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arXiv:1708.06986

Multiparticle entanglement criteria for nonsymmetric collective variances

Oliver Marty, M. Cramer, Giuseppe Vitagliano, Géza Tóth, Martin Bodo Plenio

Published Aug 23, 2017• 2 citations
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Abstract

We introduce a general scheme to detect various multiparticle entanglement structures from global non-permutationally invariant observables. In particular, we derive bounds on the variance of non-permutationally invariant and collective operators for the verification of $k$-party entanglement. For a family of observables related to the spin structure factor, we give quantitative bounds on entanglement that are independent of the total number of particles. We introduce highly non-symmetric states with genuine multipartite entanglement that is verifiable with the presented technique and discuss how they can be prepared with trapped ions exploiting the high degree of control in these systems. As a special case, our framework provides an alternative approach to obtain a tight relaxation of the entanglement criterion by Sø rensen and Mø lmer [Phys. Rev. Lett. 86, 4431 (2001)] that is free from technical assumptions and allows to calculate the bounds with an improved scaling in the detectable depth.

DOI
10.48550/arxiv.1708.06986

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We introduce a general scheme to detect various multiparticle entanglement structures from global non-permutationally invariant observables. In particular, we derive bounds on the variance of non-permutationally invariant and collective operators for the verification of kkk-party entanglement. For a family of observables related to the spin structure factor, we give quantitative bounds on entanglement that are independent of the total number of particles. We introduce highly non-symmetric states with genuine multipartite entanglement that is verifiable with the presented technique and discuss how they can be prepared with trapped ions exploiting the high degree of control in these systems. As a special case, our framework provides an alternative approach to obtain a tight relaxation of the entanglement criterion by Sø rensen and Mø lmer [Phys. Rev. Lett. 86, 4431 (2001)] that is free from technical assumptions and allows to calculate the bounds with an improved scaling in the detectable depth.
Multipartite entanglement plays an essential role in both quantum information science and many-body physics. Because of the exponentially large dimension and complex geometric structure of the state space, the detection of entanglement in many-body systems is extremely challenging in reality. Conventional means, like entanglement witness and entropy criterion, either highly depend on the prior knowledge of the studied systems or the detection capability is relatively weak. In this Letter, we propose a framework for designing multipartite entanglement criteria based on permutation moments, which have an effective implementation with either the generalized control-swap quantum circuits or the random unitary techniques. As an example, in the bipartite scenario, we develop an entanglement criterion that can detect bound entanglement and show strong detection capability in the multiqubit Ising model with a long-range XY Hamiltonian. In the multipartite case, the permutation-moment-based criteria can detect entangled states that are not detectable by any criteria extended from the bipartite case. Our framework also shows potential in entanglement quantification and entanglement structure detection.
I present methods for detecting entanglement around symmetric Dicke states. In particular, I consider N-qubit symmetric Dicke states with N/2 excitations. I show that for large N these states have the smallest overlap possible with states without genuine multipartite entanglement. Thus these states are particularly well suited for the experimental examination of multipartite entanglement. I present fidelity-based entanglement witness operators for detecting multipartite entanglement around these states. I then consider entanglement criteria, somewhat similar to the spin squeezing criterion, based on the moments or variances of the collective spin operators. Surprisingly, these criteria are based on an upper bound for variances for separable states. I present criteria detecting entanglement in general and criteria detecting only genuine multipartite entanglement. The collective operator measured for this criteria is an important physical quantity: Its expectation value essentially gives the intensity of the radiation when a coherent atomic cloud emits light.
We consider the problem of characterizing genuine multiparticle entanglement for permutationally invariant states using the approach of positive partial transpose mixtures. We show that the evaluation of this necessary biseparability criterion scales polynomially with the number of particles. In practice, it can be evaluated easily up to ten qubits and improves existing criteria significantly. Finally, we show that our approach solves the problem of characterizing genuine multiparticle entanglement for permutationally invariant three-qubit states.
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Entanglement is deemed a valuable resource for quantum computing and lies at the heart of quantum information. At the same time, current experiments are venturing into the realm of quantum systems consisting of many particles. In this book, different aspects of entanglement and non-locality in multiparticle systems are investigated. First, it presents an approach for an experimental verification of non-locality and entanglement with a high statistical significance. Second, a criterion for genuine multiparticle entanglement is introduced. Furthermore, this criterion also allows for a way of quantifying multiparticle entanglement. Analytical criteria for entanglement in a special class of quantum states, so-called graph states, are presented. These criteria turn out to be also necessary for entanglement in certain cases. Finally, ways of generalizing a non-local hidden-variable model proposed by Leggett are shown.
We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, mmm-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.