Strong monogamy conjecture for multiqubit entanglement: The four-qubit case
We investigate the distribution of bipartite and multipartite entanglement in multiqubit states. In particular we define a set of monogamy inequalities sharpening the conventional Coffman–Kundu–Wootters constraints, and we provide analytical proofs of their validity for relevant classes of states. We present extensive numerical evidence validating the conjectured strong monogamy inequalities for arbitrary pure states of four qubits.
Introduction.— Entanglement is one of the most fundamental and intriguing features of quantum mechanics. It plays a crucial role for applications in quantum information and communication, and in a variety of areas ranging from quantum field theory to condensed matter, statistical physics, thermodynamics, and biology review_horodecki (); review (); jensreview (). Despite its central importance, however, the physical understanding and mathematical description of its essential characteristics remain highly nontrivial tasks, particularly when many-particle systems are analyzed.
One of the key properties distinguishing entanglement from classical correlations is its monogamy: entanglement cannot be freely shared among multiple parties barbara (). Monogamy is a consequence of the no-cloning theorem nocloning (); gamonorev (); clonprl (), and is obeyed by several types of nonclassical correlations, including Bell nonlocality toner (), Einstein–Podolsky–Rosen steering monosteer () and contextuality monocontext (), albeit not discord-type correlations arequantum ().
In 2000 Coffman, Kundu, and Wootters (CKW) formalized the monogamy of entanglement for a three-qubit system in the form of a quantitative constraint, known as ‘monogamy inequality’ monogamy (). Given an arbitrary pure state of three qubits , , and , the CKW inequality with respect to the choice of as a focus can be written as
Here denotes the bipartite entanglement in the reduced state of the pair of qubits and , quantified by a computable entanglement monotone known as two-tangle, or simply tangle concurrence (); concurrence2 (); monogamy (). The term denotes the one-tangle, a measure of entanglement between and the rest of the system, given by the linear entropy of the marginal state of qubit ,
where is the density matrix of qubit and indicates the partial trace of over subsystem .
The meaning of Eq. (1) is clear: the entanglement between and the two other qubits taken as a group cannot be less than the sum of the individual entanglements between and each of the two remaining qubits. Similar inequalities can be written by selecting or as focus qubits. Remarkably, the difference between left and right hand side of (1) can be interpreted as a quantifier of the entanglement genuinely shared among the three qubits. Precisely, one can define the residual three-qubit tangle—or, in short, three-tangle—of as
Interestingly, this quantity does not depend on the focus qubit (e.g., ) that we privilege in the decomposition. Namely, as well monogamy (). The three-tangle is a full-fledged measure of the genuine tripartite entanglement of any three-qubit pure state dur (); review_horodecki ().
A generalization of the CKW inequality (1) to -qubit systems was only proven by Osborne and Verstraete genmon () several years after the original conjecture monogamy (). Denoting now by a general pure state of qubits, the following holds genmon (),
This means that the entanglement between and the rest is not less than the sum of the individual pairwise entanglements involving and each of the other qubits (). However, for , the difference between left and right hand side in (4) just gives a rough indicator of all the leftover entanglement not distributed in pairwise form. Attempts to construct generalized monogamy inequalities in -qubit systems have been considered gmchris (); gmgour (); gmcorn (); jensreview (), but these have not led to clear recipes to isolate the genuine -partite entanglement, nor have resulted in a general sharpening of (4) for arbitrary states.
In this Letter we propose and investigate a set of sharper monogamy constraints. We raise the intuitive hypothesis that the residual in (4) is amenable to a further decomposition into individual -partite contributions which involve qubits, in all possible combinations encompassing the focus qubit . Heuristically, one can expect that all of these multipartite contributions be independent, overall adding up to the global bipartite entanglement between and the rest of the system. This leads us to postulate a hierarchy of strong monogamy (SM) inequalities limiting the distribution of bipartite and multipartite entanglement in -qubit systems, which take in general the following form
where we have employed a short-hand notation, introducing the index vector which spans all the ordered subsets of the index set with distinct elements, and we have included in general a sequence of rational exponents , with , which can regulate the weight assigned to the different -partite contributions.
Our main conjecture is that inequality (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case), and its variants for different choices of the focus qubit, hold simultaneously for arbitrary pure states of qubits, provided one adopts a suitable definition of the -partite quantities . We remark that, for a given choice of the involved entanglement monotones (tangles), the expression in (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) yields a whole class of monogamy constraints, parameterized by the powers . Any nontrivial selection of the sequence with defines in fact a particular SM inequality, sharpening and generalizing the CKW one. Clearly, the verification of (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) given a set implies its validity for all . For this reason, in order to establish the sharpest instance, one should aim to prove the inequalities by fixing each to be as small as possible, with being the minimal choice. We will specify the adopted choices of the parameters in the subsequent analysis.
Interestingly, a constraint alike to (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) was shown to hold for the distribution of entanglement in permutationally-invariant continuous variable Gaussian states, leading to an operational quantification of genuine -partite entanglement strongmonogauss (). This gives a strong hint that a similar sharing structure should hold for entanglement in finite-dimensional systems too, although no supporting evidence was obtained prior to this work.
Setting up the notation.— Here we adopt the following prescriptions. First, we define the pure-state residual -tangle as the difference between left and right hand side in (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case),
In this way, the conjectured SM inequality (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) is recast into the nonnegativity of the residual, , where the ordering of the subscripts in (6) reflects the choice of the focus qubit, which occupies the first slot (we do not expect permutation invariance for ). Next, we extend the residual -tangle to a mixed state of qubits via a convenient and physically motivated convex roof procedure,
where the minimization is taken over all possible pure-state decompositions of the state . For , the definition (7) reduces to the mixed-state extension of the three-tangle as defined in verstraete4 (), which is an entanglement monotone monogamy (); dur (); review_horodecki () and an invariant under stochastic local operations and classical communication (SLOCC) verstraete4 (); verstraetebart (); jens (); jensreview (). For , we recover the standard pairwise tangle, , with the concurrence concurrence (); concurrence2 () defined as , where are the square-roots of the eigenvalues (in decreasing order) of the matrix , the star denoting complex conjugation in the computational basis, being the Pauli matrix, and being the marginal state of qubits and obtained by partial tracing over the remaining qubits. Finally, we use Eqs. (6)–(7) to define, in a recursive way, every -partite term (for ) appearing in the -qubit SM inequality (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case), in terms of the corresponding residual -tangle rescaled by a suitable exponent .
Proving the SM conjecture for qubits appears in general a formidable challenge. Namely, at variance with the CKW case, the -tangles defined above are not expected to enjoy a closed formula on the marginal -qubit mixed states for . Nonetheless, in the following we verify the conjecture analytically on relevant multiqubit states, and we achieve significant progress on arbitrary states of four qubits (), for which we provide a comprehensive collection of analytical and numerical evidence in support of the SM hypothesis.
Analytical example: GHZ/W superpositions.— We begin by investigating the SM constraint (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) in its sharpest form ( ), on permutationally invariant states defined as superpositions of and generalized Greenberger-Horne-Zeilinger (GHZ) states of qubits,
with , , where is the -qubit state, is the -qubit GHZ states, and denotes the string with equal symbols . Noting that we can rewrite the states as , for , where , and observing in particular that for all residual multipartite terms vanish, , we obtain the following inductive result. Assume the SM inequality (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) holds for arbitrary pure states of qubits, then for the -qubit states one has: , , , for . Substituting these into Eq. (6), one finds: , which proves the SM inequality (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) for the -qubit states of Eq. (8). As the SM clearly holds for three-qubit states, this yields a complete analytical SM proof for generalized GHZ/ superpositions of qubits, which embody archetypical representatives of genuine multiparticle entanglement.
|Normal-form states (unnormalized)||Bounds to the reduced three-tangles|
Four-qubit strong monogamy: Toolkit.— Motivated by the above result, we now analyze arbitrary pure states of four-qubit systems (). Here, a preliminary numerical exploration reveals that the choice in (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) is too strong to hold, as it leads to negative residual four-tangles on a small subset of states 111The states with negative residuals in the case were all found within the SLOCC class according to the classification of verstraete4 ().. Therefore, we focus on testing the SM inequality for a successive level of the hierarchy, namely we set (). Sticking with as focus, and according to our adopted conventions, the SM inequality (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) then specializes to (see Fig. 1 for a graphical representation)
where we omitted the state for brevity.
All the quantities in (9) are well defined. The bipartite terms with are all computable as described above, and the tripartite terms are to be evaluated on the reduced rank- mixed state of qubits , , and , via the prescription in Eq. (7). Let us recall that the three-tangle of three-qubit pure states admits the following closed expression monogamy (),
where we have expanded the state in the computational basis as . However, to date, there is no closed formula for the three-tangle of three-qubit mixed states. The minimization in Eq. (7) has been solved only for special families of states lohmayer (); lohmayernjp (); jens (); jens2 (), while a semi-analytic method to determine when vanishes is generally available for rank- states such as lohmayer (). We then resort to looking for tractable upper bounds to the tripartite terms lohmayer (); osborne (); steepest (); rdl (); jens2 (), say . A lower bound to the residual four-tangle of Eq. (6) is then, for a four-qubit state ,
and the SM inequality may then be verified by proving that . We will exploit in particular the bound recently introduced by Rodriques, Datta, and Love (RDL) rdl () in terms of the so-called best -class approximation of three-qubit states notew (). For each rank- three-qubit state , which can be written in its spectral decomposition as , one can construct an associated simplex containing states with vanishing three-tangle, obtained as mixtures of (up to) four pure -class states () notew (). The latter take the form , where are the complex roots of the fourth-order equation , defined via Eq. (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) lohmayer (). If the rank- state belongs to the simplex , then . More generally, one can bound the three-tangle from above as follows. Defining the uniform mixture , there exists a such that describes a pure three-qubit state rdl (), where denotes the trace norm. One has then
where can be computed from Eq. (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case).
Four-qubit strong monogamy: Results.— For four qubits, there are infinitely many inequivalent SLOCC classes jensreview () (unlike the case of three qubits dur ()); however, a particularly insightful classification into nine groups was derived by Verstraete et al. verstraete4 (), who showed that, up to permutations of the four qubits, any pure state can be obtained as
where are SLOCC operations with , and each denotes a normal-form family of states, representative of the corresponding class, with , see Table 1 for their definition; only class- states are generic.
We verified the proposed SM inequality (9) for the normal-form representatives of all the nine classes, by obtaining suitable analytic upper bounds to the terms in all the three-qubit marginal partitions, as presented in Table 1. Combining these bounds with the easily computable one-tangles , and with the expressions of the reduced two-tangles (not reported here), we obtained lower bounds to the residual defined as in Eq. (11), which were found to be nonnegative for all the nine families of states. These are plotted in Fig. 2(a) for some typical instances of with . The other cases are straightforward, in particular for and all the reduced three-tangles vanish, so the SM reduces to the conventional inequality (4).
We complement this collection of analytical results with a numerical exploration of arbitrary four-qubit states . Precisely, we generated them according to the prescription in Eq. (13), by the application of random SLOCC operations on states with randomized parameters (a Gaussian distribution was used to generate the matrix elements of SLOCC operations on each qubit, and a uniform distribution in a bounded interval was used to generate the complex parameters in the states ). We tested states per class, and on each state we computed the lower bound for all four independent permutations of , using the semi-analytical RDL method rdl () to bound the marginal three-tangles in all relevant three-qubit partitions via Eq. (12). Overall, this amounts to tested data points across all the different classes (class- states are excluded since for them is separable from the rest, so the SM constraint reduces to the CKW one for which needs no testing). As Fig. 2(b) shows, no negative values of were found, providing a strongly supportive evidence for the validity of the SM inequality (9) on arbitrary four-qubit states.
Conclusion.— We proposed and analyzed a novel class of monogamy inequalities for multiqubit entanglement, which extend and sharpen the existing ones monogamy (); genmon (). We proved our SM relation on relevant families of states, and verified it numerically on arbitrary pure states of four qubits spanning all the different SLOCC classes of Ref. verstraete4 ().
This Letter opens an avenue for further investigation. First, a closed formula for the three-tangle of rank- states of three qubits lohmayer (); lohmayernjp (); jens () could facilitate a general analytical proof of inequality (Strong monogamy conjecture for multiqubit entanglement: The four-qubit case) for . More generally, would other entanglement measures which satisfy conventional monogamy—such as the squashed entanglement squashed ())—obey SM-type inequalities too, for arbitrary multipartite states of qudits? The standard CKW-type monogamy monogamy (); genmon () inspired remarkable applications to quantum cryptography review_horodecki () and the characterization of quantum critical points in many-body systems review (). This work reveals more severe limitations on the sharing of multiple forms of entanglement, and is a starting point towards a quantification of those essential features of quantum correlations, which only emerge beyond the bipartite scenario. It will be fascinating to investigate the interplay between the SM trade-off and frustration phenomena in complex quantum systems frust1 (); frust2 (); frust3 ().
Acknowledgments.— We acknowledge discussions with M. Cianciaruso, N. Datta, C. Eltschka, P. Facchi, G. Florio, F. Illuminati, J. S. Kim, T. Osborne, S. Pascazio, M. Piani, J. Siewert, A. Winter, and W. K. Wootters. The authors would like to thank A. Osterloh for bringing the error in the reduced three-tangle of class-2 states in the previously published version of this manuscript to their attention (see also Ref. osterloh ()). This work has been supported by the University of Nottingham, the University of Bari, the Italian National Group of Mathematical Physics (GNFM-INdAM), the Foundational Questions Institute (FQXi-RFP3-1317), and the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2012R1A1A2003441).
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