Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2201.05622

nnn-qubit states with maximum entanglement across all bipartitions: A graph state approach

Sowrabh Sudevan, Sourin Das

Published Jan 14, 2022• 0 citations
arXivHTMLPDF

Abstract

We discuss the construction of $n$-qubit pure states with maximum bipartite entanglement across all possible choices of $k$ vs $n-k$ bi-partitioning, which implies that the Von Neumann entropy of every $k$-qubit reduced density matrix corresponding to this state should be $k \ln 2 $. Such states have been referred to as $k$-uniform, $k$-MM states. We show that a subset of the 'graph states' satisfy this condition, hence providing a recipe for constructing $k$-uniform states. Finding recipes for construction of $k$-uniform states using graph states is useful since every graph state can be constructed starting from a product state using only controlled-$Z$ gates. Though, a priori it is not clear how to construct a graph which corresponds to an arbitrary $k$-uniform state, but in particular, we show that graphs with no isolated vertices are $1$-uniform. Graphs organized as a circular linear chain corresponds to the case of $2$-uniform state, where we show that the minimum number of qubits required to host such a state is $n=5$. $3$-uniform states can be constructed by forming bi-layer graphs with $n/2$ qubits ($n=2\mathbb{Z}$) in each layer, such that each layer forms a fully connected graph while inter-layer connections are such that the vertices in one layer has a one to one connectivity to the other layer. $4$-uniform states can be formed by taking 2D lattice graphs( also referred elsewhere as a 2D cluster Ising state ) with periodic boundary conditions along both dimensions and both dimensions having at least $5$ vertices.

DOI
10.48550/arxiv.2201.05622

Related papers

Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix

S. Sharma

Graph states represent a significant class of multi-partite entangled quantum states with applications in quantum error correction, quantum communication, and quantum computation. In this work, we introduce a novel formalism called the Entanglement Matrix for quantifying and classifying entanglement in n-qubit graph states. Leveraging concepts from graph theory and quantum information, we develop a systematic approach to analyze entanglement by identifying primary and secondary midpoints in graph representations, where midpoints correspond to controlled-Z gate operations between qubits. Using Von Neumann entropy as our measure, we derive precise mathematical relationships for maximum entanglement in graph states as a function of qubit number. Our analysis reveals that entanglement follows a quadratic relationship with the number of qubits, but with distinct behaviors for odd versus even qubit systems. For odd n-qubit graph states, maximum entanglement follows $E_{\max} = n^2 - n$, while even n-qubit states exhibit higher entanglement with varying formulae depending on specific configurations. Notably, systems with qubit counts that are multiples of 12 demonstrate enhanced entanglement properties. This comprehensive classification framework provides valuable insights into the structure of multi-qubit entanglement, establishing an analytical foundation for understanding entanglement distribution in complex quantum systems that may inform future quantum technologies.

arXiv:2507.11458•Jul 15, 2025•0 citations

Exploring pure quantum states with maximally mixed reductions

Ludovic Arnaud, Nicolas J. Cerf

We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for $n$-qubit states, we express the condition that all $k$-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all $k$ vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group ${\mathcal{P}}_{n}$ that is isomorphic to ${\mathbb{Z}}_{2}^{n}$. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed $k$ for fixed $n$. In particular, the ratio $k/n$ can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed $k$-qubit reductions with $k>1$. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though $k>1$ is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.

arXiv:1211.4118•Physical Review A•Jan 18, 2013•83 citations

Controlling bi-partite entanglement in multi-qubit systems

Martin Plesch, Jaroslav Novotný, Zuzana Dzuráková +1 more

Bi-partite entanglement in multi-qubit systems cannot be shared freely. The rules of quantum mechanics impose bounds on how multi-qubit systems can be correlated. In this paper we utilize a concept of entangled graphs with weighted edges in order to analyze pure quantum states of multi-qubit systems. Here qubits are represented by vertexes of the graph while the presence of bi-partite entanglement is represented by an edge between corresponding vertexes. The weight of each edge is defined to be the entanglement between the two qubits connected by the edge, as measured by the concurrence. We prove that each entangled graph with entanglement bounded by a specific value of the concurrence can be represented by a pure multi-qubit state. In addition we present a logic network with O(N2) elementary gates that can be used for preparation of the weighted entangled graphs of N qubits.

arXiv:quant-ph/0311069•Journal of Physics A Mathematical and General•Jan 19, 2004•4 citations

Genuine entanglement and LU-equivalence of graph states in bipartition

Yuting Huang, Lin Chen

Graph states are intuitive representations of quantum states that are useful in quantum computing and quantum information. In this paper, we investigate the role of entanglement and local unitary equivalence in quantum computing through graph states in high-dimensional Hilbert spaces. We introduce bipartitions on the vertices sets and demonstrate separability and local unitary equivalence based on different choices of partitions. By performing the two-qubit controlled phase shift gates on various pairs of multiqubit system, one can generate a genuinely entangled graph state based on simple connected graphs. We extend 2-dimensional Hilbert spaces to higher and mixed dimensions and examine pure quantum states that cannot be separated by any bipartition. In addition, we generalize the analysis on classification of local unitary equivalence beyond the controlled-Z gates to controlled phase gates. By examining graphs constructed from bipartition graphs, we have shown that graphs with identical bipartition subgraph structures are local unitary equivalent.

Modern Physics Letters A•Dec 10, 2025•0 citations

Entanglement of multi-qubit quantum graph states and studies structural properties of tripartite graphs with quantum computing

Kh. P. Gnatenko

We propose a method for constructing multi-qubit entangled quantum states that represent weighted tripartite graphs, and develop approaches for investigating their structural properties using quantum computing. In the general case of multi-qubit states corresponding to arbitrary tripartite graph structures, we derive an expression for the entanglement distance. We establish a connection between entanglement and properties of the corresponding tripartite graphs. Namely, we show that the entanglement of a qubit with the rest of the system in a quantum tripartite graph state depends on the weights of the arcs in the neighborhood of the corresponding vertex, as well as on its degree with respect to the vertex sets of the tripartite graph. As an illustrative example, we consider a tripartite graph forming a triangle and evaluate the entanglement distance with quantum computing. We also compute quantum correlators for the general case of tripartite quantum graph states and relate these quantities to structural features of the underlying graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. Obtained relationships between the quantum properties of multi-qubit states and the structural features of tripartite graphs opens up the possibility of investigating classical systems, such as tripartite graphs, using quantum computing. It is worth emphasizing that tripartite graphs have applications in practical problems, including resource allocation, scheduling, and database and hypergraph modeling.

quant-pharXiv:2604.27829•May 1, 2026•0 citations

Entangled graphs: Bipartite entanglement in multiqubit systems

Martin Plesch, Vladimír Bužek

Quantum entanglement in multipartite systems cannot be shared freely. In order to illuminate basic rules of entanglement sharing between qubits, we introduce a concept of an entangled structure (graph) such that each qubit of a multipartite system is associated with a point (vertex), while a bipartite entanglement between two specific qubits is represented by a connection (edge) between these points. We prove that any such entangled structure can be associated with a pure state of a multiqubit system. Moreover, we show that a pure state corresponding to a given entangled structure is a superposition of vectors from a subspace of the ${2}^{N}$-dimensional Hilbert space, whose dimension grows linearly with the number of entangled pairs.

arXiv:quant-ph/0211020•Physical Review A•Jan 29, 2003•55 citations
We discuss the construction of nnn-qubit pure states with maximum bipartite entanglement across all possible choices of kkk vs n−kn-kn−k bi-partitioning, which implies that the Von Neumann entropy of every kkk-qubit reduced density matrix corresponding to this state should be kln⁡2k \ln 2 kln2. Such states have been referred to as kkk-uniform, kkk-MM states. We show that a subset of the 'graph states' satisfy this condition, hence providing a recipe for constructing kkk-uniform states. Finding recipes for construction of kkk-uniform states using graph states is useful since every graph state can be constructed starting from a product state using only controlled-ZZZ gates. Though, a priori it is not clear how to construct a graph which corresponds to an arbitrary kkk-uniform state, but in particular, we show that graphs with no isolated vertices are 111-uniform. Graphs organized as a circular linear chain corresponds to the case of 222-uniform state, where we show that the minimum number of qubits required to host such a state is n=5n=5n=5. 333-uniform states can be constructed by forming bi-layer graphs with n/2n/2n/2 qubits (n=2Zn=2\mathbb{Z}n=2Z) in each layer, such that each layer forms a fully connected graph while inter-layer connections are such that the vertices in one layer has a one to one connectivity to the other layer. 444-uniform states can be formed by taking 2D lattice graphs( also referred elsewhere as a 2D cluster Ising state ) with periodic boundary conditions along both dimensions and both dimensions having at least 555 vertices.
Entanglement Classification in the Graph States: The generalization to nnn-Qubits States using the Entanglement Matrix
Graph states represent a significant class of multi-partite entangled quantum states with applications in quantum error correction, quantum communication, and quantum computation. In this work, we introduce a novel formalism called the Entanglement Matrix for quantifying and classifying entanglement in n-qubit graph states. Leveraging concepts from graph theory and quantum information, we develop a systematic approach to analyze entanglement by identifying primary and secondary midpoints in graph representations, where midpoints correspond to controlled-Z gate operations between qubits. Using Von Neumann entropy as our measure, we derive precise mathematical relationships for maximum entanglement in graph states as a function of qubit number. Our analysis reveals that entanglement follows a quadratic relationship with the number of qubits, but with distinct behaviors for odd versus even qubit systems. For odd n-qubit graph states, maximum entanglement follows Emax⁡=n2−nE_{\max} = n^2 - nEmax​=n2−n, while even n-qubit states exhibit higher entanglement with varying formulae depending on specific configurations. Notably, systems with qubit counts that are multiples of 12 demonstrate enhanced entanglement properties. This comprehensive classification framework provides valuable insights into the structure of multi-qubit entanglement, establishing an analytical foundation for understanding entanglement distribution in complex quantum systems that may inform future quantum technologies.
We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for nnn-qubit states, we express the condition that all kkk-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all kkk vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group Pn{\mathcal{P}}_{n}Pn​ that is isomorphic to Z2n{\mathbb{Z}}_{2}^{n}Z2n​. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed kkk for fixed nnn. In particular, the ratio k/nk/nk/n can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed kkk-qubit reductions with k>1k>1k>1. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though k>1k>1k>1 is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.
Bi-partite entanglement in multi-qubit systems cannot be shared freely. The rules of quantum mechanics impose bounds on how multi-qubit systems can be correlated. In this paper we utilize a concept of entangled graphs with weighted edges in order to analyze pure quantum states of multi-qubit systems. Here qubits are represented by vertexes of the graph while the presence of bi-partite entanglement is represented by an edge between corresponding vertexes. The weight of each edge is defined to be the entanglement between the two qubits connected by the edge, as measured by the concurrence. We prove that each entangled graph with entanglement bounded by a specific value of the concurrence can be represented by a pure multi-qubit state. In addition we present a logic network with O(N2) elementary gates that can be used for preparation of the weighted entangled graphs of N qubits.
Graph states are intuitive representations of quantum states that are useful in quantum computing and quantum information. In this paper, we investigate the role of entanglement and local unitary equivalence in quantum computing through graph states in high-dimensional Hilbert spaces. We introduce bipartitions on the vertices sets and demonstrate separability and local unitary equivalence based on different choices of partitions. By performing the two-qubit controlled phase shift gates on various pairs of multiqubit system, one can generate a genuinely entangled graph state based on simple connected graphs. We extend 2-dimensional Hilbert spaces to higher and mixed dimensions and examine pure quantum states that cannot be separated by any bipartition. In addition, we generalize the analysis on classification of local unitary equivalence beyond the controlled-Z gates to controlled phase gates. By examining graphs constructed from bipartition graphs, we have shown that graphs with identical bipartition subgraph structures are local unitary equivalent.
We propose a method for constructing multi-qubit entangled quantum states that represent weighted tripartite graphs, and develop approaches for investigating their structural properties using quantum computing. In the general case of multi-qubit states corresponding to arbitrary tripartite graph structures, we derive an expression for the entanglement distance. We establish a connection between entanglement and properties of the corresponding tripartite graphs. Namely, we show that the entanglement of a qubit with the rest of the system in a quantum tripartite graph state depends on the weights of the arcs in the neighborhood of the corresponding vertex, as well as on its degree with respect to the vertex sets of the tripartite graph. As an illustrative example, we consider a tripartite graph forming a triangle and evaluate the entanglement distance with quantum computing. We also compute quantum correlators for the general case of tripartite quantum graph states and relate these quantities to structural features of the underlying graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. Obtained relationships between the quantum properties of multi-qubit states and the structural features of tripartite graphs opens up the possibility of investigating classical systems, such as tripartite graphs, using quantum computing. It is worth emphasizing that tripartite graphs have applications in practical problems, including resource allocation, scheduling, and database and hypergraph modeling.
Quantum entanglement in multipartite systems cannot be shared freely. In order to illuminate basic rules of entanglement sharing between qubits, we introduce a concept of an entangled structure (graph) such that each qubit of a multipartite system is associated with a point (vertex), while a bipartite entanglement between two specific qubits is represented by a connection (edge) between these points. We prove that any such entangled structure can be associated with a pure state of a multiqubit system. Moreover, we show that a pure state corresponding to a given entangled structure is a superposition of vectors from a subspace of the 2N{2}^{N}2N-dimensional Hilbert space, whose dimension grows linearly with the number of entangled pairs.