Mei-yi Wang ,Su-juan Zhang,Chen-ming Bai and Lu Liu

Department of Mathematics and Physics,Shijiazhuang Tiedao University,Shijiazhuang 050043,China

Abstract This paper focuses on quantum information masking for the quantum state in two-dimensional Hilbert space.We present a system of equations as the condition of quantum information masking.It is shown that quantum information contained in a single qubit state can be masked,if and only if the coefficients of the quantum state satisfy the given system of equations.By observing the characteristics of non-orthogonal maskable quantum states,we obtain a related conclusion,namely,if two non-orthogonal two-qubit quantum states can mask a single qubit state,they have the same number of terms and the same basis.Finally,for maskable orthogonal quantum states,we analyze two special examples and give their images for an intuitive description.

Keywords:quantum information masking,limited conditions,two-qubit

1.Introduction

Quantum mechanics is one of the most important inventions in the field of physics in the 20th century[1,2].As a crucial research object of quantum mechanics,quantum information has very great significance.In quantum information theory,the evolution in a closed quantum system is unitary and linear[3].Therefore,unlike the classical world,there are some nogo theorems in the quantum world,such as the no-deleting theorem[4–6],the no-cloning theorem[7–9],the no-broadcasting theorem[10–12]and so on.As one of the no-go theorems,the no-masking theorem[13]plays a vital role in the development of quantum information.Quantum information masking requires that the information contained in subsystems can be encoded into their correlation by some unitary operations,such that the marginal states are identical[14].In other words,one can hide the initial information contained in the subsystems[15].This principle greatly improves the security of quantum communication[16,17].As a consequence,the masking of quantum information can be used for quantum secret sharing[18–20].Besides,it also plays a major role in quantum teleportation[21,22]and key distribution[23].Therefore,it is necessary to study the quantum states whether can be masked.At present,the research method for this problem is mainly based on the definition of quantum information masking.

In recent works,many scholars have come up with a lot of new discoveries about quantum masking.Modi et al first proposed the no-masking theorem,namely,a unitary operator cannot mask any pure state[13].Based on this theorem,Ghosh et al considered two special kinds of quantum states to analyze the masking condition[24].Li and Wang showed that tripartite quantum systems can mask arbitrary quantum pure states[25].In addition,the relationship between the maskable state set and the hyperdisk was studied by Ding et al[26].Li and Modi certified that probabilistic universal masking is impossible and derived a necessary condition to make the∈-approximate universal masking possible[27].Lie et al demonstrated that the maximum number of qubits that can be masked is related to the entropy of the quantum state[28].Moreover,using time-correlated photons,Liu et al designed a quantum information masking machine[29].These above papers are important for the development of quantum information masking.

Here,we mainly study the masking of quantum information in two-dimensional Hilbert space.We express the masking conditions by the coefficients of quantum states.As a result,we obtain a system of equations as the masking conditions.Afterward,by observing some concrete maskable non-orthogonal quantum states,we find a common property of these states,i.e.they have the same number of terms and the same basis.Eventually,we analyze two special kinds of quantum states with an orthogonal basis and draw some images for an intuitive description of these examples.

2.No-masking theorem conditions

In summary,the masking conditions can be represented by the coefficients of |Ψ〉,|Ψ0〉 and |Ψ1〉,i.e.when their coefficients meet the requirements of equations(5),(8)and(9)at the same time,a single qubit state |b〉=α0|0〉+α1|1〉can be masked.

Modi et al have shown that masking the quantum information of the two systems is generally impossible,but they have also stated that there are a large number of special quantum states which can be masked[13].These quantum states which can be masked and the corresponding masker have potential applications in quantum secret sharing and quantum communication protocols.The conditions equations(5),(8)and(9)obtained by us can be used to judge given quantum states in C2whether can be masked.It is also useful for subsequent applications in quantum secret sharing to find masking quantum states more quickly by the masking conditions.

3.Masking of two-qubit quantum states

Here,we consider some non-orthogonal quantum states and orthogonal quantum states respectively,then we get some corresponding conclusions as follows.

3.1.Masking of non-orthogonal quantum states

By observing some maskable non-orthogonal quantum states in C2,we obtain the result that if two non-orthogonal quantum states can mask quantum information contained in a single qubit state,they have the same number of terms and the same basis.Below,we present the analysis process of this result.

Table 1.|Ψ0〉 consists of one term(The duplicate parts have been deleted).

Table 2.|Ψ0〉 consists of two terms(The duplicate parts have been deleted).

Table 3.|Ψ0〉 consists of three terms(The duplicate parts have been deleted).

As a result,we acquire that a2=0 or a3=0,which contradicts the condition a2,a3≠0.

In summary,the conclusion is correct,i.e.a single qubit state can be masked by two non-orthogonal quantum states,which have the same number of terms and the same basis.

Based on the general masking conditions of the quantum state obtained in the previous section,we further discuss the masking of non-orthogonal quantum states and derive a conclusion,which provides a more efficient method to judge whether the quantum states can be masked.In other words,for non-orthogonal quantum states in C2,they cannot be used to mask the quantum information in a single qubit state as long as do not have the same number of terms or the same basis.Therefore,the conclusion is useful for the study of quantum information masking.

3.2.Masking of orthogonal quantum states

Particularly,when x0=x2and y0=y2,equation(14)is further changed tocan be masked by∣Ψ0〉=

4.Conclusion

In two-dimensional Hilbert space,we expressed the quantum information masking conditions by a system of equations for the coefficients of quantum states.Moreover,by observing the characteristic of the maskable non-orthogonal two-qubit quantum states,we obtained the conclusion that if two nonorthogonal quantum states can mask a single qubit state,they have the same number of terms and the same basis.Furthermore,we considered two examples of orthogonal quantum states and calculated the masking conditions.Finally,we gave the corresponding images for an intuitive description.Our results would provide a foundation idea for further study on the masking of higher dimensional quantum states.

Acknowledgments

This work was supported by the Natural Science Foundation of Hebei Province(Grant No.A2019210057).

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