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arXiv:2009.12497

kkk-Uniform states and quantum information masking

Fei Shi, Mao-Sheng Li, Lin Chen, Xiande Zhang

Published Sep 26, 2020• 3 citations
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Abstract

A pure state of $N$ parties with local dimension $d$ is called a $k$-uniform state if all the reductions to $k$ parties are maximally mixed. Based on the connections among $k$-uniform states, orthogonal arrays and linear codes, we give general constructions for $k$-uniform states. We show that when $d\geq 4k-2$ (resp. $d\geq 2k-1$) is a prime power, there exists a $k$-uniform state for any $N\geq 2k$ (resp. $2k\leq N\leq d+1$). Specially, we give the existence of $4,5$-uniform states for almost every $N$-qudits. Further, we generalize the concept of quantum information masking in bipartite systems given by [Modi \emph{et al.} {Phys. Rev. Lett. \textbf{120}, 230501 (2018)}] to $k$-uniform quantum information masking in multipartite systems, and we show that $k$-uniform states and quantum error-correcting codes can be used for $k$-uniform quantum information masking.

DOI
10.48550/arxiv.2009.12497

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A pure state of NNN parties with local dimension ddd is called a kkk-uniform state if all the reductions to kkk parties are maximally mixed. Based on the connections among kkk-uniform states, orthogonal arrays and linear codes, we give general constructions for kkk-uniform states. We show that when d≥4k−2d\geq 4k-2d≥4k−2 (resp. d≥2k−1d\geq 2k-1d≥2k−1) is a prime power, there exists a kkk-uniform state for any N≥2kN\geq 2kN≥2k (resp. 2k≤N≤d+12k\leq N\leq d+12k≤N≤d+1). Specially, we give the existence of 4,54,54,5-uniform states for almost every NNN-qudits. Further, we generalize the concept of quantum information masking in bipartite systems given by [Modi \emph{et al.} {Phys. Rev. Lett. \textbf{120}, 230501 (2018)}] to kkk-uniform quantum information masking in multipartite systems, and we show that kkk-uniform states and quantum error-correcting codes can be used for kkk-uniform quantum information masking.
A pure quantum state of nnn parties associated with the Hilbert space $\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}$ is called kkk-uniform if all the reductions to kkk-parties are maximally mixed. The nnn partite system is called homogenous if the local dimension d1=d2=⋯=dnd_1=d_2=\cdots=d_nd1​=d2​=⋯=dn​, while it is called heterogeneous if the local dimension are not all equal. kkk-uniform sates play an important role in quantum information theory. There are many progress in characterizing and constructing kkk-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the kkk-uniform states in heterogeneous systems for k>3k>3k>3. We present two general methods to construct kkk-uniform states in the heterogeneous systems for general kkk. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new kkk-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix HHH meeting the condition that HA×Aˉ+HAˉ×ATH_{A\times \bar{A}}+H^T_{\bar{A}\times A}HA×Aˉ​+HAˉ×AT​ has full rank for any row index set AAA of size kkk. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer kkk, one can construct kkk-uniform states of a heterogeneous system in many different Hilbert spaces.
A pure quantum state ofnnnparties associated with the Hilbert spaceCd1⊗Cd2⊗⋯⊗Cdn\mathbb {C}^{d_{1}}\otimes \mathbb {C} ^{d_{2}}\otimes \cdots \otimes \mathbb {C} ^{d_{n}}Cd1​⊗Cd2​⊗⋯⊗Cdn​is calledkkk-uniform if all the reductions tokkk-parties are maximally mixed. Thennnpartite system is called homogenous if the local dimensionsd1=d2=⋯=dnd_{1}=d_{2}=\cdots =d_{n}d1​=d2​=⋯=dn​, while it is called heterogeneous if the local dimensions are not all equal.kkk-uniform sates play an important role in quantum information theory. There are much progress in characterizing and constructingkkk-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for thekkk-uniform states in heterogeneous systems fork>3k>3k>3. We present two general methods to constructkkk-uniform states in the heterogeneous systems for generalkkk. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many newkkk-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrixHHHmeeting the condition thatHA×Aˉ+HAˉ×ATH_{A\times \bar {A}}+H^{T}_{\bar {A}\times A}HA×Aˉ​+HAˉ×AT​has full rank for any row index setAAAof sizekkk. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integerkkk, one can constructkkk-uniform states of a heterogeneous system in many different Hilbert spaces.
A pure quantum state of NNN subsystems with local dimension ddd is called a kkk-uniform state if every reduction to kkk qudits is maximally mixed. Based on a special class of combinatorial design, namely irredundant orthogonal arrays, we show the existence of 2-uniform NNN-qudit states when $N\ensuremath{\ge}4$ and ddd is a prime power other than 2. In addition, given any Hadamard matrix of order greater than 4, we demonstrate how it can be used to construct 3-uniform multiqubit states. In fact, we find that there exists some 3-uniform NNN-qubit when $N\ensuremath{\ge}8$ except case N=9N=9N=9. These give an answer to a question posed by Goyeneche et al. [Phys. Rev. A 90, 022316 (2014)]. Furthermore, starting from a minimal support kkk-uniform state, we show how to generate an orthogonal basis consisting of kkk-uniform states. At last, we derive a series of $(k\ensuremath{-}1)$-uniform $(N\ensuremath{-}1)$-qudit states from a single kkk-uniform NNN-qudit state. The kkk-uniform states are good candidates among the multipartite entangled states.
It is impossible to mask an arbitrary quantum state into the correlations between two subsystems such that the original information is completely unknown to each local system. This is the no-masking theorem proposed by Modi et al. [K. Modi, A. K. Pati, A. Sen(De), and U. Sen, Phys. Rev. Lett. 120, 230501 (2018)]. In this work, we propose the concept of kkk-uniform quantum information masking in multipartite systems and indicate the relation between quantum error-correcting codes (QECCs) in heterogeneous systems and quantum information masking. As a consequence, we show that the no-masking theorem is a special case of the quantum Singleton bound for QECCs in heterogeneous systems essentially, and we give a more general no-masking theorem in multipartite systems based on the quantum Singleton bound. We also solve two open questions proposed by Li and Wang [M.-S. Li and Y.-L. Wang, Phys. Rev. A 98, 062306 (2018)]. That is, an arbitrary state of level ddd cannot be masked in a tripartite system of level ddd such that the marginal states are not proportional to identity, and an arbitrary state of level ddd cannot be masked in a tripartite system of level nnn with n<dn<dn<d. Further, we give some methods for constructing new QECCs from old QECCs in heterogeneous systems.
A pure quantum state of NNN subsystems with ddd levels each is called kkk-multipartite maximally entangled state, which we call a kkk-uniform state, if all its reductions to kkk qudits are maximally mixed. These states form a natural generalization of NNN-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and kkk-uniform states and prove the existence of several classes of such states for NNN-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of N>5N>5N>5 qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of kkk-uniform states and classical and quantum error correction codes and provide a graph representation for such states.
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 correction codes.To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes.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 the so-called maximum distance separable error correction codes.Furthermore, we use our method to construct several examples of absolutely maximally entangled states whose existence was open so far.