Constructing new k-uniform and absolutely maximally entangled states

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Constructing new k-uniform and absolutely maximally entangled states

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Constructing new k-uniform and absolutely maximally entangled states

Zahra Raissi, Adam Teixido, Christian Gogolin, Antonio Acín

28 Oct 2019-arXiv: Quantum Physics-

TL;DR: The only known systematic construction of k-uniform quantum states is based on a class of classical error correction codes known as maximum distance separable codes as mentioned in this paper, and it is shown that the states derived through their construction are not equivalent to any kuniform state constructed from maximum distance separationable codes.

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Abstract: Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols and the construction of quantum error correcting codes. To our knowledge, the only known systematic construction of these states is based on a class of classical error correction codes known as maximum distance separable. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from maximum distance separable codes. Furthermore, we used our method to construct several examples of absolutely maximally entangled states whose existence was open so far.

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Strong quantum nonlocality with entanglement

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Fei Shi¹, Mengyao Hu², Lin Chen², Xiande Zhang¹•Institutions (2)

University of Science and Technology of China¹, Beihang University²

02 Oct 2020-Physical Review A

TL;DR: In this article, a Rubik's cube-based construction of strongly nonlocal orthogonal bases with entanglement has been proposed, and two protocols for local discrimination of these bases have been proposed.

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Abstract: Strong quantum nonlocality was introduced recently as a stronger manifestation of nonlocality in multipartite systems through the notion of local irreducibility in all bipartitions. Known existing results for sets of strongly nonlocal orthogonal states are limited to product states. In this paper, based on the Rubik's cube, we give a construction of such sets consisting of entangled states in $d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d$ for all $d\ensuremath{\ge}3$. Consequently, we answer an open problem given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)], that is, orthogonal entangled bases that are strongly nonlocal do exist. Furthermore, we propose two entanglement-assisted protocols for local discrimination of our results. Each protocol consumes less entanglement resources than the teleportation-based protocol on average. Our results exhibit the phenomenon of strong quantum nonlocality with entanglement.

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43 citations

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Constructions of $k$-uniform states from mixed orthogonal arrays

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Fei Shi, Yi Shen, Lin Chen, Xiande Zhang

07 Jun 2020-arXiv: Quantum Physics

TL;DR: Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthoglobal arrays and $k$-uniform states, this article presented two constructions of $2$-uneiform states in heterogeneous systems.

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Abstract: We study $k$-uniform states in heterogeneous systems whose local dimensions are mixed. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and $k$-uniform states, we present two constructions of $2$-uniform states in heterogeneous systems. We also construct a family of $3$-uniform states in heterogeneous systems, which solves a question posed in [D. Goyeneche et al., Phys. Rev. A 94, 012346 (2016)]. We also show two methods of generating $(k-1)$-uniform states from $k$-uniform states. Some new results on the existence and nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting of $k$-uniform states with minimum support. Moreover, we show that some $k$-uniform bases can not be distinguished by local operations and classical communications, and this shows quantum nonlocality with entanglement.

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8 citations

Cites background from "Constructing new k-uniform and abso..."

...In [16, 21], they showed that if there exists a k-uniform state with minimum support in (Cd)⊗N , then there exists an orthogonal basis consisting of k-uniform states with minimum support in (Cd)⊗N ....
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Journal Article•DOI•

Modifying Method of Constructing Quantum Codes From Highly Entangled States

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Zahra Raissi

04 May 2020-IEEE Access

TL;DR: This work provides explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes.

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Abstract: There is a connection between classical codes, highly entangled pure states (called $k$ -uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from a $k$ -uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the method to produce another set of stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. This construction produces quantum codes starting from an AME state without tracing out any party. Therefore, quantum stabilizer codes with larger codespace can be constructed.

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5 citations

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Journal Article•DOI•

Quantum secret sharing

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Mark Hillery¹, Vladimír Bužek², André Berthiaume³•Institutions (3)

City University of New York¹, Slovak Academy of Sciences², DePaul University³

01 Mar 1999-Physical Review A

TL;DR: This work shows how GHZ states can be used to split quantum information into two parts so that both parts are necessary to reconstruct the original qubit.

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Abstract: Secret sharing is a procedure for splitting a message into several parts so that no subset of parts is sufficient to read the message, but the entire set is. We show how this procedure can be implemented using Greenberger-Horne-Zeilinger (GHZ) states. In the quantum case the presence of an eavesdropper will introduce errors so that his presence can be detected. We also show how GHZ states can be used to split quantum information into two parts so that both parts are necessary to reconstruct the original qubit.

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2,789 citations

"Constructing new k-uniform and abso..." refers background in this paper

...[1] This can be established by checking the stabilizer formalism and graph states representation....
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...The state AME(7, 4) can be constructed by using MDS code with parameters [5, 3, 3]4 and showing that all the terms can be classified into 4 2 many boxes with terms forming an MDS code [5, 1, 5]4....
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...Considering the connection between the codewords of the original code and its dual, one can check that the states |ψ〉 and |ψ⊥〉 can be transformed into each other by transforming the basis using Fourier gates, i.e., from Z-eigenbasis to X-eigenbasis [1]....
[...]
..., from Z-eigenbasis to X-eigenbasis [1]....
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Journal Article•DOI•

A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States

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Roman Orus¹•Institutions (1)

University of Mainz¹

01 Oct 2014-Annals of Physics

TL;DR: This is a partly non-technical introduction to selected topics on tensor network methods, based on several lectures and introductory seminars given on the subject, that should be a good place for newcomers to get familiarized with some of the key ideas in the field.

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1,584 citations

"Constructing new k-uniform and abso..." refers background or methods in this paper

...Therefore, many efforts have focused on the study of relevant sets of states such as, for instance, graph states [7, 8] or tensor network states [9]....
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...n = 5 [3, 2, 2]q Bell basis, q 2 states q ≥ 2 q ≥ 4 n = 6 [4, 2, 3]q Bell basis, q 2 states q ≥ 3 q ≥ 4 k = 2 n = 7 [5, 2, 4]q Bell basis, q 2 states q ≥ 4 q ≥ 7 n = 8 [5, 3, 3]q GHZ basis, q 3 states q ≥ 4 q ≥ 7 n = 9 [6, 3, 4]q GHZ basis, q 3 states q ≥ 4 q ≥ 8 n = 10 [7, 3, 5]q GHZ basis, q 3 states q ≥ 7 q ≥ 9 n = 11 [7, 4, 4]q AME(4, q) basis, q 4 states q ≥ 7 q ≥ 11 n = 12 [8, 4, 5]q AME(4, q) basis, q 4 states q ≥ 7 q ≥ 11 k = 3 n = 13 [9, 4, 6]q AME(4, q) basis, q 4 states q ≥ 8 q ≥ 13 n = 14 [9, 5, 5]q AME(5, q) basis, q 5 states q ≥ 8 q ≥ 13 n = 15 [10, 5, 6]q AME(5, q) basis, q 5 states q ≥ 9 q ≥ 16 n = 16 [11, 5, 7]q AME(5, q) basis, q 5 states q ≥ 11 q ≥ 16...
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...For the state AME(11, 8) we employ MDS code [9, 5, 5]8 such that it can be classified to 8 2 boxes of MDS codes with parameters [9, 3, 7]8....
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Journal Article•DOI•

Multiparty entanglement in graph states

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M. Hein¹, Jens Eisert², Jens Eisert³, Hans J. Briegel⁴, Hans J. Briegel¹ - Show less +1 more•Institutions (4)

Ludwig Maximilian University of Munich¹, Imperial College London², University of Potsdam³, Austrian Academy of Sciences⁴

09 Jun 2004-Physical Review A

TL;DR: This work characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which it provides upper and lower bounds in graph theoretical terms.

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Abstract: Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.

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843 citations

"Constructing new k-uniform and abso..." refers background or methods in this paper

...The graph state associated with a given graph G is the +1 eigenstate of the following set of stabilizer operators [7, 8, 23, 24]...
[...]
...Therefore, many efforts have focused on the study of relevant sets of states such as, for instance, graph states [7, 8] or tensor network states [9]....
[...]
...n = 5 [3, 2, 2]q Bell basis, q 2 states q ≥ 2 q ≥ 4 n = 6 [4, 2, 3]q Bell basis, q 2 states q ≥ 3 q ≥ 4 k = 2 n = 7 [5, 2, 4]q Bell basis, q 2 states q ≥ 4 q ≥ 7 n = 8 [5, 3, 3]q GHZ basis, q 3 states q ≥ 4 q ≥ 7 n = 9 [6, 3, 4]q GHZ basis, q 3 states q ≥ 4 q ≥ 8 n = 10 [7, 3, 5]q GHZ basis, q 3 states q ≥ 7 q ≥ 9 n = 11 [7, 4, 4]q AME(4, q) basis, q 4 states q ≥ 7 q ≥ 11 n = 12 [8, 4, 5]q AME(4, q) basis, q 4 states q ≥ 7 q ≥ 11 k = 3 n = 13 [9, 4, 6]q AME(4, q) basis, q 4 states q ≥ 8 q ≥ 13 n = 14 [9, 5, 5]q AME(5, q) basis, q 5 states q ≥ 8 q ≥ 13 n = 15 [10, 5, 6]q AME(5, q) basis, q 5 states q ≥ 9 q ≥ 16 n = 16 [11, 5, 7]q AME(5, q) basis, q 5 states q ≥ 11 q ≥ 16...
[...]
...For the state AME(11, 8) we employ MDS code [9, 5, 5]8 such that it can be classified to 8 2 boxes of MDS codes with parameters [9, 3, 7]8....
[...]
...A graph G = (V,Γ) is composed of a set V of n vertices and a set of weighted edges specified by the adjacency matrix Γ [7, 8, 23, 24], an n × n symmetric matrix such that Γi,j = 0 if vertices i and j are not connected and Γi,j > 0 otherwise....
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Journal Article•DOI•

Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

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Fernando Pastawski¹, Beni Yoshida¹, Daniel Harlow², John Preskill¹•Institutions (2)

California Institute of Technology¹, Princeton University²

20 Mar 2015-arXiv: High Energy Physics - Theory

TL;DR: In this article, a tensor network is proposed to capture key features of entanglement in the AdS/CFT correspondence, in particular the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases.

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Abstract: We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.

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526 citations

Journal Article•DOI•

Maximum distance q -nary codes

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R. Singleton¹•Institutions (1)

SRI International¹

01 Apr 1964-IEEE Transactions on Information Theory

TL;DR: Examples and construction methods are given to show that these codes exist for a number of values of q, k, and r and with the restriction that the codes be linear.

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Abstract: A q -nary error-correcting code with N = q^{k} code words of length n = k + r can have no greater minimum distance d than r+1 . The class of codes for which d = r+1 is studied first in general, then with the restriction that the codes be linear. Examples and construction methods are given to show that these codes exist for a number of values of q, k , and r .

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525 citations