Polynomial Monogamy Relations for Entanglement Negativity

Abstract

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 sufficient inequality which, for the first time, is a non-linear higher degree polynomial. For very large quantum systems, we prove that the negativity can be distributed at least linearly for the tightest constraint and conjecture that it is at most linear.The prototypical quantum correlation, entanglement, accounts for many of the effects seen in quantum theory, and there is a significant effort to enslave it as a resource to be manipulated and exploited by quantum engineers [1]. The most striking property of entanglement is its distributability, or lack there of, exemplified by its compliance with, for example, monogamy laws [2] and area laws [3], laws which constrain the shareability of correlations. The former law states that the closer two parties are to being maximally entangled, the more the pair become separated and hidden from all other parties. The latter law requires that for certain types of many-particle ground states split in twain, the entanglement between the parts scales according to the size of the separation boundary. Both laws set the stage for entanglement to play a central role in physics, and an active research community continues to strengthen this role [4].The secludedness of maximally entangled states has specifically impacted quantum key distribution [5], ground state frustration [6], and even black holes [7]. These perspectives on monogamy have traditionally been understood in terms of multi-qubit networks. In the case of tripartite systems, the monogamy law can be written in terms of concurrence [8], as the following,where C is the concurrence, subscripts label the parties, and the vertical bar denotes the bipartite split across which it is computed. In this way, if C 2 A|BC ≈ C 2 A|B ≈ 1, i.e., A and B are maximally entangled, then C 2 A|C ≈ 0, hence the personification of entanglement.The elegance and simplicity of the monogamy inequality has made it the paragon for entanglement shareability. It has since been shown that other entanglement measures such as entanglement of formation [9], squashed entanglement [10], and entanglement negativity [11], all satisfy the same monogamy relation. This raises two issues: the monogamy inequality may just be a first order approximation to the shareability of correlational resources, so to what extent can we quantify the exact amount possible to share? Second, is the limited shareability a consequence of limited systems, i.e., qubits?The first issue has recent...

Multipartite entanglement measure and complete monogamy relation

“…One of the most important issues closely related to entanglement measure is the monogamy relation of entanglement [23], which states that, unlike classical correlations, if two parties A and B are maximally entangled, then neither of them can share entanglement with a third party C. An important question in this field is to determine whether a given entanglement measure is monogamous. Considerable efforts have been devoted to this task in the last two decades [19,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] ever since Coffman, Kundu, and Wootters (CKW) presented the first quantitative monogamy relation in Ref. [19] for threequbit states.…”

Monogamy of entanglement without inequalities

“…In contrast with the classical world, it is not possible to prepare three qubits A, B, C in a way that any two qubits are maximally entangled [1]. In fact, if qubit A is maximally entangled with qubit B, then it must be uncorrelated (not even classically) with qubit C. This phenomenon of monogamy of entanglement was first quantified in a seminal paper by Coffman, Kundu, and Wootters (CKW) [1] for three qubits, and later on studied intensively in more general settings [2,3,4,5,14,15,6,16,17,18,7,19,20,21,22,23,24,25,8,9,26,27,10,11,28,29,30,12,13,31,32,33,34,35,36,37,38,39,40].…”

“…This was already apparent in the seminal work of [1] in which E was taken to be the square of the concurrence and not the concurrence itself. More recently, it was shown that many other measures of entanglement satisfy the monogamy relation (1) if E is replaced by E α for some α > 1 [13,8,12,5,7,11,9,6,10].…”

Exponentially many entanglement and correlation constraints for multipartite quantum states

“…In order to prove the functional indepedence for the right-hand sides of the constraints in Eq. (17) in the main text, we need to show that if for all ρ,…”

“…Compatibility of marginals.-Finally we want to highlight the relation of our results with the quantum-marginal problem, that is, the question whether or not a given set of reduced states is compatible with a joint global state [53]. Clearly, our linear-entropy constraints (17) represent necessary conditions for the reduced states ρ S to be compatible with the global state ρ. However, we can make new statements even at the operator level.…”