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Physical Review A

Entanglement measures based on the complete information of reduced states

Zhi‐Xiang Jin, Xianqing Li‐Jost, Shao-Ming Fei, Cong‐Feng Qiao

Published Jan 6, 2023• 19 citations
Publisher

Abstract

Quantum entanglement has been verified experimentally and applied in quantum computing, quantum sensing, and quantum networks. It is of great significance to find measures to characterize the quantum entanglement faithfully. In this work, by exploiting the Schmidt decomposition of bipartite states, we first establish a one-to-one correspondence between the characteristic polynomial of the reduced state of a bipartite pure state and the trace of the reduced state. We introduce a family of entanglement measures based on the eigenvalues of the reduced density matrices. Specific measures called informationally complete entanglement measures (ICEMs) are presented. It is demonstrated that such ICEMs can characterize better the entanglement than the existing well-known entanglement measures. The ICEMs also give rise to criteria of state transformations under local operation and classical communication. Moreover, it is shown that the ICEMs can be efficiently estimated on a quantum computer. The full separability, entanglement, and genuine multipartite entanglement can be detected faithfully on quantum devices.

DOI
10.1103/physreva.107.012409

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Quantum entanglement has been verified experimentally and applied in quantum computing, quantum sensing, and quantum networks. It is of great significance to find measures to characterize the quantum entanglement faithfully. In this work, by exploiting the Schmidt decomposition of bipartite states, we first establish a one-to-one correspondence between the characteristic polynomial of the reduced state of a bipartite pure state and the trace of the reduced state. We introduce a family of entanglement measures based on the eigenvalues of the reduced density matrices. Specific measures called informationally complete entanglement measures (ICEMs) are presented. It is demonstrated that such ICEMs can characterize better the entanglement than the existing well-known entanglement measures. The ICEMs also give rise to criteria of state transformations under local operation and classical communication. Moreover, it is shown that the ICEMs can be efficiently estimated on a quantum computer. The full separability, entanglement, and genuine multipartite entanglement can be detected faithfully on quantum devices.
Although quantum entanglement has already been verified experimentally and applied in quantum computing, quantum sensing and quantum networks, most of the existing measures cannot characterize the entanglement faithfully. In this work, by exploiting the Schmidt decomposition of a bipartite state ∣ψ⟩AB|ψ\rangle_{AB}∣ψ⟩AB​, we first establish a one-to-one correspondence relation between the characteristic polynomial of the reduced state ρAρ_AρA​ and the polynomials its trace. Then we introduce a family of entanglement measures which are given by the complete eigenvalues of the reduced density matrices of the system. Specific measures called informationally complete entanglement measures (ICEMs) are presented to illustrate the advantages. It is demonstrated that such ICEMs can characterize finer and distinguish better the entanglement than existing well-known entanglement measures. They also give rise to criteria of state transformations under local operation and classical communication. Moreover, we show that the ICEMs can be efficiently estimated on a quantum computer. The fully separability, entanglement and genuine multipartite entanglement can detected faithfully on quantum devices.
Abstract Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.
Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, a new genuine multipartite entanglement measure based on informationally complete entanglement is proposed, termed the informationally complete genuine multipartite entanglement measure(ICGMEM). A set of entanglement inequalities for pure states in ‐partite systems is first established based on constraints on entanglement distribution. These inequalities reveal that the pairwise entanglement between any individual part of the system cannot exceed the collective entanglement of the remaining parts. Building on these inequalities, a series of geometric structures called informationally complete entanglement measure triangles is constructed and the ICGMEM is defined based on the geometric mean area of these triangles. These results demonstrate that the ICGMEM provides a more refined characterization of entanglement compared to existing well‐known measures. Furthermore, it is shown that the ICGMEM significantly enhances computational efficiency for analyzing genuinely entangled pure states, achieving up to a twofold improvement when the number of subsystems is sufficiently large. Through detailed examples, the advantages of the ICGMEM are illustrated, highlighting its effectiveness in distinguishing and quantifying entanglement in multipartite systems. This work can better advance the theoretical understanding of genuine multipartite entanglement.
Multipartite entanglement is an essential resource for quantum communication, quantum computing, quantum sensing, and quantum networks. The utility of a quantum state |ψ⟩ for these applications is often directly related to the degree or type of entanglement present in |ψ⟩. Therefore, efficiently quantifying and characterizing multipartite entanglement is of paramount importance. In this work, we introduce a family of multipartite entanglement measures, called concentratable entanglements. Several well-known entanglement measures are recovered as special cases of our family of measures, and hence we provide a general framework for quantifying multipartite entanglement. We prove that the entire family does not increase, on average, under local operations and classical communications. We also provide an operational meaning for these measures in terms of probabilistic concentration of entanglement into Bell pairs. Finally, we show that these quantities can be efficiently estimated on a quantum computer by implementing a parallelized SWAP test, opening up a research direction for measuring multipartite entanglement on quantum devices.
The Schmidt number represents the genuine entanglement dimension of a bipartite quantum state. We derive simple criteria for the Schmidt number of a density matrix in arbitrary local dimensions, given that certain symmetric measurements exist. They are based on the trace norm of correlations obtained from seminal families of quantum measurements, specifically symmetric informationally complete measurements and mutually unbiased bases. Our criteria are strictly stronger than both the well-known fidelity witness criterion and the computable cross-norm or realignment criterion.
The entanglement of a pure bipartite state is uniquely measured by the von-Neumann entropy of its reduced density matrices. Though it cannot specify all the non-local characteristics of pure entangled states. It was proven that for every possible value of entanglement of a bipartite system, there exists an infinite number of equally entangled pure states, not comparable(satisfies Nielsen’s criteria) to each other. In this work, we investigate other correlation measures of pure bipartite states that are able to differentiate the quantum correlations of the states with entropy of entanglement. In Schmidt rank 3, we consider the whole set of states having same entanglement and compare how minutely such states can be distinguished by other correlation measures. Then for different values of entanglement we compare the sets of states belonging to the same entanglement and also investigate the graphs of different correlation measures. We extend our search to Schmidt rank 4 and 5 also.