Physical Review A

Classifying entanglement in the superposition of Greenberger-Horne-Zeilinger and W states

Xuan Wang, Xian Shi

Published Oct 14, 2015• 1 citation

Abstract

In this paper, we study the entanglement classification of the two-dimensional subspace spanned by $n$-qubit Greenberger-Horne-Zeilinger (GHZ) and $W$ 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 $n$-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if ${(\frac{x}{y})}^{n}=1$.

DOI: 10.1103/physreva.92.042318

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Physical Review A

Classifying entanglement in the superposition of Greenberger-Horne-Zeilinger and W states

Xuan Wang, Xian Shi

Published Oct 14, 2015• 1 citation

Publisher

Abstract

In this paper, we study the entanglement classification of the two-dimensional subspace spanned by

n

-qubit Greenberger-Horne-Zeilinger (GHZ) and

W

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

n

-qubit

{|W\ensuremath{\rangle}}_{n}

and

{|\mathrm{GHZ}\ensuremath{\rangle}}_{n}

are SLOCC equivalent if and only if

{(\frac{x}{y})}^{n}=1

DOI: 10.1103/physreva.92.042318

Related papers

Locality of three-qubit Greenberger-Horne-Zeilinger-symmetric states

Dian Zhu, Gang-Gang He, Fulin Zhang

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, $W$ , and Greenberger-Horne-Zeilinger (GHZ) classes. The $W$ 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 $W$ and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.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, $W$, and Greenberger-Horne-Zeilinger (GHZ) classes. The $W$ 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 $W$ and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.

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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.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.

arXiv:2406.10662Physical Review ASep 23, 20249 citations

In this paper, we study the entanglement classification of the two-dimensional subspace spanned by

n

-qubit Greenberger-Horne-Zeilinger (GHZ) and

W

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

n

-qubit

{|W\ensuremath{\rangle}}_{n}

and

{|\mathrm{GHZ}\ensuremath{\rangle}}_{n}

are SLOCC equivalent if and only if

{(\frac{x}{y})}^{n}=1

W

, and Greenberger-Horne-Zeilinger (GHZ) classes. The

W

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

W

and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.

We study the interconversion of multipartite symmetric

N

-qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected with a nonsymmetric invertible local operation (ILO), then they belong necessarily to the separable, W, or Greenberger-Horne-Zeilinger (GHZ) entanglement class, establishing a practical method of discriminating subsets of entanglement classes. Furthermore, we prove that there always exists a symmetric ILO connecting any pair of symmetric

N

-qubit states equivalent under SLOCC, simplifying the requirements for experimental implementations of local interconversion of those states.

{L}_{ab{c}_{2}}

and

{G}_{abcd}

. Extension to the multiqubit system is briefly discussed.

We analyze entanglement classes for permutation-symmetric states for

n

qudits (i.e.,

d

-level systems), with respect to local unitary operations (LU equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the Jordan form of such one-particle operators is a SLOCC invariant. We find, as representatives of those classes, a discrete set of entangled states that generalize the Greenberger-Horne-Zeilinger and

W

states for the many-particle qudit case. In the latter case, we introduce the excitation state as a natural generalization of the

W

state for

d>2

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.