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:1510.06673

The superposition invariance of unitary operators and maximally entangled state

Xin-Wei Zha, Yunguang Zhang, Jianxia Qi

Published Oct 15, 2015• 0 citations
arXivHTMLPDF

Abstract

In this paper, we study the superposition invariance of unitary operators and maximally entangled state respectively. Furthermore, we discuss the set of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.

DOI
10.48550/arxiv.1510.06673

Related papers

The superposition invariance of unitary operators and maximally entangled state

Xin-Wei Zha, Ning Miao, Xiaoyuan Yu

In this paper, we study unitary operators and the superposition of unitary operators. We calculate the the superposition of unitary operators and find that some unitary operators superposition is also unitary operator. Furthermore, via this property, we discuss the set of orthogonal maximally entangled states. For 2,3,4,5-qubit, we introduce the complete sets of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.

•Dec 18, 2019•0 citations

Entanglement of Superpositions of Orthogonal Maximally Entangled States

Zhang Dao-Hua, Duan-Lu Zhou, Fan Heng

We study the entanglement properties of the superposed state of orthogonal maximally entangled states. It is shown that the superposed state is maximally entangled and the superposed state is separable. The relation between the superposed state and the mutually unbiased state is discussed.

Chinese Physics Letters•Sep 1, 2010•2 citations

Typical Entanglement of Superpositions

Damien Quinn, Joshuah T. Heath, Graham Kells

We investigate universal entanglement properties inherent to superpositions of randomized states. We find that an $m$-fold superposition of typical states may be classified into two distinct entanglement classes via the 2nd Rényi entropy density $s_2$. The maximally entangled regime is defined by $s_2 \sim \ln (2)$, for which superposition adds no additional entanglement. The sub-maximally entangled regime, $s_2<\ln 2$, instead constrains the reduced density matrices of independent components to be orthogonal in the thermodynamic limit, which fixes the entanglement of the superposition to a logarithmic enhancement $ΔS(m)=\ln (m)$. As a consequence, an exponentially large number of superpositions is required to transition from the sub-maximally entangled class to maximal entanglement. We explicitly calculate $s_2$ and the logarithmic enhancement, and demonstrate orthogonality for two canonical examples of the sub-maximally entangled regime (superpositions of pure Gaussian states and of random matrix-product states). We also examine the entanglement of superpositions of random stabilizer states, and discuss their relaxation to the Haar limit.

quant-pharXiv:2607.06474•Jul 8, 2026•0 citations

Degree of entanglement for two qubits

Ayman F. Abouraddy, Bahaa E. A. Saleh, Alexander V. Sergienko +1 more

We demonstrate that any pure bipartite state of two qubits may be decomposed into a superposition of a maximally entangled state and an orthogonal factorizable one. Although there are many such decompositions, the weights of the two superposed states are, remarkably, unique. We propose a measure of entanglement based on this decomposition. We also demonstrate that this measure is connected to three measures of entanglement previously set forth: maximal violation of Bell's inequality, concurrence, and two-particle visibility.

arXiv:quant-ph/0109081•Physical Review A•Oct 8, 2001•81 citations

Entanglement of subspaces and error-correcting codes

Gilad Gour, Nolan R. Wallach

We introduce the notion of entanglement of subspaces as a measure that quantifies the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error-correction codes. We discuss both degenerate and nondegenerate codes and show that the subspace spanned by the logical code words of a nondegenerate code is a $k$-totally- (maximally) entangled subspace. As for non-degenerate codes, we provide a mathematical definition in terms of subspaces and, as an example, we analyze Shor's 9-qubit code in terms of 22 mutually orthogonal subspaces.

arXiv:0704.0251•Physical Review A•Oct 5, 2007•32 citations

Entanglement in the Symmetric Sector of n Qubits

Pedro Ribeiro, Rémy Mosseri

We discuss the entanglement properties of symmetric states of n qubits. The Majorana representation maps a generic such state into a system of n points on a sphere. Entanglement invariants, either under local unitaries (LU) or stochastic local operations and classical communication (SLOCC), can then be addressed in terms of the relative positions of the Majorana points. In the LU case, an overcomplete set of invariants can be built from the inner product of the radial vectors pointing to these points; this is detailed for the well-documented three-qubits case. In the SLOCC case, a cross ratio of related Möbius transformations are shown to play a central role, exemplified here for four qubits. Finally, as a side result, we also analyze the manifold of maximally entangled 3 qubit state, both in the symmetric and generic case.

arXiv:1101.2828•Physical Review Letters•May 3, 2011•78 citations
In this paper, we study the superposition invariance of unitary operators and maximally entangled state respectively. Furthermore, we discuss the set of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.
In this paper, we study unitary operators and the superposition of unitary operators. We calculate the the superposition of unitary operators and find that some unitary operators superposition is also unitary operator. Furthermore, via this property, we discuss the set of orthogonal maximally entangled states. For 2,3,4,5-qubit, we introduce the complete sets of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.
We investigate universal entanglement properties inherent to superpositions of randomized states. We find that an mmm-fold superposition of typical states may be classified into two distinct entanglement classes via the 2nd Rényi entropy density s2s_2s2​. The maximally entangled regime is defined by s2∼ln⁡(2)s_2 \sim \ln (2)s2​∼ln(2), for which superposition adds no additional entanglement. The sub-maximally entangled regime, s2<ln⁡2s_2<\ln 2s2​<ln2, instead constrains the reduced density matrices of independent components to be orthogonal in the thermodynamic limit, which fixes the entanglement of the superposition to a logarithmic enhancement ΔS(m)=ln⁡(m)ΔS(m)=\ln (m)ΔS(m)=ln(m). As a consequence, an exponentially large number of superpositions is required to transition from the sub-maximally entangled class to maximal entanglement. We explicitly calculate s2s_2s2​ and the logarithmic enhancement, and demonstrate orthogonality for two canonical examples of the sub-maximally entangled regime (superpositions of pure Gaussian states and of random matrix-product states). We also examine the entanglement of superpositions of random stabilizer states, and discuss their relaxation to the Haar limit.
We demonstrate that any pure bipartite state of two qubits may be decomposed into a superposition of a maximally entangled state and an orthogonal factorizable one. Although there are many such decompositions, the weights of the two superposed states are, remarkably, unique. We propose a measure of entanglement based on this decomposition. We also demonstrate that this measure is connected to three measures of entanglement previously set forth: maximal violation of Bell's inequality, concurrence, and two-particle visibility.
We introduce the notion of entanglement of subspaces as a measure that quantifies the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error-correction codes. We discuss both degenerate and nondegenerate codes and show that the subspace spanned by the logical code words of a nondegenerate code is a kkk-totally- (maximally) entangled subspace. As for non-degenerate codes, we provide a mathematical definition in terms of subspaces and, as an example, we analyze Shor's 9-qubit code in terms of 22 mutually orthogonal subspaces.
We discuss the entanglement properties of symmetric states of n qubits. The Majorana representation maps a generic such state into a system of n points on a sphere. Entanglement invariants, either under local unitaries (LU) or stochastic local operations and classical communication (SLOCC), can then be addressed in terms of the relative positions of the Majorana points. In the LU case, an overcomplete set of invariants can be built from the inner product of the radial vectors pointing to these points; this is detailed for the well-documented three-qubits case. In the SLOCC case, a cross ratio of related Möbius transformations are shown to play a central role, exemplified here for four qubits. Finally, as a side result, we also analyze the manifold of maximally entangled 3 qubit state, both in the symmetric and generic case.