Semidefinite Tests for Quantum Network Topologies

Johan Åberg, Ranieri Nery, Cristhiano Duarte, and Rafael Chaves

Phys. Rev. Lett. 125, 110505 – Published 10 September 2020

Abstract

Quantum networks play a major role in long-distance communication, quantum cryptography, clock synchronization, and distributed quantum computing. Generally, these protocols involve many independent sources sharing entanglement among distant parties that, upon measuring their systems, generate correlations across the network. The question of which correlations a given quantum network can give rise to remains almost uncharted. Here we show that constraints on the observable covariances, previously derived for the classical case, also hold for quantum networks. The network topology yields tests that can be cast as semidefinite programs, thus allowing for the efficient characterization of the correlations in a wide class of quantum networks, as well as systematic derivations of device-independent and experimentally testable witnesses. We obtain such semidefinite tests for fixed measurement settings, as well as parties that independently choose among collections of measurement settings. The applicability of the method is demonstrated for various networks, and compared with previous approaches.

Received 28 February 2020
Accepted 19 August 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.110505

Physics Subject Headings (PhySH)

Nonlocality Quantum communication Quantum correlations in quantum information Quantum networks Quantum simulation

Quantum Information, Science & Technology

Authors & Affiliations

Johan Åberg¹, Ranieri Nery², Cristhiano Duarte ³, and Rafael Chaves^2,4

¹Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, D-50937 Cologne, Germany
²International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil
³Schmid College of Science and Technology, Chapman University, 1 University Drive, Orange, California 92866, USA
⁴School of Science and Technology, Federal University of Rio Grande do Norte, 59078-970 Natal, Brazil

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Vol. 125, Iss. 11 — 11 September 2020

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Images

Figure 1
Triangular quantum network. (a) Triangular quantum DAG with three children $α$ , $β$ , $γ$ (white disks). Each parent (gray disks) has two children. The parents are uncorrelated, but distribute (possibly entangled) quantum systems (colored disks) to all of its children, and each child performs a measurement on its systems. (b) The covariance matrix of the observed random variables can be decomposed into a sum of positive semidefinite blocks; one for each parent, where the support of the block is determined by the children of the parent. For the triangular DAG, there are three positive semidefinite components (red, yellow, blue), each with bipartite supports. If a given covariance matrix cannot be decomposed in this manner, then the triangular quantum scenario is rejected as an explanation of the observed covariance. (c) The witness $W_{GHZ}$ [Eq. (4)] rejects the triangular quantum DAG as an explanation of the distributions $P_{p, q}$ in Eq. (5), establishing the analytic expression $q > p + (4 - \sqrt{1 + 48 p}) / 3$ for the incompatibility region determined by our method. Remarkably, $W_{GHZ}$ is optimal for all points above the solid blue curve (verified numerically by comparison with the general semidefinite results in Proposition 1). For comparison we also consider two versions of the Finner inequality [35] (red and magenta dashed curves) that differ in terms of local postprocessings of the observable variables. First, by using the variables directly without any postprocessing, the test rejects all cases above the dashed red curve. Then, by resorting to a numerical optimization over the possible processings, we obtain the magenta curve, that approximates our curve from below and rejects a slightly larger region. Details of the comparison with the Finner inequality are provided in the Supplemental Material [38]. We consider also the inflation bound (green dot-dashed) recently obtained in Ref. [36] and the entropic bound (black dot-dashed) [10, 41, 42] (see Ref. [38] for more details).
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Figure 2
Tests for four-partite scenarios. Analogous to Fig. 1, we consider the family of four-partite distributions $P_{p, q} = p δ_{0000} + q δ_{1111} + (1 - p - q) (1 - δ_{0000} - δ_{1111}) / 14$ , with $p$ , $q \geq 0$ and $p + q \leq 1$ . We test for values of $p$ and $q$ for which the resulting covariance matrix is compatible with two quantum scenarios: (a) Six parents (gray disks), each with two children (white disks). (b) Four parents, each with three children. (c) For all values of $p$ , $q$ above the solid blue curve, our test rejects scenario (a) as an explanation of $P_{p, q}$ . Above the dot-dashed curve, the test rejects scenario (b) as an explanation of $P_{p, q}$ . In scenario (a) we moreover compare again with two versions of the Finner test [35], with and without optimization over local postprocessings of the observable variables. For the latter, we obtain the dashed red curve, while for the former, we obtain a curve that approximately coincides with our curve from above, shown superimposed in a light dashed line on top of the solid blue curve. See the Supplemental Material for details [38].
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Figure 3
Triangular quantum network with inputs. (a) For the same triangular quantum DAG as in Fig. 1, we allow each child $m = α$ , $β$ , $γ$ to choose among two different measurement settings (POVMs). Child $α$ chooses between settings $0_{α}$ and $1_{α}$ , and analogous for child $β$ and $γ$ . (b) For all possible measurement choices we form a covariance matrix of the random variables of the measurement outcomes, where we need to take into account that, e.g., child $α$ cannot simultaneously measure the POVMs corresponding to $0_{α}$ and $1_{α}$ . These unobservable covariances correspond to the black squares, while all white squares correspond to measurable covariances. Any scenario as in (a) results in observable covariances that can be completed into a matrix that can be decomposed into positive semidefinite blocks, analogous to Fig. 1.
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