartite entangled states (MMES) is given via n-tangle. The condition shows that the n-tangle equal zero for the four-, and eight-qubit of MMESs and the n-tangle equal 1 for two- and six- qubits of MMESs. Furthermore,we give the theoretical limitation for maximally ten-qubit and twelve-qubit entangled states. We conjecture that 4n-qubit states of MMES should have n-tangle equal zero and 4n+2-qubit states of MMES should have n-tangle equal one.
In this paper, we derive a general formula of the tangle for pure states of three qubits, and present three explicit local unitary (LU) polynomial invariants. Our result goes beyond the classical work of tangle, 3-tangle and von Neumann entropy of entanglement for Ac\'ın et al.' Schmidt decomposition (ASD) of three qubits by connecting the tangle, 3-tangle, and von Neumann entropy for ASD with Ac\'ın et al.'s LU invariants. In particular, our result reveals a general relation among tangle, 3-tangle, and von Neumann entropy, together with a relation among their averages. The relations can help …
A new form of local unitary (LU) transformation invariant is given for multi-qubit states . The general relation between tangle and the LU transformation invariant of pure three and four-qubit states is given. We find that the tangle actually is a special LU transformation invariant. Furthermore, the tangle is a important local unitary (LU) transformation invariant which is a nessesary condition for maximally multi-qubit entangled state.
We show that $n$-tangle, the generalization of the 3-tangle to even $n$ qubits, is the square of the SLOCC polynomial invariant of degree 2. We find that the $n$-tangle is not the residual entanglement for any even $n\geq 4$\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence $C_{1(2...n)}$. The condition implies that the concurrence $% C_{1(2...n)}$ is always positive for any entangled states while the $n$% -tangle vanishes for some entangled states. We argue that for even $n$\ qubits, the concurrence $C_{1(2...n)}$\ is equal to or greater than the $n$% …
We present a new tripartite entanglement measure for three-qubit mixed states. The new measure $t_{\mathrm{r}}(ρ)$, which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of $ t_{\mathrm{r}}(ρ)$ for a three-qubit mixed state $ρ$, we can also derive $t_{\mathrm{r}}(ρ')$ analytically for any states $ρ'$ which are SLOCC-equivalent to the state $ρ$. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle $t_{\mathrm{r}}$ is the app…
We obtain, analytically, the global negativity, partial $K-$way negativities (K=2, 3), Wooter's tangle and three tangle for the generic three qubit canonical state. It is found that the product of global negativity and partial three way negativity is equal to three tangle, while the partial two way negativity is related to tangle of qubit pairs. We also calculate similar quantities for the state canonical to a single parameter (0<q<1) pure state which is a linear combination of a GHZ state and a W state. In this case for q=0.62685, the state has zero three tangle and zero three-way negativity,…
We analyze mixed multi-qubit states composed of a W class state and a product state with all qubit in |0>. We find the optimal pure state decomposition and convex roofs for higher-tangle with bipartite partition between one qubit and the rest qubits for those mixed states. The optimality of the decomposition is ensured by the Coffman-Kundu-Wootters (CKW) inequality which describes the monogamy of quantum entanglement. The generalized monogamy inequality is found to be true for W class states with arbitrary partitions between one qubit and multi-qubit.
We report a four qubit polynomial invariant that quantifies genuine four-body correlations. The four qubit invariants are obtained from transformation properties of three qubit invariants under a local unitary on the fourth qubit.
We provide a complete analysis of mixed three-qubit states composed of a GHZ state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find th…
Closed formulae for upper bound on three tangles of three-qubit reduced states in terms of three-qubit invariant polynomials of pure four-qubit states are obtained. Our results offer tighter constraints on total three-way entanglement of a given qubit with the rest of the system than those used in ref. [PRL 113, 110501 (2014), PRL 116, 049902(E) (2016)] to verify monogamy of four-qubit quantum entanglement.
Our main result is a monogamy inequality satisfied by the entanglement of a focus qubit (one-tangle) in a four-qubit pure state and entanglement of subsystems. Analytical relations between three-tangles of three-qubit marginal states, two-tangles of two-qubit marginal states and unitary invariants of four-qubit pure state are used to obtain the inequality. The contribution of three-tangle to one-tangle is found to be half of that suggested by a simple extension of entanglement monogamy relation for three qubits. On the other hand, an additional contribution due to a two-qubit invariant which i…
The three-tangle is a measure of three-way entanglement in a system of three qubits. For a pure state, it can be understood as the residual entanglement not accounted for by pairwise entanglements between individual qubits. Here we define and evaluate the analogous quantity for three rebits (that is, binary systems in the real-amplitude variant of quantum theory). We find that the resulting formula is the same as in the complex case, except that an overall absolute value sign is missing. As a result, the rebit three-tangle can be negative, expressing the possibility of non-monogamous entanglem…
Majorana stars, the $2j$ spin coherent states that are orthogonal to a spin-$j$ state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement - the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state, is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot applied to general non-symmetric $n$-qubit states. We show that after a series of SL$(2,\mathbb{C})$ transformations, non…
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement mea…
We report a set of monogamy constraints on one-tangle, two-tangles, three-tangles and four-way correlations of a general four-qubit pure state. It is found that given a two-qubit marginal state $ρ$ of a four qubit pure state $\left\vert Ψ_{4}\right\rangle $, the non-Hermitian matrix $ρ\widetildeρ$ where $\widetildeρ$ $=\left( σ_{y} \otimesσ_{y}\right) ρ^{\ast}\left( σ_{y}\otimesσ_{y}\right) $, contains information not only about the entanglement properties of the two-qubits in state $ρ$ but also about three tangles involving the selected pair as well as four-way correlations of the pair of qub…
We discuss entanglement of multiparticle quantum systems. We propose a potential measure of a type of entanglement of pure states of n qubits, the n-tangle. For a system of two qubits the n-tangle is equal to the square of the concurrence, and for systems of three qubits it is equal to the ''residual entanglement''. We show that the n -tangle, is also equal to the generalization of concurrence squared for even n, and use this fact to prove that the n-tangle is an entanglement monotone. However, the n-tangle is undefined for odd n>3. Finally we propose a measure related to the n-tangle for mixe…
The $k$-ME concurrence as a measure of multipartite entanglement (ME) unambiguously detects all $k$-nonseparable states in arbitrary dimensions, and satisfies many important properties of an entanglement measure. Negativity is a simple computable bipartite entanglement measure. Invariant and tangle are useful tools to study the properties of the quantum states. In this paper we mainly investigate the internal relations among the $k$-ME concurrence, negativity, polynomial invariants, and tangle. Strong links between $k$-ME concurrence and negativity as well as between $k$-ME concurrence and pol…
Based on quantum complementary relations (QCRs) and a purification scenario, we analyze a class of N-qubit mixed states that are entangled but do not have two-, and genuine three-, four-, ..., N-qubit entanglements. It is shown that entanglement (one-tangle or negativity) in these mixed states is closely related to the QCR entanglement of their purified states. In particular, it is elaborated that when the mixed state does not have multipartite tangles (two- and higher tangles), its entanglement is actually a kind of genuine multipartite QCR entanglement between the system and its environment.
We define an entanglement measure, called the partial tangle, which represents the residual two-qubit entanglement of a three-qubit pure state. By its explicit calculations for three-qubit pure states, we show that the partial tangle is closely related to the faithfulness of a teleportation scheme over a three-qubit pure state.
We explore the connection between quantum brachistochrone (time-optimal) evolution of a three-qubit system and its residual entanglement called three-tangle. The result shows that the entanglement between two qubits is not required for some brachistochrone evolutions of a three-qubit system. However, the evolution between two distinct states cannot be implemented without its three-tangle, except for the trivial cases in which less than three qubits attend evolution. Although both the probability density function of the time-averaged three-tangle and that of the time-averaged squared concurrenc…
We investigate the entanglement measures of tripartite W-State and GHZ-state in noninertial frame through the coordinate transformation between Minkowski and Rindler. First it is shown that all three qubits undergo in a uniform acceleration $a$ of W-State, we find that the one-tangle, two-tangle, and $π$-tangle decrease when the acceleration parameter $r$ increases, and the two-tangle cannot arrive to infinity of the acceleration. Next we show that the one qubit goes in a uniform acceleration $a_{1}$ and the other two undergo in a uniform acceleration $a$ of GHZ-state, we find that the two-tan…
The multi-qubit GHZ state possesses tangles with elegant transformation properties under stochastic local operations and classical communication. Since almost all pure 3-qubit states are connected to the GHZ state via SLOCC, we derive a necessary and sufficient achievability inequality on arbitrary 3-qubit tangles, which is a strictly stronger constraint than both the monogamy inequality and the marginal eigenvalue inequality. We then show that entanglement shared with any single party in the n-qubit GHZ SLOCC equivalence class is precisely accounted for by the sum of its k-tangles, recently c…
In this paper we describe a set of circuits that can measure the concurrence of a two qubit density matrix without requiring the deliberate addition of noise. We then extend these methods to obtain a circuit to measure one type of three qubit entanglement for pure states, namely the 3-tangle.
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…
A family of rank-n (n=5,6,7,8) three-qubit mixed states are constructed. The explicit expressions for the three-tangle and optimal decompositions for all these states are given. The CKW relations for these states are also discussed.
Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the "tangle," which is related to the entanglement of formation. Specifically, we show that the tangle between A and B, plus the tangle between A and C, cannot be greater than the tangle between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the tangles satisfying the corresponding equality, one can find …
In this paper we establish a relation between quantum relative phase, $m$-tangle, and multi-local Lorentz-group invariant or $SL(2,\mathbb{C})^{\times m}$-invariant $S^{2}_{(m)}$. Our construction is based on the orthogonal complement of a positive operator valued measure on quantum phase. In particular, we propose a quantity based on the quantum relative phase of a multi-qubit operator that coincides with $m$-tangle, and multi-local Lorentz-group invariant.
The two-qubit canonical decomposition SU(4) = [SU(2) \otimes SU(2)] Delta [SU(2) \otimes SU(2)] writes any two-qubit quantum computation as a composition of a local unitary, a relative phasing of Bell states, and a second local unitary. Using Lie theory, we generalize this to an n-qubit decomposition, the concurrence canonical decomposition (C.C.D.) SU(2^n)=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any computation in K preserves the tangle |<phi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi>|^2 for n even. Thus, the C.C.D. shows that any n-qubit quantum compu…
We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized GHZ state, a|000>+b|111>, and a generalized W state, c|001>+d|010>+f|100>. Using the methods introduced by Lohmayer et al. we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.
Five-Dimensional Compatibility and Stratified Local-Unitary Structure of Pure Three-Qubit States
School of Physics and Materials Engineering, Hefei Normal University（合肥师范学院物理与材料工程学院）；School of Physics, Anhui University（安徽大学物理学院）；Leibniz International Joint Research Center of Materials Sciences of Anhui Province, Anhui University（安徽大学安徽省材料科学 Leibniz 国际联合研究中心）
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…
For two qubits in a pure state there exists a one-to-one relation between the entanglement measure (the concurrence ${\cal C}$) and the maximal violation ${\cal M}$ of a Bell inequality. No such relation exists for the three-qubit analogue of ${\cal C}$ (the tangle $τ$), but we have found that numerical data is consistent with a simple set of upper and lower bounds for $τ$ given ${\cal M}$. The bounds on $τ$ become tighter with increasing ${\cal M}$, so they are of practical use. The Svetlichny form of the Bell inequality gives tighter bounds than the Mermin form. We show that the bounds can b…
For pure symmetric 3-qubit states there are only three algebraically independent entanglement measures; one choice is the pairwise concurrence $\mathcal C$, the 3-tangle $τ$, and the Kempe invariant $κ$. Using a canonical form for symmetric $N$-qubit states derived from their Majorana representation, we derive the explicit achievable region of triples $(\mathcal C,τ,κ)$.
A general method in constructing a complete set of wave functions for multipartite identical qubits is presented based on the irreducible representations of the permutation group and the nth rank tensors. Particular examples for n =2, 3, and 4 are derived and the entanglement behavior for each state is examined from several criteria. It is found that the states so constructed are all bound entangled states. For the case of even n, all the states are found to have maximum "n-tangle". The symmetry in spin space is found to increase the n-tangle in general. The "n-tangle" for n = 4 is found not a…
Let Alice, Bob, and Charlie initially share an arbitrary fermionic three-qubit pure state, whose three-tangle is $τ_3^{(0)}$. It is shown within the single-mode approximation that if one party among the three of them moves with a uniform acceleration with respect to the other parties, the three-tangle reduces to $τ_3^{(0)} \cos^2 r$, where $r$ denotes a statistical factor in Fermi-Dirac statistics.
The role of SU(2) invariants for the classification of multiparty entanglement is discussed and exemplified for the Kempe invariant I_5 of pure three-qubit states. It is found to being an independent invariant only in presence of both W-type entanglement and threetangle. In this case, constant I_5 admits for a wide range of both threetangle and concurrences. Furthermore, the present analysis indicates that an SL^3 orbit of states with equal tangles but continuously varying I_5 must exist. This means that I_5 provides no information on the entanglement in the system in addition to that containe…
We investigate the relation between the entanglement and the robustness of a multipartite system to a depolarization noise. We find that the robustness of a two-qubit system in an arbitrary pure state depends completely on its entanglement. However, this is not always true in a three-qubit system. There is a residual effect on the robustness of a three-qubit system in an arbitrary superposition of Greenberger-Horne-Zeilinger state and W state. Its entanglement determines the trend of its robustness. However, there is a splitting on its robustness under the same entanglement. Its robustness not…
We analytically investigate the role of entanglement in time-optimal state evolution as an appli- cation of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly cou- pled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initia…
We investigate the lower bound obtained from experimental data of a quantum state $ρ$, as proposed independently by Gühne et al. and Eisert et al. for mixed states of three qubits. The measure we consider is the convex-roof extended three-tangle. Our findings highlight an intimate relation to lower bounds obtained recently from so-called characteristic curves of a given entanglement measure. We apply the bounds to estimate the three-tangle present in recently performed experiments aimed at producing a three-qubit GHZ state. A non-vanishing lower bound is obtained if the GHZ-fidelity of the pro…
Using the single-mode approximation, we first calculate entanglement measures such as negativity ($1-3$ and $1-1$ tangles) and von Neumann entropy for a tetrapartite W-Class system in noninertial frame and then analyze the whole entanglement measures, the residual $π_{4}$ and geometric $Π_{4}$ average of tangles. Notice that the difference between $π_{4}$ and $Π_{4}$ is very small or disappears with the increasing accelerated observers. The entanglement properties are compared among the different cases from one accelerated observer to four accelerated observers. The results show that there sti…
Usually, the three-tangle of a tripartite pure state of qubits can be directly measured with the simultaneous preparation of a not-less-than-four-fold copy of the state. We show that the exact genuine tripartite entanglement for ($2\otimes2\otimes n$)-dimensional pure quantum states can be measured in a similar manner, provided that only two simultaneous copies of the state are available. Lower bounds are also proposed for more convenient experimental operations. As an example, a comprehensive demonstration of the scheme is provided for the three-tangle of a three-qubit state.
We find a single parameter family of genuinely entangled three qubit pure states, called the maximally Bell inequality violating states (MBV), which exhibit maximum Bell inequality violation by the reduced bipartite system for a fixed amount of genuine tripartite entanglement quantified by the so called tangle measure. This in turn implies that there holds a complementary relation between the Bell inequality violation by the reduced bipartite systems and the tangle present in the three qubit states, not necessarily pure. The MBV states also exhibit maximum Bell inequality violation by the redu…
Based on the geometry of entangled three and two qubit states, we present the connection between the entanglement measure of the three-qubit state defined using the last Hopf fibration and the entanglement measures known as two- and three-tangle. Moreover, the generalization of the geometric representation of four qubit state and a potential entanglement measure is studied using sedenions for the simplification of the Hilbert space S^31 of the four qubit system. An entanglement measure is proposed and the degree of entanglement is calculated for specific states. The difficulties of a possible …
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…
We report the discovery of two new invariants for three-qubit states which, similarly to the 3-tangle, are invariant under local unitary transformations and permutations of the parties. These quantities have a direct interpretation in terms of the anisotropy of pairwise spin correlations. Applications include a universal ordering of pairwise quantum correlation measures for pure three-qubit states; tradeoff relations for anisotropy, 3-tangle and Bell nonlocality; strong monogamy relations for Bell inequalities, Einstein-Podolsky-Rosen steering inequalities, geometric discord and fidelity of re…
Employing the Pauli matrices, we have constructed a set of operators, which can be used to distinguish six inequivalent classes of entanglement under SLOCC (stochastic local operation and classical communication) for three-qubit pure states. These operators have very simple structure and can be obtained from the Mermin's operator with suitable choice of directions. Moreover these operators may be implemented in an experiment to distinguish the types of entanglement present in a state. We show that the measurement of only one operator is sufficient to distinguish GHZ class from rest of the clas…
Methods for measuring an integral of a classical field via local interaction of classical bits or local interaction of qubits passing through the field one at a time are analyzed. A quantum method, which has an exponentially better precision than any classical method we could see, is described.
Bipartite quantum entanglement for qutrits and higher-dimensional objects is considered. We analyze the possibility of violation of monogamy inequality, introduced by Coffman, Kundu, and Wootters, for some systems composed of such objects. An explicit counterexample with a three-qutrit totally antisymmetric state is presented. Since three-tangle has been confirmed to be a natural measure of entanglement for qubit systems, our result shows that the three-tangle is no longer a legitimate measure of entanglement for states with three qutrits or higher dimensional objects.
We present a scheme for a projective measurement of the parity operator $P_z=\prod_{i=1}^N σ_z^{(i)}$ of $N$-qubits. Our protocol uses a single ancillary qubit, or a probe qubit, and involves manipulations of the total spin of the $N$ qubits without requiring individual addressing. We illustrate our protocol in terms of an experimental implementation with atomic ions in a two-zone linear Paul trap, and further discuss its extensions to several more general cases.
We show that bipartite concurrence for rank-2 mixed states of qubits is written by an observable which can be exactly and directly measurable in experiment by local projective measurements, provided that four copies of the composite quantum system are available. In addition, for a tripartite quantum pure state of qubits, the 3-tangle is also shown to be measurable only by projective measurements on the reduced density matrices of a pair of qubits conditioned on four copies of the state.
