Multipartite Entanglement Measures Research Articles

Exact Time Evolution of Genuine Multipartite Correlations for N-Qubit Systems in a Common Thermal Reservoir

We investigate the dynamical evolution of genuine multipartite correlations for N-qubits in a common reservoir considering a non-dissipative qubits-reservoir model. We derive an exact expression for the time-evolved density matrix by modeling the reservoir as a set of infinite harmonic oscillators with a bilinear form of interaction Hamiltonian. Interestingly, we find that the choice of two-level systems corresponding to an initially correlated multipartite state plays a significant role in potential robustness against environmental decoherence. In particular, the generalized W-class Werner state shows robustness against the decoherence for an equivalent set of qubits, whereas a certain generalized GHZ-class Werner state shows robustness for inequivalent sets of qubits. It is shown that the genuine multipartite concurrence (GMC), a measure of multipartite entanglement of an initially correlated multipartite state, experiences an irreversible decay of correlations in the presence of a thermal reservoir. For the GHZ-class Werner state, the region of mixing parameters for which there exists GMC, shrinks with time and with increase in the temperature of the thermal reservoir. Furthermore, we study the dynamical evolution of the relative entropy of coherence and von-Neumann entropy for the W-class Werner state.

Computable and Operationally Meaningful Multipartite Entanglement Measures.

Multipartite entanglement is an essential resource for quantum communication, quantum computing, quantum sensing, and quantum networks. The utility of a quantum state |ψ⟩ for these applications is often directly related to the degree or type of entanglement present in |ψ⟩. Therefore, efficiently quantifying and characterizing multipartite entanglement is of paramount importance. In this work, we introduce a family of multipartite entanglement measures, called concentratable entanglements. Several well-known entanglement measures are recovered as special cases of our family of measures, and hence we provide a general framework for quantifying multipartite entanglement. We prove that the entire family does not increase, on average, under local operations and classical communications. We also provide an operational meaning for these measures in terms of probabilistic concentration of entanglement into Bell pairs. Finally, we show that these quantities can be efficiently estimated on a quantum computer by implementing a parallelized SWAP test, opening up a research direction for measuring multipartite entanglement on quantum devices.

Quantum speed limit and geometric measure of entanglement

Using the approach offered by quantum speed limit, we show that geometric measure of multipartite entanglement for pure states [Phys. Rev. A 68, 042307(2003)] can be interpreted as the minimal time necessary to unitarily evolve a given quantum state to a separable one.

Hyperthreads in holographic spacetimes

We generalize bit threads to hyperthreads in the context of holographic spacetimes. We define a “k-thread” to be a hyperthread which connects k different boundary regions and posit that it may be considered as a unit of k-party entanglement. Using this new object, we show that the contribution of hyperthreads to calculations of holographic entanglement entropy are generically finite. This is accomplished by constructing a surface whose area determines their maximum allowed contribution. We also identify surfaces whose area is proportional to the maximum number of k-threads, motivating a possible measure of multipartite entanglement. We use this to make connections to the current understanding of multipartite entanglement in holographic spacetimes.

Entanglement of local operators and the butterfly effect

We study the robustness of quantum and classical information to perturbations implemented by local operator insertions. We do this by computing multipartite entanglement measures in the Hilbert space of local operators in the Heisenberg picture. The sensitivity to initial conditions that we explore is an illuminating manifestation of the butterfly effect in quantum many-body systems. We present a ``membrane theory'' in Haar random unitary circuits to compute the mutual information, logarithmic negativity, and reflected entropy in the local operator state by mapping to a classical statistical mechanics problem and find that any local operator insertion delocalizes information as fast as is allowed by causality after taking the large local Hilbert space dimension limit. Identical behavior is found for conformal field theories admitting holographic duals where the bulk geometry is described by the eternal black hole with a local object situated at the horizon. In contrast to these maximal scramblers, only an $O(1)$ amount of information is found to be delocalized by local operators in free fermionic systems and random Clifford circuits.

The role of localizable concurrence in quantum teleportation protocols

Teleporting an unknown qubit state is a paradigmatic quantum information processing task revealing the advantage of quantum communication protocols over their classical counterpart. For a teleportation protocol using a Bell state as quantum channel, the resource has been identified to be the concurrence. However, for mixed multipartite states the lack of computable entanglement measures has made the identification of the quantum resource responsible for this advantage more challenging. Here, by building on previous results showing that localizable concurrence is the necessary resource for controlled quantum teleportation, we show that any teleportation protocol using an arbitrary multipartite state, that includes a Bell measurement, requires a nonvanishing localizable concurrence between two of its parties to perform better than the classical protocol. By first analyzing Greenberger–Horne–Zeilinger (GHZ) channel and GHZ measurement teleportation protocol, in the presence of GHZ-symmetric-preserving noise, we compare different multipartite entanglement measures with the fidelity of teleportation, and we find that the protocol performs better than the classical protocol when all multipartite entanglement measures vanish, except for the localizable concurrence. Finally, we extend our proof to an arbitrary teleportation protocol with an arbitrary multipartite entangled channel.

The verification of a requirement of entanglement measures

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 $\rho$ ought to be a lower bound on entanglement measure of the original state $\rho$, 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 has to prove. In this work, we show that most known entanglement measures of bipartite quantum systems satisfy the new criterion, include all convex-roof entanglement measures, the relative entropy of entanglement, the negativity, the logarithmic negativity and the logarithmic convex-roof extended negativity. Our results give a refinement in quantifying entanglement and provide new insights into a better understanding of entanglement properties of quantum systems.

Separability of heterogeneous quantum systems using multipartite concurrence and tangle

As heterogeneous quantum systems become a high-impact topic in the state of the art through hybrid entanglement, none of the many competing measures of multipartite entanglement has yet been shown to provide a complete characterization. In this paper, for the first time, we connect two of the most popular research directions on entanglement of heterogeneous systems: correlation tensors on the one hand, and concurrence and tangle on the other. We find the first ever separability criteria for heterogeneous quantum systems under any arbitrary partition in terms of multipartite concurrence and tangle, respectively, generalizing and refining previous work on m-separability of homogeneous systems, and we derive the first relations between the Frobenius norms of correlation tensors, concurrence and tangle, generalizing previous work for homogeneous systems.

Metrologically resourceful multipartite entanglement under quantum many-body effects

In traditional quantum metrology protocols, the initial multipartite entangled pure quantum probes are considered to be isolated, i.e., free of quantum many-body effects. Here, we study the impact of inherent many-body effects such as interaction with noisy environment and nonlocal interactions among particles on metrologically resourceful multipartite entanglement of initially mixed quantum probes. In this regard, we employ an information-theoretic multipartite entanglement measure as a figure-of-merit. The inevitable interaction with the noisy environment leads to disentanglement in multipartite quantum probes which restricts its metrological advantage. For this, we use entanglement dissociation to derive bounds on the multipartite entanglement measure that can identify the relevant entanglement structure under global as well as local noisy evolution. Furthermore, we investigate nonlocal interactions in terms of their entangling capability in a multipartite quantum probe. We show that such nonlocal interactions can be exploited as a valuable resource that exhibits better precision scaling in mixed-state quantum metrology. Moreover, we numerically observe these results for GHZ-W class states.

Entanglement in first excited states of some many-body quantum spin systems: indication of quantum phase transition in finite size systems

We compute concurrence, a measure of bipartite entanglement, of the first excited state of the 1-D Heisenberg frustrated J 1-J 2 spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength α = J 2/J 1 ≈ 0.241, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with 8 sites to 16 sites, and the value of α at which the change in entanglement is observed, asymptotically tends to a value α c ≈ 0.24116. We have calculated the finite-size scaling exponents for spin chains with even and odd spins. It may be noted that bipartite as well as multipartite entanglement measures applied on the ground state of the system, fail to detect any quantum phase transition from the gapless to the gapped phase in the 1-D Heisenberg frustrated J 1-J 2 spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like 2-D Heisenberg J 1-J 2 model and Shastry-Sutherland model and find similar indications of quantum phase transitions.

A brief note on the Scott measure as a multipartite entanglement criterion

In recent decades, various multipartite entanglement measures have been proposed by many researchers, with different characteristics. Meanwhile, Scott studied various interesting aspects of multipartite entanglement measures and he has defined a class of related multipartite entanglement measures in an obvious manner. Recently, Jafarpour et al (2015 Int. J. Quantum Inform. 13 1550 047) have calculated the entanglement quantity of a two-dimensional five-site spin system by means of the Scott measure. Their calculation presented . In this note, we would like to point out that in a five-qudit system, Q2 does not provide a stronger entanglement than Q3, while there is a simple relation between them.

Experimentally Accessible Lower Bounds for Genuine Multipartite Entanglement and Coherence Measures

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 genuine multipartite entanglement and coherence. On the one hand, the method works for genuine multipartite entanglement measures including the convex-roof extended negativity, the concurrence, the G-concurrence, and the geometric measure for genuine multipartite entanglement. On the other hand, the method also delivers observable lower bounds for the convex roof of the $l_{1}$-norm of coherence, the geometric measure of coherence, and the coherence of formation. Furthermore, all the lower bounds are based on the fidelity between the chosen pure state and the target state, and we obtain the lower bounds of several real experimental states as examples of our results.

Multipartite entanglement measure and complete monogamy relation

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 are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis $q$-entropy of entanglement, R\'{e}nyi $\alpha$-entropy of entanglement, the convex-roof extension of negativity and negativity, respectively. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis $q$-entropy of entanglement are genuine MEMs, (ii) multipartite extensions of R\'{e}nyi $\alpha$-entropy of entanglement, negativity and the convex-roof extension of negativity are unified MEMs but not genuine MEMs, and (iii) all these multipartite extensions are completely monogamous and the ones which are defined by the convex-roof structure (except for the R\'{e}nyi $\alpha$-entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, we find a class of tripartite states that one part can maximally entangled with other two parts simultaneously according to the definition of maximally entangled mixed state (MEMS) in [Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that there is no MEMS and that the only MES is the pure MES.

Relations among k-ME concurrence, negativity, polynomial invariants, and tangle

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 polynomial invariants are derived. We obtain the quantitative relation between $k$-ME ($k$=$n$) concurrence and negativity for all $n$-qubit states, give a exact value of the $n$-ME concurrence for the mixture of $n$-qubit GHZ states and white noise, and derive an connection between $k$-ME concurrence and tangle for $n$-qubit W state. Moreover, we find that for any $3$-qubit pure state the $k$-ME concurrence ($k$=2, 3) is related to negativity, tangle and polynomial invariants, while for $4$-qubit states the relations between $k$-ME concurrence (for $k$=2, 4) and negativity, and between $k$-ME concurrence and polynomial invariants also exist. Our work provides clear quantitative connections between $k$-ME concurrence and negativity, and between $k$-ME concurrence and polynomial invariants.

Construction of Complete Orthogonal Genuine Multipartite Entanglement State**Supported by the National Natural Science Foundation of China under Grant Nos 11775177, 11775178, 11647057 and 11705146, the Special Research Funds of the Department of Education of Shaanxi Province under Grant No 16JK1759, the Basic Research Plan of Natural Science in Shaanxi Province under Grant No 2018JQ1014, the Major Basic Research Program of

With the development of quantum information processing, multipartite entanglement measures are needed in many cases. However, there are still no complete orthogonal genuine multipartite entanglement (GME) bases available as Bell states to bipartite systems. To achieve this goal, we find a method to construct complete orthogonal GME states, and we exclude many equivalent states by leveraging the group theory. We also provide the case of a 3-order 3-dimensional Hilbert space as an example and study the application of general results in the dense coding scheme as an application. Moreover, we discuss some open questions and believe that this work will enlighten extensive studies in this field.

Multipartite entanglement and quantum phase transitions in the XXZ model with Dzyaloshinskii–Moriya interaction

With the aid of the quantum renormalization group (QRG) method, we study the multipartite entanglement and quantum phase transitions (QPTs) in the XXZ model with Dzyaloshinskii–Moriya (DM) interaction. The measures employed here are genuine multipartite entanglement (GME-concurrence) and global entanglement. The results show that both multipartite entanglement measures can accurately detect QPTs at the critical points after sufficient iteration steps. By the difference analysis, moreover, we derive an inequality such that the global entanglement is exactly an upper bound of the square GME-concurrence. Actually, this inequality is established whether or not the QRG method is carried out, which discloses some intrinsic features of long-scale multipartite entanglement. Based on the first derivative of GME-concurrence, the non-analytic phenomenon and scaling behaviors with different theoretical exponents are analyzed in detail, showing that the ground states of the XXZ model with DM interaction are asymmetrical.

Periodicity of quantum correlations in the quantum kicked top

Quantum kicked top is a fundamental model for time-dependent, chaotic Hamiltonian system and has been realized in experiments as well. As the quantum kicked top can be represented as a system of qubits, it is also popular as a testbed for the study of measures of quantum correlations such as entanglement, quantum discord and other multipartite entanglement measures. Further, earlier studies on kicked top have led to a broad understanding of how these measures are affected by the classical dynamical features. In this work, relying on the invariance of quantum correlation measures under local unitary transformations, it is shown exactly these measures display periodic behaviour either as a function of time or as a function of the chaos parameter in this system. As the kicked top has been experimentally realised using cold atoms as well as superconducting qubits, it is pointed out that these periodicities must be factored in while choosing of experimental parameters so that repetitions can be avoided.

Entanglement of purification for multipartite states and its holographic dual

We introduce a new information-theoretic measure of multipartite correlations ΔP, by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjecture a holographic dual of multipartite entanglement of purification ΔW, as a sum of minimal areas of codimension-2 surfaces which divide the entanglement wedge into multi-pieces. We prove that this geometrical quantity satisfies all properties we proved for the multipartite entanglement of purification. These agreements strongly support the ΔP = ΔW conjecture. We also show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement. We discuss potential saturation of multipartite squashed entanglement onto multipartite mutual information in holographic CFTs and its applications.

A Brief Overview of Bipartite and Multipartite Entanglement Measures

Measuring entanglement is a demanding task in the field of quantum computation and quantum information theory. Recently, some authors experimentally demonstrated an embedding quantum simulator, using it to efficiently measure two-qubit entanglement. Here, we are reviewing some measures of entanglement which are used for pure and mixed states. Furthermore, we have reported the efficient bipartite and multipartite entanglement measures.

Quantum information measures for restricted sets of observables

We study measures of quantum information when the space spanned by the set of accessible observables is not closed under products, i.e., we consider systems where an observer may be able to measure the expectation values of two operators, $\langle O_1 \rangle$ and $\langle O_2 \rangle$, but may not have access to $\langle O_1 O_2 \rangle$. This problem is relevant for the study of localized quantum information in gravity since the set of approximately-local operators in a region may not be closed under arbitrary products. While we cannot naturally associate a density matrix with a state in this setting, it is still possible to define a modular operator for a state, and distinguish between two states using a relative modular operator. These operators are defined on a little Hilbert space, which parameterizes small deformations of the system away from its original state, and they do not depend on the structure of the full Hilbert space of the theory. We extract a class of relative-entropy-like quantities from the spectrum of these operators that measure the distance between states, are monotonic under contractions of the set of available observables, and vanish only when the states are equal. Consequently, these distance-measures can be used to define measures of bipartite and multipartite entanglement. We describe applications of our measures to coarse-grained and fine-grained subregion dualities in AdS/CFT and provide a few sample calculations to illustrate our formalism.