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arXiv:2005.11169 · Europhysics Letters (EPL)

Quantum multipartite maskers vs. quantum error-correcting codes

Kanyuan Han, Zhihua Guo, Huaixin Cao, Yuxing Du, Chuan Fu Yang

Published Aug 27, 2020• 14 citations
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Abstract

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 $Q$ of pure states of a system $K$ can be masked by an operator $S$ into a multipartite system $\H^{(n)}$ if all of the image states $S|ψ\>$ of states $|ψ\>$ in $Q$ have the same marginal states on each subsystem. We call such an $S$ a QMM of $Q$. 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)].

DOI
10.1209/0295-5075/131/30005

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arXiv:2009.12497•Sep 26, 2020•3 citations
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 \C2\C^2\C2 into \C2⊗\C2⊗\C2\C^2\otimes\C^2\otimes\C^2\C2⊗\C2⊗\C2 and then the states of \C3\C^3\C3 can not be masked isometrically into \C2⊗\C2⊗\C2\C^2\otimes\C^2\otimes\C^2\C2⊗\C2⊗\C2. 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 \Cd\C^d\Cd can be completely hidden in correlations between any two subsystems of the tripartite system \Cd+1⊗\Cd+1⊗\Cd+1\C^{d+1}\otimes\C^{d+1}\otimes\C^{d+1}\Cd+1⊗\Cd+1⊗\Cd+1, while arbitrary quantum states cannot be completely hidden in the correlations between subsystems of a bipartite system [PRL 98, 080502 (2007)].
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)].
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.
Masking information into quantum correlations is a cornerstone of many quantum information applications. While there exist the no-hiding and no-masking theorems, approximate quantum information masking (AQIM) offers a promising means of circumventing the constraints. Despite its potential, AQIM still remains underexplored, and constructing explicit approximate maskers remains a challenge. In this work, we investigate AQIM from multiple perspectives and propose using random isometries to construct approximate maskers. First, different notions of AQIM are introduced and we find there are profound intrinsic connections among them. These relationships are characterized by a set of figures of merit, which are introduced to quantify the deviation of AQIM from exact QIM. We then explore the possibility of realizing AQIM via random isometries in bipartite and multipartite systems. In bipartite systems, we identify a fundamental lower bound for a key figure of merit, implying that almost all random isometries fail to realize AQIM. This surprising result generalizes the original no-masking theorem to the no-random-AQIM theorem for bipartite systems. In contrast, in multipartite systems, we show almost all random isometries can realize AQIM. Remarkably, the number of physical qubits required to randomly mask a single logical qubit scales only linearly. We further explore the implications of these findings. In particular, we show that, under certain conditions, approximate quantum error correction is equivalent to AQIM. Consequently, AQIM naturally gives rise to approximate quantum error correction codes with constant code rates and exponentially small correction inaccuracies. Overall, our results establish quantum information masking as a central concept in quantum information theory, bridging diverse notions across multiple domains.
Masking of quantum information means that information is hidden from a subsystem and spread over a composite system. Modi et al. proved in [Phys. Rev. Lett. 120, 230501 (2018)] that this is true for some restricted sets of nonorthogonal quantum states and it is not possible for arbitrary quantum states. In this paper, we discuss the problem of masking quantum information encoded in pure and mixed states, respectively. Based on an established necessary and sufficient condition for a set of pure states to be masked by an operator, we find that there exists a set of four states that can not be masked, which implies that to mask unknown pure states is impossible. We construct a masker S♯S^\sharpS♯ and obtain its maximal maskable set, leading to an affirmative answer to a conjecture proposed in Modi's paper mentioned above. We also prove that an orthogonal (resp. linearly independent) subset of pure states can be masked by an isometry (resp. injection). Generalizing the case of pure states, we introduce the maskability of a set of mixed states and prove that a commuting subset of mixed states can be masked by an isometry S⋄S^{\diamond}S⋄ while it is impossible to mask all of mixed states by any operator. We also find the maximal maskable sets of mixed states of the isometries S♯{S^{\sharp}}S♯ and S⋄{S^{\diamond}}S⋄, respectively.
The no-masking theories show that it is impossible to mask the set of all qubit states into the quantum correlation of bipartite qubit system or tripartite qubit system. In this paper, we give a new proof of the no-masking situation of the tripartite qubit system. Recent work has shown that there exists a universal masker which can mask an arbitrary set of qubit states in four-qubit systems perfectly by means of the maximum entangled states. Here we show that there exist more than one masking scheme even for the same multipartite qubit system. Basing on the maximum entangled states we give the deterministic masking scenario for N-qubit system. In practice, decoherence hinders us from obtaining the maximum entangled states. From this viewpoint, the masking scenario based on non-maximum entangled states becomes more universal. Furthermore, we provide an approximate quantum masking scenario and investigate the relation between approximate masking and quantum entanglement.
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.