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arXiv:2406.10662 · Physical Review A

Entanglement classification and non- k -separability certification via Greenberger-Horne-Zeilinger-class fidelity

Marcin Płodzień, Jan Chwedeńczuk, Maciej Lewenstein, Grzegorz Rajchel-Mieldzioć

Published Sep 23, 2024• 9 citations
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Abstract

Many-body quantum systems can be characterized using the notions of $k$-separability and entanglement depth. A quantum state is $k$-separable if it can be expressed as a mixture of $k$ entangled subsystems, and its entanglement depth is given by the size of the largest entangled subsystem. In this paper we propose a multipartite entanglement measure that satisfies the following criteria: (i) It can be used with both pure and mixed states; (ii) it is encoded in a single element of the density matrix, so it does not require knowledge of the full spectrum of the density matrix; (iii) it can be applied to large systems; and (iv) it can be experimentally verified. The proposed method allows the certification of non-$k$-separability of a given quantum state. We show that the proposed method successfully classifies three-qubit systems into known stochastic local operations and classical communication (SLOCC) classes, namely, bipartite, $W$-, and Greenberger-Horne-Zeilinger-type entanglement. Furthermore, we characterize the non-$k$-separability in nine known SLOCC classes of four-qubit states and absolutely maximally entangled states for five and six qubits and for arbitrary-size-qubit Dicke states.

DOI
10.1103/physreva.110.032428

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Many-body quantum systems can be characterized using the notions of kkk-separability and entanglement depth. A quantum state is kkk-separable if it can be expressed as a mixture of kkk entangled subsystems, and its entanglement depth is given by the size of the largest entangled subsystem. In this paper we propose a multipartite entanglement measure that satisfies the following criteria: (i) It can be used with both pure and mixed states; (ii) it is encoded in a single element of the density matrix, so it does not require knowledge of the full spectrum of the density matrix; (iii) it can be applied to large systems; and (iv) it can be experimentally verified. The proposed method allows the certification of non-kkk-separability of a given quantum state. We show that the proposed method successfully classifies three-qubit systems into known stochastic local operations and classical communication (SLOCC) classes, namely, bipartite, WWW-, and Greenberger-Horne-Zeilinger-type entanglement. Furthermore, we characterize the non-kkk-separability in nine known SLOCC classes of four-qubit states and absolutely maximally entangled states for five and six qubits and for arbitrary-size-qubit Dicke states.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
In this paper, we study the entanglement classification of the two-dimensional subspace spanned by nnn-qubit Greenberger-Horne-Zeilinger (GHZ) and WWW states under stochastic local operations and classical communication (SLOCC). We show that for $n\ensuremath{\ge}4$ such a system can have infinite SLOCC inequivalent classes. Specifically, two states, ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+x{|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+y{|W\ensuremath{\rangle}}_{n}$, from the subspace spanned by nnn-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if (xy)n=1{(\frac{x}{y})}^{n}=1(yx​)n=1.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.
We propose a method of classifying nnn-qubit states into stochastic local operations and classical communication inequivalent families in terms of the rank of the square matrix $C{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}$, where CCC is the rectangular coefficient matrix of the state and ${\ensuremath{\sigma}}_{y}$ is the Pauli operator. The rank of the square matrix $C{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}$ is capable of distinguishing between nnn-qubit Greenberger-Horne-Zeilinger and WWW states. The determinant of the matrix gives rise to a family of polynomial invariants for nnn qubits which include as special cases well-known polynomial invariants in the literature. In addition, explicit expressions can be given for these polynomial invariants and this allows us to investigate the properties of entanglement measures built upon the absolute values of polynomial invariants for product states.
The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states σAB=p∣χ1⟩⟨χ1∣⊗ρ1+(1−p)∣χ2⟩⟨χ2∣⊗ρ2σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}σAB​=p∣χ1​⟩⟨χ1​∣⊗ρ1​+(1−p)∣χ2​⟩⟨χ2​∣⊗ρ2​, where ∣χ1⟩=α∣0⟩+β∣1⟩|χ_{1}\rangle=α|0\rangle+β|1\rangle∣χ1​⟩=α∣0⟩+β∣1⟩, ∣χ2⟩=β∣0⟩+(−1)nα∣1⟩|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle∣χ2​⟩=β∣0⟩+(−1)nα∣1⟩. We have shown that the state σABσ_{AB}σAB​ represent a classical state when nnn is odd while it represent a non-classical state when nnn is even. The purification of the state σABσ_{AB}σAB​ is studied and found that the purification is possible if the spectral decomposition of the density matrices ρ1ρ_{1}ρ1​ and ρ2ρ_{2}ρ2​ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity ⟨χ1∣χ2⟩\langleχ_{1}|χ_{2}\rangle⟨χ1​∣χ2​⟩, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.