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Matrix Pencils and Entanglement Classification1

Eric Chitambar, Carl A. Miller, Yaoyun Shi

Published Aug 15, 2016• 15 citations
OpenAlex

Abstract

Quantum entanglement plays a central role in quantum information processing. A main objective of the theory of quantum entanglement is to classify different types of entanglement according to their inter-convertibility through manipulations that do not require quantum communication. While bipartite entanglement is well understood in this framework, the classification of entangle-ments among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2 ⊗ m ⊗ n. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two 2⊗m ⊗ n states are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC equivalence classes in 2 ⊗ 3 ⊗ n systems, we also determine the hierarchy between these classes.

Related papers

Matrix pencils and entanglement classification

Eric Chitambar, Carl A. Miller, Yaoyun Shi

Quantum entanglement plays a central role in quantum information processing. A main objective of the theory is to classify different types of entanglement according to their interconvertibility through manipulations that do not require additional entanglement to perform. While bipartite entanglement is well understood in this framework, the classification of entanglements among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2⊗m⊗n. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two 2⊗m⊗n states are SLOCC equivalent. We then proceed to present a canonical form for all 2⊗m⊗n states based on the matrix pencil construction such that two states are equivalent if and only if they have the same canonical form. Besides recovering the previously known 26 distinct SLOCC equivalence classes in 2⊗3⊗n systems, we also determine the hierarchy between these classes.

arXiv:0911.1803•Journal of Mathematical Physics•Jul 1, 2010•22 citations

Comment on "Matrix Pencils and Entanglement Classification"

Eric Chitambar, Carl A. Miller, Yaoyun Shi

In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension $2\otimes m\otimes n$ are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in $2\otimes 3\otimes n$ systems.

arXiv:0911.4058•Nov 20, 2009•2 citations

Graphical Classification of Entangled Qutrits

Kentaro Honda

A multipartite quantum state is entangled if it is not separable. Quantum entanglement plays a fundamental role in many applications of quantum information theory, such as quantum teleportation. Stochastic local quantum operations and classical communication (SLOCC) cannot essentially change quantum entanglement without destroying it. Therefore, entanglement can be classified by dividing quantum states into equivalence classes, where two states are equivalent if each can be converted into the other by SLOCC. Properties of this classification, especially in the case of non two-dimensional quantum systems, have not been well studied. Graphical representation is sometimes used to clarify the nature and structural features of entangled states. SLOCC equivalence of quantum bits (qubits) has been described graphically via a connection between tripartite entangled qubit states and commutative Frobenius algebras (CFAs) in monoidal categories. In this paper, we extend this method to qutrits, i.e., systems that have three basis states. We examine the correspondence between CFAs and tripartite entangled qutrits. Using the symmetry property, which is required by the definition of a CFA, we find that there are only three equivalence classes that correspond to CFAs. We represent qutrits graphically, using the connection to CFAs. We derive equations that characterize the three equivalence classes. Moreover, we show that any qutrit can be represented as a composite of three graphs that correspond to the three classes.

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Classification of the Entangled States of 2×N ×N

Shuo Cheng, Junli Li, Cong‐Feng Qiao

We develop a novel method in classifying the multipartite entanglement state of 2×N ×N under stochastic local operations and classical communication (SLOCC). In this method, all inequivalent classes of a genuine entangled state can be assorted directly without knowing the classification information of lower dimension ones for any given dimension N. It also gives a nature explanation for the non-local parameters remaining in the entanglement classes when N ≥ 4. Entanglement is at the heart of the quantum information theory (QIT) and is now thought as a physical resource to realize quantum information tasks, such as quantum cryptography [1, 2], superdense coding [3, 4], and quantum computation [5]. The study of entanglement improves our further understanding of quantum non-locality [6] and also is of particular interest in QIT. One such case is the classification of multipartite entanglement. Two quantum states can be employed to implement the same tasks in QIT and are thought to be equivalent while they are mutually convertible by Stochastic Local Operations and Classical Communication (SLOCC) [7]. Nevertheless, in practice the classification of multipartite entanglement and in high dimensions in the Hilbert space seems to be a formidable task [8]. Matrix decomposition keeps to be a useful tool as in the two-partite case [9, 10]. A widely adopted philosophy in dealing with this issue is first to know the classification of the state in lower dimension (or less partite) and then extend to the higher dimension case [11, 12] (or more partite [13, 14]) in an inductive way. However, nontrivial aspect emerges as the dimension increases, i.e. some non-local parameters may nest in the entangled states [15, 16]. In literature, in

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Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by [Formula: see text]-secant varieties of the Segre variety and [Formula: see text]-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.

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Entanglement classification from a topological perspective

Dmitry Melnikov

Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the $W$ entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.

arXiv:2208.13901•Physical review. D/Physical review. D.•Jun 12, 2023•9 citations
Quantum entanglement plays a central role in quantum information processing. A main objective of the theory of quantum entanglement is to classify different types of entanglement according to their inter-convertibility through manipulations that do not require quantum communication. While bipartite entanglement is well understood in this framework, the classification of entangle-ments among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2 ⊗ m ⊗ n. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two 2⊗m ⊗ n states are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC equivalence classes in 2 ⊗ 3 ⊗ n systems, we also determine the hierarchy between these classes.
Quantum entanglement plays a central role in quantum information processing. A main objective of the theory is to classify different types of entanglement according to their interconvertibility through manipulations that do not require additional entanglement to perform. While bipartite entanglement is well understood in this framework, the classification of entanglements among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2⊗m⊗n. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two 2⊗m⊗n states are SLOCC equivalent. We then proceed to present a canonical form for all 2⊗m⊗n states based on the matrix pencil construction such that two states are equivalent if and only if they have the same canonical form. Besides recovering the previously known 26 distinct SLOCC equivalence classes in 2⊗3⊗n systems, we also determine the hierarchy between these classes.
In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension 2⊗m⊗n2\otimes m\otimes n2⊗m⊗n are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in 2⊗3⊗n2\otimes 3\otimes n2⊗3⊗n systems.
A multipartite quantum state is entangled if it is not separable. Quantum entanglement plays a fundamental role in many applications of quantum information theory, such as quantum teleportation. Stochastic local quantum operations and classical communication (SLOCC) cannot essentially change quantum entanglement without destroying it. Therefore, entanglement can be classified by dividing quantum states into equivalence classes, where two states are equivalent if each can be converted into the other by SLOCC. Properties of this classification, especially in the case of non two-dimensional quantum systems, have not been well studied. Graphical representation is sometimes used to clarify the nature and structural features of entangled states. SLOCC equivalence of quantum bits (qubits) has been described graphically via a connection between tripartite entangled qubit states and commutative Frobenius algebras (CFAs) in monoidal categories. In this paper, we extend this method to qutrits, i.e., systems that have three basis states. We examine the correspondence between CFAs and tripartite entangled qutrits. Using the symmetry property, which is required by the definition of a CFA, we find that there are only three equivalence classes that correspond to CFAs. We represent qutrits graphically, using the connection to CFAs. We derive equations that characterize the three equivalence classes. Moreover, we show that any qutrit can be represented as a composite of three graphs that correspond to the three classes.
We develop a novel method in classifying the multipartite entanglement state of 2×N ×N under stochastic local operations and classical communication (SLOCC). In this method, all inequivalent classes of a genuine entangled state can be assorted directly without knowing the classification information of lower dimension ones for any given dimension N. It also gives a nature explanation for the non-local parameters remaining in the entanglement classes when N ≥ 4. Entanglement is at the heart of the quantum information theory (QIT) and is now thought as a physical resource to realize quantum information tasks, such as quantum cryptography [1, 2], superdense coding [3, 4], and quantum computation [5]. The study of entanglement improves our further understanding of quantum non-locality [6] and also is of particular interest in QIT. One such case is the classification of multipartite entanglement. Two quantum states can be employed to implement the same tasks in QIT and are thought to be equivalent while they are mutually convertible by Stochastic Local Operations and Classical Communication (SLOCC) [7]. Nevertheless, in practice the classification of multipartite entanglement and in high dimensions in the Hilbert space seems to be a formidable task [8]. Matrix decomposition keeps to be a useful tool as in the two-partite case [9, 10]. A widely adopted philosophy in dealing with this issue is first to know the classification of the state in lower dimension (or less partite) and then extend to the higher dimension case [11, 12] (or more partite [13, 14]) in an inductive way. However, nontrivial aspect emerges as the dimension increases, i.e. some non-local parameters may nest in the entangled states [15, 16]. In literature, in
Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by [Formula: see text]-secant varieties of the Segre variety and [Formula: see text]-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.
Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the WWW entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.