Entanglement Classification Research Articles

Quantum Entanglement and Its Classification Protocols

Quantum entanglement has its own major role in quantum information theory. Its application in numerous areas namely quantum computing, quantum cryptography and quantum teleportation are proven vital and essential. Over the last decades, the interests in quantum entanglement have grown and significant progress in quantum computing has been revealed. However, the classification of entanglement was proved to be challenging especially in a higher qubit-dimensional system setting. In this review, indexed literature as the secondary resource was chosen by specific keywords from several database. In reference to the literatures review, there exists several entanglement classifications protocols that will be presented in this paper namely local unitary (LU), local operations and classical communication (LOCC), and stochastic local operations and classical communication (SLOCC). This study will offer a better understanding of quantum entanglement and the entanglement classification protocols.

Enhancing Quantum Information Processing – SU(2) Operator Model Development for Three-Qubit Quantum Systems Entanglement Classification

The study introduces the development of the operator model within the Local Unitary (LU) protocol for entanglement classification, specifically of the pure three-qubit quantum systems. Addressing the challenge of accurately distinguishing different classes of entanglement, this research aims to enhance the understanding and utilization of entangled quantum states in quantum information processing. A systematic approach was employed in the development of the model, designed to effectively distinguish different classes of entanglement. This study contributes significantly to quantum information processing, by providing valuable insights into the nature of entangled quantum states and enabling researchers to gain a better understanding, utilizing them effectively in various quantum applications. The developed operator model holds significant potential for the advancements in entanglement classification and broaderscope of quantum information processing. The model marks a notable achievement in enabling a more precise and efficient entanglement of entangled quantum systems, which is crucial for advancing various quantum technologies. Future research should extends this model to a higher-qubit and higher-dimensional quantum systems and explore its integration with several other entanglement classification protocols and Lie groups to further advance the field.

Frozen condition of quantum coherence

Quantum coherence as an important physical resource plays the key role in implementing various quantum tasks, whereas quantum coherence is generally nonincreasing under incoherent operations. In this paper, we analyse under which dynamical conditions the l 1-norm or the relative entropy of coherence can remain unchanged under strictly incoherent operations (freezing coherence). We provide a detailed analysis of their structure together with exact geometric conditions of freezing coherence. It reveals a remarkable feature: any strictly incoherent operation freezing coherence can be decomposed as a convex combination of unitary operations. This partially answers an open question named unitary decomposition of doubly-stochastic quantum operations [M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000)]. Based on this analysis, we also give a complete classification of coherent states from operational coherence theory. This builds the counterpart of entanglement classification under LOCC.

Phase sensitivity of entanglement in the Quantum Phase Estimation algorithm

We study entanglement in the steps before Quantum Fourier Transform (pre-QFT) part of the Quantum Phase Estimation and the Quantum Counting algorithms (QPEA—QCA) with the use of three entanglement detection tools. In particular we focus on the sensitivity of entanglement to the input value (the phase ϕ and the ratio of marked elements MN ) in some basic cases. One starts from numerical observations and deduce some general results in particular regarding the classes of entanglement. More precisely, when the second register of both algorithms (i.e. the register on which a specific unitary operator act, see section ) is initialized in the non-entangled superposition of two (separable) eigenvectors, one proves that the QPEA and QCA curves of entanglement evolution are the same up to a scalar multiplication of the parameter ϕ. One demonstrates that a local minimum is obtained and corresponds to an EPR (Einstein-Podolsky-Rosen) state and finally one proves that, up to Stochastic Local Operation and Classical Communication (SLOCC), all states, except for a few values of ϕ, are equivalent to the product of a separable state and a generalized GHZ (Greenberger-Horen-Zeilinger) state , i.e. GHZn+1=12(00...0+11...1) .

Entanglement detection with classical deep neural networks

In this study, we introduce an autonomous method for addressing the detection and classification of quantum entanglement, a core element of quantum mechanics that has yet to be fully understood. We employ a multi-layer perceptron to effectively identify entanglement in both two- and three-qubit systems. Our technique yields impressive detection results, achieving nearly perfect accuracy for two-qubit systems and over 90%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90\\%$$\\end{document} accuracy for three-qubit systems. Additionally, our approach successfully categorizes three-qubit entangled states into distinct groups with a success rate of up to 77%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$77\\%$$\\end{document}. These findings indicate the potential for our method to be applied to larger systems, paving the way for advancements in quantum information processing applications.

Decoherence and the classes of maximally entangled states

Self-interactions and interaction with the environment inevitably push quantum systems towards highly mixed states, which is the definition of decoherence. However, if decoherence is completely suppressed, then any quantum system evolves unitarily through pure states towards maximally entangled configurations. We argue that these maximally entangled states of the systems fall into the well-defined classes that can be uniquely described by the values of certain entanglement invariants. After discussing these ideas, we present examples of maximally entangled states for a number of generic systems, construct compact states in the most entangled classes for tripartite systems, and suggest how they may be constructed for other [Formula: see text]-partite systems. We study random walks through the space of entanglement classes to see how unitary evolution along paths to the maximally entangled states, i.e. entanglement growth, works in practice.

Partial-transpose-guided entanglement classes and minimum noise filtering in many-body Gaussian quantum systems

Partial-transpose-guided entanglement classes and minimum noise filtering in many-body Gaussian quantum systems

Multi-Unitary Complex Hadamard Matrices

We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of [Formula: see text] subsystems with [Formula: see text] levels each to the set of complex Hadamard matrices of order [Formula: see text]. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ([Formula: see text] or [Formula: see text]), two-unitary ([Formula: see text] and [Formula: see text] are unitary), or [Formula: see text]-unitary. Here [Formula: see text] denotes reshuffling of a matrix [Formula: see text] describing a bipartite system, and [Formula: see text] its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in [Formula: see text] dimensions.

Classification of large black holes

The black-hole--qubit correspondence has been proven to be ``useful for obtaining additional insight into one of the string black hole theory and quantum information theory by exploiting approaches of the other"[Phys. Rev. D 82, 026003 (2010)]. Though different classes of stringy black holes can be related to the well-known stochastic local operations and classical communication (SLOCC) entanglement classes of pure states, the string theory requires a more detailed classification than the SLOCC classification of three qubits. In this paper, we derive the entanglement family of three qubits under local unitary operations (LU), and use the black-hole--qubit correspondence to classify \textquotedblleft large\textquotedblright\ black holes into seven inequivalent families. In particular, we show that two black holes with 4 non-vanishing charges ($q_{0}$, $p^{1}$, $p^{2}$, and $p^{3}$) are LU equivalent if their difference is only in the signs of charges. Thus, the classification of black holes is independent of the signs of charges and is only related to the ratio of the absolute values of charges. This observation simplifies the classification task, as one would only need to consider either the classification of non-BPS black holes or the classification of BPS black holes, but not both. Moreover, through the LU classification, the physical basis for this black-hole--qubit correspondence can be observed, and a relation between the black-hole entropy and the von Neumann entanglement entropy is revealed. Therefore, the LU classification offers a more straightforward physical connection than the SLOCC classification. Based on the LU classification, we further study the properties of von Neumann entanglement entropy for each of the seven families, and find the black holes with the maximal von Neumann entanglement entropy.

Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication

Recent advances have led towards first prototypes of quantum networks in which entanglement is distributed by sources producing bipartite entangled states. This raises the question of which states can be generated in quantum networks based on bipartite sources using local operations and classical communication. In this work, we study state transformations under finite rounds of local operations and classical communication (LOCC) in networks based on maximally entangled two-qubit states. We first derive the symmetries for arbitrary network structures, as these determine which transformations are possible. Then, we show that contrary to tree graphs, for which it has already been shown that any state within the same entanglement class can be reached, there exist states which can be reached probabilistically but not deterministically if the network contains a cycle. Furthermore, we provide a systematic way to determine states which are not reachable in networks consisting of a cycle. Moreover, we provide a complete characterization of the states which can be reached in a cycle network with a protocol where each party measures only once, and each step of the protocol results in a deterministic transformation. Finally, we present an example which cannot be reached with such a simple protocol, and constitutes, up to our knowledge, the first example of a LOCC transformation among fully entangled states requiring three rounds of classical communication.

Certifying quantum separability with adaptive polytopes

The concept of entanglement and separability of quantum states is relevant for several fields in physics. Still, there is a lack of effective operational methods to characterise these features. We propose a method to certify quantum separability of two- and multiparticle quantum systems based on an adaptive polytope approximation. This leads to an algorithm which, for practical purposes, conclusively recognises two-particle separability for small and medium-size dimensions. For multiparticle systems, the approach allows to characterise full separability for up to five qubits or three qutrits; in addition, different classes of entanglement can be distinguished. Finally, our methods allow to identify systematically quantum states with interesting entanglement properties, such as maximally robust states which are separable for all bipartitions, but not fully separable.

The entanglement criteria based on equiangular tight frames

Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.

A hyperdeterminant on fermionic Fock space

Twenty years ago, Cayley's hyperdeterminant, the degree four invariant of the polynomial ring \mathbb{C}[\mathbb{C}^2\otimes\mathbb{C}^2\otimes \mathbb{C}^2]^{{\operatorname{SL}_2(\mathbb{C})}^{\times 3}} , was popularized in modern physics as it separates genuine entanglement classes in the three-qubit Hilbert space and is connected to entropy formulas for special solutions of black holes. In this note, we compute the analogous invariant on the fermionic Fock space for N=8 , i.e., spin particles with four different locations, and show how this invariant projects to other well-known invariants in quantum information. We also give combinatorial interpretations of these formulas.

Entanglement classification via witness operators generated by support vector machine

Although entanglement is a basic resource for reaching quantum advantage in many computation and information protocols, we lack a universal recipe for detecting it, with analytical results obtained for low-dimensional systems and few special cases of higher-dimensional systems. In this work, we use a machine learning algorithm, the support vector machine with polynomial kernel, to classify separable and entangled states. We apply it to two-qubit and three-qubit systems, and we show that, after training, the support vector machine is able to recognize if a random state is entangled with an accuracy up to 92% for the two-qubit system and up to 98% for the three-qubit system. We also describe why and in what regime the support vector machine algorithm is able to implement the evaluation of an entanglement witness operator applied to many copies of the state, and we describe how we can translate this procedure into a quantum circuit.

Symmetry classification of typical quantum entanglement

Symmetry classification of typical quantum entanglement

Classification and quantification of entanglement through wedge product and geometry

The wedge product of post-measurement vectors of a two-qubit state gives rise to a parallelogram, whose ‘area’ has been shown to be equivalent to the generalized I-concurrence measure of entanglement. In multi-qudit systems, the wedge product of post-measurement vectors naturally leads to a higher dimensional parallelepiped which yields a modified faithful entanglement measure. Our new measure fine grains the entanglement monotone, wherein different entangled classes manifest with different geometries. We present a complete analysis of the bipartite qutrit case considering all possible geometric structures where three entanglement classes of pure bipartite qutrit states can be identified with different geometries of post-measurement vectors: three planar vectors; three mutually orthogonal vectors; and three vectors that are neither planar and not all of them are mutually orthogonal. It is further demonstrated that the geometric condition of area and volume maximization naturally leads to the maximization of entanglement. The wedge product approach uncovers an inherent geometry of entanglement and is found to be very useful for the characterization and quantification of entanglement in higher dimensional systems.

Entanglement classification from a topological perspective

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.

Towards a classification of entanglements of Galois representations attached to elliptic curves

Let E/\mathbb{Q} be an elliptic curve, let \overline{\mathbb{Q}} be a fixed algebraic closure of \mathbb{Q} , and let G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) be the absolute Galois group of \mathbb{Q} . The action of G_{\mathbb{Q}} on the adelic Tate module of E induces the adelic Galois representation \rho_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}). The goal of this paper is to explain how the image of \rho_E can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over \mathbb{Q} where the entanglement occurs over an abelian extension.

Partition GHZ SLOCC class of three qubits into ten families under LU

In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. Ac\'{\i}n et al. proposed the generalized Schmidt decomposition for three qubits and partitioned pure states of three qubits into five types. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.

Genuine Tripartite Non-Gaussian Entanglement.

Triple-photon states generated by three-mode spontaneous parametric down-conversion are the paradigm of unconditional non-Gaussian states, essential assets for quantum advantage. How to fully characterize their non-Gaussian entanglement remains however elusive. We propose here a set of sufficient and necessary conditions for separability of the broad family of spontaneously generated three-mode non-Gaussian states. We further derive state-of-the-art conditions for genuine tripartite non-Gaussian entanglement, the strongest class of entanglement. We apply our criteria to triple-photon states revealing that they are fully inseparable and genuinely entangled in moments of order 3n. Our results establish a systematic framework for characterizing the entanglement of triple-photon states and thus fostering their application in quantum information protocols.