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arXiv:1902.07978 · Physical Review A

Masking quantum information in multipartite scenario

Mao-Sheng Li, Yan-Ling Wang

Published Dec 5, 2018• 54 citations
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Abstract

Recently, Modi et al. [Phys. Rev. Lett. 120, 230501 (2018)] found that masking quantum information is impossible in a bipartite scenario. This adds another item to the no-go theorems. In this paper, we present some schemes different from error correction codes, which show that quantum states can be masked when more participants are allowed in the masking process. Moreover, using a pair of mutually orthogonal Latin squares of dimension $d$, we show that all the $d$ level quantum states can be masked into tripartite quantum systems whose local dimensions are $d$ or $d+1$. This highlights some differences between the no-masking theorem and the classical no-cloning theorem or no-deleting theorem.

DOI
10.1103/physreva.98.062306

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Recently, Modi et al. [Phys. Rev. Lett. 120, 230501 (2018)] found that masking quantum information is impossible in a bipartite scenario. This adds another item to the no-go theorems. In this paper, we present some schemes different from error correction codes, which show that quantum states can be masked when more participants are allowed in the masking process. Moreover, using a pair of mutually orthogonal Latin squares of dimension ddd, we show that all the ddd level quantum states can be masked into tripartite quantum systems whose local dimensions are ddd or d+1d+1d+1. This highlights some differences between the no-masking theorem and the classical no-cloning theorem or no-deleting theorem.
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.
The no-masking theorem says that masking quantum information is impossible in a bipartite scenario. However, there exist schemes to mask quantum states in multipartite systems. In this work, we show that, the joint measurement in the teleportation is really a masking process, when the apparatus is regarded as a quantum participant in the whole system. Based on the view, we present two four-partite maskers and a tripartite masker. One of the former provides a generalization in arbitrary dimension of the four-qubit scheme given by Li and Wang [Phys. Rev. A 98, 062306 (2018)], and the latter is precisely their tripartite scheme. The occupation probabilities and coherence of quantum states are masked in two steps of our schemes. And the information can be extracted naturally in their reverse processes.
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.
Since masking of quantum information was introduced by Modi et al. in [PRL 120, 230501 (2018)], many discussions on this topic have been published. In this paper, we consider relationship between quantum multipartite maskers (QMMs) and quantum error-correcting codes (QECCs). We say that a subset QQQ of pure states of a system KKK can be masked by an operator SSS into a multipartite system $\H^{(n)}$ if all of the image states S∣ψ S|ψ\>S∣ψ of states ∣ψ |ψ\>∣ψ in QQQ have the same marginal states on each subsystem. We call such an SSS a QMM of QQQ. By establishing an expression of a QMM, we obtain a relationship between QMMs and QECCs, which reads that an isometry is a QMM of all pure states of a system if and only if its range is a QECC of any one-erasure channel. As an application, we prove that there is no an isometric universal masker from $\C^2$ into $\C^2\otimes\C^2\otimes\C^2$ and then the states of $\C^3$ can not be masked isometrically into $\C^2\otimes\C^2\otimes\C^2$. This gives a consummation to a main result and leads to a negative answer to an open question in [PRA 98, 062306 (2018)]. Another application is that arbitrary quantum states of $\C^d$ can be completely hidden in correlations between any two subsystems of the tripartite system $\C^{d+1}\otimes\C^{d+1}\otimes\C^{d+1}$, while arbitrary quantum states cannot be completely hidden in the correlations between subsystems of a bipartite system [PRL 98, 080502 (2007)].
The structure of quantum mechanics forbids a bipartite scenario for masking quantum information, however, it allows multipartite maskers. The Latin squares are found to be closely related to a series of tripartite maskers. This adds another item, significantly different from the original no-cloning theorem, to the no-go theorems. On the other hand, anyonic excitations in two dimensions exhibit exotic collective behaviors of quantum physics, and open the avenue of fault-tolerant topological quantum computing. Here, we give the Latin-square construction of Abelian and Ising anyons in the Kitaev model and study the maskable space configuration in anyonic space. The circling and braiding of Kitaev anyons are masking operations on extended hyperdisks in anyonic space. We also realize quantum information masking in a teleportation way in the Kitaev Ising anyon model.
Since masking of quantum information was introduced by Modi et al. in [PRL 120, 230501 (2018)], many discussions on this topic have been published. In this paper, we consider relationship between quantum multipartite maskers (QMMs) and quantum error-correcting codes (QECCs). We say that a subset QQQ of pure states of a system KKK can be masked by an operator SSS into a multipartite system $\H^{(n)}$ if all of the image states S∣ψ S|ψ\>S∣ψ of states ∣ψ |ψ\>∣ψ in QQQ have the same marginal states on each subsystem. We call such an SSS a QMM of QQQ. By establishing an expression of a QMM, we obtain a relationship between QMMs and QECCs, which reads that an isometry is a QMM of all pure states of a system if and only if its range is a QECC of any one-erasure channel. As an application, we prove that there is no an isometric universal masker from $\C^2$ into $\C^2\otimes\C^2\otimes\C^2$ and then the states of $\C^3$ can not be masked isometrically into $\C^2\otimes\C^2\otimes\C^2$. This gives a consummation to a main result and leads to a negative answer to an open question in [PRA 98, 062306 (2018)]. Another application is that arbitrary quantum states of $\C^d$ can be completely hidden in correlations between any two subsystems of the tripartite system $\C^{d+1}\otimes\C^{d+1}\otimes\C^{d+1}$, while arbitrary quantum states cannot be completely hidden in the correlations between subsystems of a bipartite system [PRL 98, 080502 (2007)].