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Recently, a kind of deterministic all-versus-nothing proof of Bell nonlocality induced from the qubit non-stabilizer state was proposed, breaking the tradition that deterministic all-versus-nothing proofs are always derived from stabilizer states. A trivial generalization to the qudit (d is even) version is by using a special basis map, but such a proof can still be reduced to the qubit version. So far, whether high dimensional non-stabilizer states can induce nontrivial deterministic all-versus-nothing proofs of Bell nonlocality remains unknown. Here we present an example induced from a specific four-qudit non-stabilizer state (with d = 4), showing that such proofs can be constructed in high dimensional scenarios as well.

In the literature, bilocality and $n$-locality of correlation tensors (CTs) are described by integration local hidden variable models (called C-LHVMs) rather than by summation LHVMs (called D-LHVMs). Obviously, C-LHVMs are easier to be constructed than D-LHVMs, while the later are easier to be used than the former, e.g., in discussing on the topological and geometric properties of the sets of all bilocal and of all $n$-local CTs. In this context, one may ask whether the two descriptions are equivalent. In the present work, we first establish some equivalent characterizations of bilocality of a tripartite CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$, implying that the two descriptions of bilocality are equivalent. As applications, we prove that all bilocal CTs with the same size form a compact path-connected set that has many star-convex subsets. Secondly, we introduce and discuss the bilocality of a tripartite probability tensor (PT) ${\bf{P}}=\Lbrack P(abc)\Rbrack$, including equivalent characterizations and properties of bilocal PTs. Lastly, we obtain corresponding results about $n$-locality of $n+1$-partite CTs ${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ and PTs ${\bf{P}}=\Lbrack P({\bf{a}}b)\Rbrack$, respectively.

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 $S|\psi\>$ of states $|\psi\>$ 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)]., Comment: This is a revision about arXiv:2004.14540. In the present version, $k$ and $j$ old Eq. (2.2) have been exchanged and the followed three equations have been corrected

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 $S^\sharp$ 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 $S^{\diamond}$ 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 ${S^{\sharp}}$ and ${S^{\diamond}}$, respectively.

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 $S|\psi\>$ of states $|\psi\>$ 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)]., Comment: Some errors have been found in the original sumbssion

Bell nonlocality and Einstein-Podolsky-Rosen (EPR) steering are every important quantum correlations of a composite quantum system. Bell nonlocality of a bipartite state is a quantum correlation demonstrated by some local quantum measurements, while EPR steering is another form of quantum correlations, observed firstly by Schrodinger in the context of famous EPR paradox. In this paper, we give some remarks on Bell nonlocality and EPR steering of bipartite states, including mathematical definitions and characterizations of these two quantum correlations, the convexity and closedness of the set of all Bell local states and the set of all EPR unsteerable states. We also derive a EPR-steering criteria, with which the EPR steerability of the maximally entangled states are checked.

In this paper, we discuss time evolution and adiabatic approximation in $PT$-symmetric quantum mechanics. we give the time evolving equation for a class of $PT$-symmetric Hamiltonians and some conditions of the adiabatic approximation for the class of $PT$-symmetric Hamiltonians.