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.
Quantum information leverages properties of quantum behaviors in order to perform useful tasks such as secure communication and randomness certification. Nevertheless, not much is known about the intricate geometric features of the set quantum behaviors. In this paper we study the structure of the set of quantum correlators using semidefinite programming. Our main results are (i) a generalization of the analytic description by Tsirelson-Landau-Masanes, (ii) necessary and sufficient conditions for extremality and exposedness, and (iii) an operational interpretation of extremality in the case of…
We develop numerous results that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, we develop a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices. We then explore applications of our work in the theory of quantum resources. For example, we characterize exactly which quantum states have modified trace distance of coherence equal t…
Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $ω(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. cond…
Unsharp quantum measurements provide a resource in scenarios where one faces the trade-off between information gain and disturbance. In this work we introduce a prepare-transform-measure scenario in which two-outcome unsharp measurements outperform their sharp counterparts, as well as any stochastic strategy involving dichotomic projective measurements. Based on that, we propose a scheme for semi-device-independent self-testing of unsharp measurements and show that all two-outcome qubit measurements can be characterized in a robust way. Along with the main result, in this work we introduce a m…
Semidefinite Programming for Quantum Channel Learning
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics（罗蒙诺索夫莫斯科国立大学，力学与数学系）；Autretech Group, Skolkovo Innovation Center, Nobel Street, Building 7, Moscow, 121205, Russia（Autretech集团，斯克尔科沃创新中心，诺贝尔街，7号大楼，莫斯科，121205，俄罗斯）；Peter the Great St. Petersburg Polytechnic University（彼得大帝圣彼得堡国立理工大学）；Ioffe Institute（伊万诺夫研究所）
We present a technique for reducing the computational requirements by several orders of magnitude in the evaluation of semidefinite relaxations for bounding the set of quantum correlations arising from finite-dimensional Hilbert spaces. The technique, which we make publicly available through a user-friendly software package, relies on the exploitation of symmetries present in the optimisation problem to reduce the number of variables and the block sizes in semidefinite relaxations. It is widely applicable in problems encountered in quantum information theory and enables computations that were …
