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Journal of Optimization Theory and Applications

A Practical Approach to SOS Relaxations for Detecting Quantum Entanglement

Abhishek Bhardwaj

Published Jun 29, 2023• 1 citation
Publisher

Abstract

Abstract This article introduces PnCP, a MATLAB toolbox for constructing positive maps which are not completely positive. We survey optimization and sum of squares relaxation techniques to find the most numerically efficient methods and establish some benchmarks for this construction. We also show how this package can be applied to the problem of classifying entanglement in quantum states.

DOI
10.1007/s10957-023-02258-5

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Abstract This article introduces PnCP, a MATLAB toolbox for constructing positive maps which are not completely positive. We survey optimization and sum of squares relaxation techniques to find the most numerically efficient methods and establish some benchmarks for this construction. We also show how this package can be applied to the problem of classifying entanglement in quantum states.
Positive non-Completely Positive (PnCP) maps are an essential tool to detect entanglement since their characterization is a dual aspect of the separability problem. A recent algorithm proposed by Kelp et al. explains how to generate PnCP maps based on the construction of certain positive non-Sum-of-Squares polynomials. We implement this algorithm in a numerically robust way and propose a working version on GitHub. We theoretically demonstrate that the maps produced by the algorithm are indecomposable, localized on the boundary of the positive cone and show that they are inequivalent with most other known PnCP maps. We numerically investigate their entanglement power, demonstrating notably that they are capable of detecting PPT entangled states that most criteria fail to detect.
Although the quantum state of a many-body system (e.g, a system consisting of several atoms) is described in terms of its degrees of freedom, i.e., a large number of variables representing a point on the compact space of quantum states, there is usually a small number of parameters that can be actually measured experimentally. We show that the problem of deciding whether the underlying quantum state is entangled is equivalent to the problem of deciding whether the set of measured parameters are the moments of some measure defined over the compact space of states of the many-body system. By considering quantum entanglement of n identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of n copies of unit balls in R^d. We consider moments up to degree 2 since these are most readily available experimentally. We derive necessary and sufficient conditions for belonging to the moment cone, which can be expressed via a linear matrix inequality (LMI) of size at most 2d+2, which is independent of n. We can convert the LMIs into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone, and show that the two conditions approach each other in the limit of large n. The inequalities are derived via considering the dual cone of nonnegative polynomials, and its sum-of-squares (SOS) relaxation. We show that the SOS relaxation of the dual cone is asymptotically exact, and using symmetry reduction techniques (J. Pure Appl. Algebra 192, no. 1-3, 95, arXiv:1205.3102), it can be written as a small linear matrix inequality of size at most 2d+2, which is independent of n. For the cone of nonnegative polynomials we also prove an analogue of the half degree principle for globally nonnegative symmetric polynomials (J. Math. Anal. Appl. 284, no. 1, 174, J. Pure Appl. Algebra 216, no. 4, 850).
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
Abstract The problem of entanglement detection is a long standing problem in quantum information theory. One of the primary procedures of detecting entanglement is to find the suitable positive but non-completely positive maps. Here we try to give a generic prescription to construct a positive map that can be useful for such scenarios. We study a class of positive maps arising from Lindblad structures. We show that two famous positive maps viz. transposition, reduction map and Choi map can be obtained as a special case of a class of positive maps having Lindblad structure. Generalizing the transposition map to a one parameter family we have used it to detect genuine multipartite entanglement. Finally being motivated by the negativity of entanglement, we have defined a similar measure for genuine multipartite entanglement.
Recently, a toolkit of highly symmetric techniques employing matrix inequalities has been developed to detect entanglement in various ways. Here we unifiedly explain in detail these methods, and expand them to a new family of positive maps with further detection capabilities. In the simplest case, we generalize the reduction map to detect more generic states using both multiple copies and local filters. Through the Choi-Jamiołkowski isomorphism, this family of maps leads to a construction of multipartite entanglement witnesses. Discussions and examples are provided regarding the detection of states with local positive partial transposition and the use of multiple copies.