Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
AI Summary
The paper asks how to bound multipartite entanglement quantitatively without full state tomography. It uses a previously introduced definition in which entanglement up to degree $k$ is the maximum, over prescribed particle-removal sequences, of sums of single-particle-versus-cluster bipartite entanglement. Degree-$k$ entanglement is the difference between consecutive totals, rather than a separability-based classification.
For any bipartite state, it derives $E_{\rm sq}\geq\max\{L(P_A),L(P_B)\}-U(P_{AB},d_{AB})$, clipped at zero, where $P_X=\operatorname{Tr}\rho_X^2$ and $L,U$ are extremal entropies compatible with purity. A complementary bound is $E_{\rm sq}\geq\max\{C(M_A,M_B)^2/(4\|M_A\|^2\|M_B\|^2)-U(P_{AB},d_{AB})/2,0\}$. These follow from the Carlen–Lieb conditional-mutual-information inequality and a covariance bound on mutual information. The Koashi–Winter relation supplies $E_F\leq\sum_i p_iS(\rho_{A,i})$ after measuring an extension. Since $E_{\rm sq}\leq E_F$, these bracket both measures. Maximizing sums of bipartite bounds, then subtracting opposite bounds at adjacent degrees, yields multipartite bounds.
Examples include dephased GHZ states, W and Dicke states through ten qubits, and 100 sparse random five-qubit pure states. For dephased GHZ states, lower-degree upper bounds vanish; the exact formation entanglement is $h_2((1+\sqrt{1-p^2})/2)$.
The purity lower bound is informative primarily for highly entangled, nearly pure states. Generic multipartite optimization remains cumbersome; a subset-based dynamic program avoids enumerating complete sequences. Upper-bound evaluation requires conditional-state entropies; single-qubit pivots simplify this, but a mixed full-system cluster may require introducing a purification.
Little sign of AI-generated textIncrementalMissing comparisonOverclaimsStrong assumptions
Central claim may not hold: The stated multipartite faithfulness property fails: an explicitly biseparable three-qubit mixture has E^3=1 under the paper's definitions, undermining the claimed genuine-entanglement interpretation.
The proposed purification-based upper-bound protocol for a full mixed system assumes experimental access to a purifier that the paper does not establish, although a trivial accessible entropy upper bound survives.
The claimed equality between Dicke-state and GHZ N-partite entanglement at N=2m fails already for D_{4,2}, whose entangled three-qubit marginal makes its defined E_F^4 strictly less than one bit.
The random-state dynamic program still evaluates exponentially many particle subsets and is not an efficient algorithm in system size in the usual polynomial-time sense.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
The paper asks how to bound multipartite entanglement quantitatively without full state tomography. It uses a previously introduced definition in which entanglement up to degree k is the maximum, over prescribed particle-removal sequences, of sums of single-particle-versus-cluster bipartite entanglement. Degree-k entanglement is the difference between consecutive totals, rather than a separability-based classification.
For any bipartite state, it derives Esq≥max{L(PA),L(PB)}−U(PAB,dAB), clipped at zero, where PX=TrρX2 and L,U are extremal entropies compatible with purity. A complementary bound is Esq≥max{C(MA,MB)2/(4∥MA∥2∥MB∥2)−U(PAB,dAB)/2,0}. These follow from the Carlen–Lieb conditional-mutual-information inequality and a covariance bound on mutual information. The Koashi–Winter relation supplies EF≤∑ipiS(ρA,i) after measuring an extension. Since Esq≤EF, these bracket both measures. Maximizing sums of bipartite bounds, then subtracting opposite bounds at adjacent degrees, yields multipartite bounds.
Examples include dephased GHZ states, W and Dicke states through ten qubits, and 100 sparse random five-qubit pure states. For dephased GHZ states, lower-degree upper bounds vanish; the exact formation entanglement is h2((1+1−p2)/2).
The purity lower bound is informative primarily for highly entangled, nearly pure states. Generic multipartite optimization remains cumbersome; a subset-based dynamic program avoids enumerating complete sequences. Upper-bound evaluation requires conditional-state entropies; single-qubit pivots simplify this, but a mixed full-system cluster may require introducing a purification.
Multipartite entanglement determines the strength and range of interactions in many-body quantum systems. Yet, it is hard to evaluate it, due to the complex structures of quantum states. Here, we introduce a generic method to quantify the k <= N-partite entanglement of an N-particle system, by maximizing an arbitrary bipartite entanglement measure within subsystems of size up to k. The resulting classification of multipartite states captures their experimental cost: creating a k-partite entangled state requires at least k-1 two-particle entangling gates. Further, we analytically calculate the newly defined k-partite entanglement of formation, which generalizes an important bipartite entanglement measure, in several classes of states, including the W states of any dimension.
The focal point of this thesis is strongly based on quantum entanglement within Quantum Information Science. We provide an indepth analysis of entanglement quantification in both bipartite and multipartite quantum systems, with an emphasis on many-body entanglement. We examine the definitions of entanglement, introduce several entanglement measures, and examine the challenges in expanding these measures to multipartite scenarios. For bipartite systems, we detail entropy-based measures such as von Neumann and Rényi entropies, the entanglement spectrum and important algebraic metrics such as negativity and concurrence. We then elaborate on multipartite entanglement, introducing modern measures and detection techniques including geometric and global entanglement measurements, generalised multipartite concurrence, multipartite negativity, fidelity-based entanglement benchmarks, and entanglement witnesses. Some examples include Greenberger-Horne-Zeilinger (GHZ), and the W states to illustrate the distinction between biseparable and truly multi-partite-entangled states, as well as the SLOCC classification of families of entangled states. We also briefly introduce practical ways of measuring entanglement, such as the SWAP test, quantum state tomography for experimental estimation of entanglement and consequently randomised measurements. We apply these measures to specific quantum states (GHZ, W , graph states, and random states), analysing how entanglement evolves under operations and how different metrics capture distinct features of multi-qubit entanglement. Finally, we conclude with a summary of findings, a discussion of current limitations such as scalability of measures. The thesis is organised into five chapters: Introduction, Entanglement, Multipartite Entanglement Measures, Negativity vs. Three-Tangle, and Conclusion.
Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500 atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.
The theory of the asymptotic manipulation of pure bipartite quantum systems can be considered completely understood: the rates at which bipartite entangled states can be asymptotically transformed into each other are fully determined by a single number each, the respective entanglement entropy. In the multipartite setting, similar questions of the optimally achievable rates of transforming one pure state into another are notoriously open. This seems particularly unfortunate in the light of the revived interest in such questions due to the perspective of experimentally realizing multipartite quantum networks. In this Letter, we report substantial progress by deriving simple upper and lower bounds on the rates that can be achieved in asymptotic multipartite entanglement transformations. These bounds are based on ideas of entanglement combing and state merging. We identify cases where the bounds coincide and hence provide the exact rates. As an example, we bound rates at which resource states for the cryptographic scheme of quantum secret sharing can be distilled from arbitrary pure tripartite quantum states. This result provides further scope for quantum internet applications, supplying tools to study the implementation of multipartite protocols over quantum networks.
In this paper, we investigate the characterization of multipartite entanglement based on the unified (q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure Eq,sA∣B(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures Aq,sk(ρ) and Gq,sk(ρ) for quantifying k-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters q and s are in certain ranges, Aq,sk(ρ) and Gq,sk(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures Aq,sk(ρ) and Gq,sk(ρ) with respect to the parameters q and s for fixed quantum states. In addition, for fixed q and s, we also give their order relations for any two pure states.
We investigate parametrized multipartite entanglement measures from the perspective of k nonseparability in this paper. We present two types of entanglement measures in n-partite systems, $q\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$ and $\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$, which unambiguously detect all k-nonseparable states in arbitrary n-partite systems. Rigorous proofs show that the proposed k-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all k-separable states, and being strictly greater than zero for all k-nonseparable states. In particular, the $q\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}$ concurrence, renamed as $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}$ concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition (k=2). The lower bounds of the two classes k-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between $q\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence defined by us to other genuine entanglement measures.