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arXiv:2404.19308 · Annalen der Physik

A Characterization of Entangled Two‐Qubit States via Partial‐Transpose‐Moments

Lin Zhang, Ming‐Jing Zhao, Lin Chen, Hua Xiang, Yi Shen

Published Sep 10, 2022• 5 citations

Abstract

Abstract Although quantum entanglement is an important resource, its characterization is quite challenging. The partial transposition is a common method to detect bipartite entanglement. In this paper, the authors study the partial‐transpose(PT)‐moments of two‐qubit states, and completely describe the whole region, composed of the second and third PT‐moments, for all two‐qubit states. Furthermore, they determine the accurate region corresponding to all entangled two‐qubit states. The states corresponding to those boundary points of the whole region, and to the border lines between separable and entangled states are analyzed. As an application, they characterize the entangled region of PT‐moments for the two families of Werner states and Bell‐diagonal states. The relations between entanglement and the pairs of PT‐moments are revealed from these typical examples. They also numerically plot the whole region of possible PT‐moments for all two‐qubit X‐states, and find that this region is almost the same as the whole region of PT‐moments for all two‐qubit states. Moreover, they extend their results to detect the entanglement of multiqubit states. By utilizing the PT‐moment‐based method to characterize the entanglement of the multiqubit states mixed by the GHZ and W states, they propose an operational way of verifying the genuine entanglement in such states.

DOI: 10.1002/andp.202200289

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Abstract Although quantum entanglement is an important resource, its characterization is quite challenging. The partial transposition is a common method to detect bipartite entanglement. In this paper, the authors study the partial‐transpose(PT)‐moments of two‐qubit states, and completely describe the whole region, composed of the second and third PT‐moments, for all two‐qubit states. Furthermore, they determine the accurate region corresponding to all entangled two‐qubit states. The states corresponding to those boundary points of the whole region, and to the border lines between separable and entangled states are analyzed. As an application, they characterize the entangled region of PT‐moments for the two families of Werner states and Bell‐diagonal states. The relations between entanglement and the pairs of PT‐moments are revealed from these typical examples. They also numerically plot the whole region of possible PT‐moments for all two‐qubit X‐states, and find that this region is almost the same as the whole region of PT‐moments for all two‐qubit states. Moreover, they extend their results to detect the entanglement of multiqubit states. By utilizing the PT‐moment‐based method to characterize the entanglement of the multiqubit states mixed by the GHZ and W states, they propose an operational way of verifying the genuine entanglement in such states.

Efficiently detecting entanglement based on measurable quantities is a basic problem for quantum information processing. Recently, the measurable quantities called partial-transpose (PT) moments have been proposed to detect and characterize entanglement. Recently [L. Zhang et al., Ann. Phys. (Berlin) 534, 2200289 (2022)], we have already identified the two-dimensional region, comprised of the second and third PT moments, corresponding to two-qubit entangled states, and described the whole region for all two-qubit states. In the present paper, we visualize the three-dimensional (3D) region corresponding to all two-qubit states by further involving the fourth PT moment (the last one for two-qubit states). The characterization of this 3D region can finally be achieved by optimizing some polynomials. Furthermore, we identify the dividing surface which separates the two parts of the whole 3D region corresponding to entangled and separable states, respectively. Due to the measurability of PT moments, we obtain a complete and operational criterion for the detection of two-qubit entanglement.

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.

For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.

We first show that partial transposition for pure and mixed two-particle states in a discrete

N

-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter

r

of a Bell state

|Φ^{+} \rangle

and the completely incoherent state,

\hatρ_r = r |Φ^{+} \rangle \langle Φ^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}

: we find that for

r_0 < r < 1

, where

r_0

is obtained as a function of the dimensionality

N

, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.

Detection of entanglement in bipartite states is a fundamental task in quantum information. The first method to verify entanglement in mixed states was the partial-transpose criterion. Subsequently, numerous quantifiers for bipartite entanglement were introduced, among them concurrence and negativity. Surprisingly, these quantities are often treated as distinct or independent of each other. The aim of this contribution is to highlight the close relations between these concepts, to show the connections between seemingly independent results, and to present various estimates for the mixed-state concurrence within the same framework.

Detection of entanglement through partial knowledge of the quantum state is a challenge to implement efficiently. Here we propose a separability criterion for detecting bipartite entanglement in arbitrary-dimensional quantum states using partial moments of the realigned density matrix. Our approach enables the detection of both distillable and bound entangled states through a common framework. We illustrate the significance of our method through examples of states belonging to both the above categories, which are not detectable using comparable other schemes relying on partial state information. The formalism of employing partial realigned moments proposed here is further shown to be effective for two-qubit systems too, with a slight modification of our separability criterion.