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arXiv:2101.08848 · Physical Review A

Entanglement detection in quantum many-body systems using entropic uncertainty relations

Bjarne Bergh, Martin Gärttner

Published May 12, 2021• 24 citations
arXivHTMLPDF

Abstract

The authors study experimentally accessible lower bounds on entanglement measures for quantum many-body systems based on entropic uncertainty relations. Rather than requiring the full reconstruction of the quantum state, the bound derived here requires measuring joint probability distributions in only two different measurement settings per subsystem.

DOI
10.1103/physreva.103.052412

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•Dec 9, 2016•1 citation
The authors study experimentally accessible lower bounds on entanglement measures for quantum many-body systems based on entropic uncertainty relations. Rather than requiring the full reconstruction of the quantum state, the bound derived here requires measuring joint probability distributions in only two different measurement settings per subsystem.
Abstract Uncertainty relations and quantum entanglement are pivotal concepts in quantum theory. Beyond their fundamental significance in shaping our understanding of the quantum world, they also underpin crucial applications in quantum information theory. In this article, we investigate entropic uncertainty relations and entanglement detection with an emphasis on quantum measurements with design structures. On the one hand, we derive improved Rényi entropic uncertainty relations for design-structured measurements, exploiting the property that the sum of powered (e.g. squared) probabilities of obtaining different measurement outcomes is now invariant under unitary transformations of the measured system and can be easily computed. On the other hand, the above property essentially imposes a state-independent upper bound, which is achieved at all pure states, on one’s ability to predict local outcomes when performing a set of design-structured measurements on quantum systems. Realizing this, we also obtain criteria for detecting multipartite entanglement with design-structured measurements.
Entanglement is not only the resource that fuels many quantum technologies but also plays a key role for some of the most profound open questions of fundamental physics. Experiments controlling quantum systems at the single quantum level may shed light on these puzzles. However, measuring, or even bounding, entanglement experimentally has proven to be an outstanding challenge, especially when the prepared quantum states are mixed. We use entropic uncertainty relations for bipartite systems to derive measurable lower bounds on distillable entanglement. We showcase these bounds by applying them to physical models realizable in cold-atom experiments. The derived entanglement bounds rely on measurements in only two different bases and are generically applicable to any quantum simulation platform.
We present the generalized state-dependent entropic uncertainty relations in a multiple measurements setting and the optimal lower bound is obtained by considering different measurement sequences. We then apply this uncertainty relation to witness entanglement and give the experimentally accessible lower bounds on both bipartite and tripartite entanglements. This method of detecting entanglement is applied to a physical system of two particles on a one-dimensional lattice and Greenberger-Horne-Zeilinger (GHZ)-Werner state. It is shown that, for measurements that are not in mutually unbiased bases, this generalized entropic uncertainty relation is superior to the previous state-independent one in entanglement detection. Furthermore, we conduct a demonstration of multipartite entanglement detection of GHZ states up to ten qubits on the Quafu cloud quantum computation platform. Our results might play important roles in detecting multipartite entanglement experimentally.
For a given pure state of a composite quantum system we analyze the product of its projections onto a set of locally orthogonal separable pure states. We derive a bound for this product analogous to the entropic uncertainty relations. For bipartite systems the bound is saturated for maximally entangled states and it allows us to construct a family of entanglement measures, we shall call collectibility. As these quantities are experimentally accessible, the approach advocated contributes to the task of experimental quantification of quantum entanglement, while for a three-qubit system it is capable to identify the genuine three-party entanglement.
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.