We study the problem of information masking through nonzero linear operators that distribute information encoded in single qubits to the correlations between two qubits. It is shown that a nonzero linear operator can not mask any nonzero measure set of qubit states. We prove that the maximal maskable set of states on the Bloch sphere with respect to any masker is the ones on a spherical circle. Any states on a spherical circle on the Bloch sphere are maskable, which also proves the conjecture on maskable qubit states given in [Phys. Rev. Lett. 120, 230501 (2018)]. Moreover, we provide explicitly operational unitary maskers for all maskable sets. As applications, new protocols for secret sharing are introduced. PACS numbers: 03.67.-a, 03.65.Ud, 03.65.YzIntroduction. Due to the properties of linearity (unitarity) of the evolution of a closed quantum system in quantum mechanics, it is well known that there are several no-go theorems such as the no-cloning theorem [1-3], the no broadcasting theorem and the no-deleting theorem [4][5][6][7]. Recently, Kavan Modi et. al. [8] considered the problem of quantum information masking based on unitary operators, and obtained the so-called no-masking theorem: it is impossible to mask all arbitrary qubit states by the same unitary operator. Different from the decoherence of open systems due to interactions between the system and the environment [9-12], the quantum masking means that the information in subsystems are transferred into the correlations of bipartite systems by unitary operations, such that the final reduced states of any subsystems are identical. Namely, the subsystems themselves contain no longer the initial information. Nomasking theorem is also different from other no-go theorems such that non-orthogonal states cannot be perfectly cloned or deleted. In fact, there are many sets containing infinitely many nonorthogonal quantum states which can be masked [8].
In a recent paper, M. F. Sacchi [Phys. Rev. A 96, 042325 (2017)] addressed the general problem of approximating an unavailable quantum state by the convex mixing of different available states. For the case of qubit mixed states, we show that the analytical solutions in some cases are invalid. In this Comment, we present complete analytical solutions for the optimal convex approximation. Our solutions can be viewed as correcting and supplementing the results in the aforementioned paper.PACS numbers: 03.65.Ud, 03.65.Yz In Sec. III of Ref.[1], the problem of optimally generating a desired quantum state ρ by the given set of the eigenstates of all Pauli matrices was provided. Namely, consider the optimal convex approximation of a quantum state with respect to the setThe optimal convex approximation of ρ with respect to, the minimum is taken over all possible probability weights {p i }, and A 1 denotes the trace norm of A, that is,The optimal convex approximate set is given byHere we point out that the analytical solution given in [Phy. Rev. A 96, 042325(2017)] is invalid in some cases. We first provide a simple example. Consider the target qubit ρ given bywith a ∈ [0, 1], φ ∈ [0, 2π], and k ∈ [0, 1]. If we set a = 1/2, k = 1, φ = π/4, it is easily verified that the point belongs to the region of case (i) in Ref.[1], that is, k th ≡ a/( a(1 − a)(cos φ + sin φ)) < k ≤ a/ ( a(1 − a)). Then the optimal convex approximation and the corresponding optimal weights are given by Eq.and (19) in [1], respectively. However, if one substitutes a = 1/2, k = 1 and φ = π/4 into Eq. (19) in [1], one has p 0 = 1 − 4a/3 − 2k a(1 − a)(cos φ + sin φ)/3 = (1 − √ 2)/3 < 0, which implies that the optimal probability is negative and this solution is invalid.In the following, in terms of the method used in [2] (see also the Karush-Kuhn-Tucker theorem and its conclusion * Electronic address: libobeijing2008@163.com in [3], p46-60), we provide the complete analytical solution for the optimal convex approximation of a quantum state under B 3 distance and the corresponding optimal weights.For simplicity, we denote u = k a(1 − a) cos φ, v = k a(1 − a) sin φ, where k ∈ [0, 1], a ∈ [0, 1 2 ] and φ ∈ [0, π/2]. When a − u − v ≥ 0, one has D B3 (ρ) = 0. The pertaining weights corresponding to ρ i are given bywhere t 1 and t 2 are arbitrary non-negative arguments such that p 1 ≥ 0. If t 1 = t 2 = 0, then Eq.(2) reduce to Eq. (14) in Ref. [1]. However, if one sets t 1 = a − u − v, t 2 = 0 in (2), one gets p 0 = 1−2a, p 1 = 0, p 2 = a+u−v, p 3 = a−u−v, p 4 = 2v and p 5 = 0. This is another kind of decomposition which is different from the one in Ref.[1]. Thus, our decompositions can be viewed as a complete supplement to the results in Ref.[1]. The previous complete analytical solution can be classified into the following four cases, see proof in Supplemental Material: i) If a < u+v ≤ (3−4a)/2, a−v+2u ≥ 0 and a−u+2v ≥ 0, the optimal convex approximation of ρ is given bywith corresponding optimal weights p 0 = 1 − 4a/3 − 2u/3 − 2v/3, p 2 = 2a/3 − 2v/3 + ...
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