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arXiv:1407.0314 · Physical Review A

Entanglement detection using mutually unbiased measurements

Bin Chen, Teng Ma, Shao-Ming Fei

Published Jun 30, 2014• 61 citations
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Abstract

We study entanglement detection by using mutually unbiased measurements and provide a quantum separability criterion that can be experimentally implemented for arbitrary $d$-dimensional bipartite systems. We show that this criterion is more effective than the criterion based on mutually unbiased bases. For isotropic states our criterion becomes both necessary and sufficient.

DOI
10.1103/physreva.89.064302

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We study entanglement detection by using mutually unbiased measurements and provide a quantum separability criterion that can be experimentally implemented for arbitrary ddd-dimensional bipartite systems. We show that this criterion is more effective than the criterion based on mutually unbiased bases. For isotropic states our criterion becomes both necessary and sufficient.
We investigate correlations among complementary observables. In particular, we show how to take advantage of mutually unbiased bases for the efficient detection of entanglement in arbitrarily high-dimensional, multipartite, and continuous-variable quantum systems. The introduced entanglement criteria are relatively easy to implement experimentally since they require only a few local measurement settings. In addition, we establish a link between the separability problem and the maximum number of mutually unbiased bases---opening an additional avenue in this long-standing open problem.
Mutually unbiased measurements (MUMs) are generalized from the concept of mutually unbiased bases (MUBs) and include the complete set of MUBs as a special case, but they are superior to MUBs as they do not need to be rank one projectors. We investigate entanglement detection using sets of MUMs and derive separability criteria for multipartite qudit systems, arbitrary high-dimensional bipartite systems of a d1-dimensional subsystem and a d2-dimensional subsystem, and multipartite systems of multi-level subsystems. These criteria are of the advantages of more effective and wider application range than previous criteria. They provide experimental implementation in detecting entanglement of unknown quantum states.
Abstract The detection of quantum entanglement represents a fundamental challenge in quantum measurement theory within the context of quantum information science. In this work, we propose a family of separability criteria based on mutually unbiased equiangular tight frames [IEEE Trans. Inf. Theory 67 , 1656–1667 (2021)], which constitute a natural generalization of both mutually unbiased bases and equiangular tight frames. The proposed criteria are applicable to a broad class of quantum systems, including bipartite systems with equal or unequal subsystem dimensions, as well as multipartite systems with heterogeneous dimensional components. In the examined cases, namely isotropic, Bell-diagonal, and Werner states, the proposed entanglement detection criteria demonstrate superior performance relative to several established methods. Crucially, these criteria rely solely on a small set of local measurements, which significantly enhances their experimental feasibility for the characterization of entanglement in unknown quantum states. Overall, the MUETF-based criteria provide more robust and versatile tools for entanglement verification across diverse quantum architectures.
Based on the mutually unbiased bases, the mutually unbiased measurements and the general symmetric informationally complete positive-operator-valued measures, we propose three separability criteria for ddd-dimensional bipartite quantum systems, which are more powerful than the corresponding ones introduced previously [C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B. C. Hiesmayr, Phys. Rev. A 86, 022311 (2012); B. Chen, T. Ma, and S. M. Fei, Phys. Rev. A 89, 064302 (2014); and B. Chen, T. Li, and S. M. Fei, arXiv:1406.7820v1 [quant-ph] (2014)]. Some states such as Werner states and Bell-diagonal states are used to further illustrate the efficiency of the presented criteria.
Mutually unbiased bases, mutually unbiased measurements and general symmetric informationally complete measurements are three related concepts in quantum information theory. We investigate multipartite systems using these notions and present some criteria detecting entanglement of arbitrary high dimensional multi-qudit systems and multipartite systems of subsystems with different dimensions. It is proved that these criteria can detect the k -nonseparability ( k is even) of multipartite qudit systems and arbitrary high dimensional multipartite systems of m subsystems with different dimensions. We show that they are more efficient and wider of application range than the previous ones. They provide experimental implementation in detecting entanglement without full quantum state tomography.
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.