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arXiv:2203.08391 · Physical Review Letters

Detecting Entanglement in Quantum Many-Body Systems via Permutation Moments

Zhenhuan Liu, Yifan Tang, Hao Dai, Pengyu Liu, Shu Chen, Xiongfeng Ma

Published Dec 23, 2022• 41 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.129.260501

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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.
Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.
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.
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.
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.
Multipartite entanglement detection is crucial for the develop of quantum information science and quantum computation, communication, simulation and metrology tasks. In contrast to experiments, where several handreds of qubits have been entangled to build a quantum computer or a quantum simulator, theoretically only for lower dimensional quantum system such as two qubit system or two-mode Gaussian state and some special quantum states, efficient criteria have been developed to detect the entanglement. An efficient and practical entanglement detection method is anticipated to appear for current scalable quantum system. Based on the matched entanglement witness scheme and integrated uncertainty relations, we propose a quite general entanglement detection method for all kinds of multipartite entanglement of multimode continuous variable systems. With the covariance matrix of any randomly generated multimode continuous variable quantum state, our criterion can detect entanglement, genuine entanglement and other kinds of inseparabilities almost imidiately.
Entanglement detection is essential in quantum information science and quantum many-body physics. It has been proved that entanglement exists almost surely for a random quantum state, while the realizations of effective entanglement criteria usually consume exponentially many resources with regard to system size or qubit number, and efficient criteria often perform poorly without prior knowledge. This fact implies a fundamental limitation might exist in the detectability of entanglement. In this work, we formalize this limitation as a fundamental trade-off between the efficiency and effectiveness of entanglement criteria via a systematic method to evaluate the detection capability of entanglement criteria theoretically. For a system coupled to an environment, we prove that any entanglement criterion needs exponentially many observables to detect the entanglement effectively when restricted to single-copy operations. Otherwise, the detection capability of the criterion will decay double exponentially. Furthermore, if multicopy joint measurements are allowed, the effectiveness of entanglement detection can be exponentially improved, which implies a quantum advantage in entanglement detection problems. Our results may shed light on why quantum phenomena are difficult to observe in large noisy systems.