A scheme that probabilistically realizes hierarchical quantum state sharing of an arbitrary unknown qubit state with a four-qubit non-maximally entangled |χ state is presented in this paper. In the scheme, the sender...A scheme that probabilistically realizes hierarchical quantum state sharing of an arbitrary unknown qubit state with a four-qubit non-maximally entangled |χ state is presented in this paper. In the scheme, the sender Alice distributes a quantum secret with a Bell-state measurement and publishes her measurement outcomes via a classical channel to three agents who are divided into two grades. One agent is in the upper grade, while the other two agents are in the lower grade. Then by introducing an ancillary qubit, the agent of the upper grade only needs the assistance of any one of the other two agents for probabilistically obtaining the secret, while an agent of the lower grade needs the help of both the other two agents by using a controlled-NOT operation and a proper positive operator-valued measurement instead of the usual projective measurement. In other words, the agents of two different grades have different authorities to reconstruct Alice's secret in a probabilistic manner. The scheme can also be modified to implement the threshold-controlled teleportation.展开更多
The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space ef...The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.展开更多
In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm ineq...In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.展开更多
In this paper,we consider the splitting operators into the product of two positive operators.For given operators A and T∈B(H)with A≥0 and R(A)=R(T),if T can be splitted as T=AX for some positive operator X,then the ...In this paper,we consider the splitting operators into the product of two positive operators.For given operators A and T∈B(H)with A≥0 and R(A)=R(T),if T can be splitted as T=AX for some positive operator X,then the existence of X and the properties of T are studied.The related research are the solutions of operator equations T=XAX and TX=XAX,which have some particular properties and broad applications in many fields.The conditions for the existence of solutions or idempotent solutions of these kinds of equations are studied and new representations of the general solutions are given.展开更多
Quantum teleportation provides a ＂bodiless＂ way of transmitting the quantum state from one object to another, at a distant location, using a classical communication channel and a previously shared entangled state. I...Quantum teleportation provides a ＂bodiless＂ way of transmitting the quantum state from one object to another, at a distant location, using a classical communication channel and a previously shared entangled state. In this paper, we present a tripartite scheme for probabilistic teleportation of an arbitrary single qubit state, without losing the information of the state being teleported, via a fourqubit cluster state of the form ｜Ф） 1234 = α｜10000） ＋β｜1010） ＋ γ｜0101） - η｜1111）, as the quantum channel, where the nonzero real numbers α, β, γ, and η satisfy the relation ｜α｜^2 ＋ ｜β｜^2 ＋ ｜γ｜^2 ＋ ｜η｜^2 = 1. With the introduction of an auxiliary qubit with state ｜0}, using a suitable unitary transformation and a positive-operator valued measure （POVM）, the receiver can recreate the state of the original qubit. An important advantage of the teleportation scheme demonstrated here is that, if the teleportation fails, it can be repeated without teleporting copies of the unknown quantum state, if the concerned parties share another pair of entangled qubit. We also present a protocol for quantum information splitting of an arbitrary two-particle system via the aforementioned cluster state and a Bell-state as the quantum channel. Problems related to security attacks were examined for both the cases and it was found that this protocol is secure. This protocol is highly efficient and easy to implement.展开更多
A nonlinear version of Krein Rutman Theorem is established.This paper presents aunified proof of the Krein Rutman Theorem for linear operators and for nonlinear operators,and ofthe Perron-Frobenius theorem for nonnega...A nonlinear version of Krein Rutman Theorem is established.This paper presents aunified proof of the Krein Rutman Theorem for linear operators and for nonlinear operators,and ofthe Perron-Frobenius theorem for nonnegative matrices and for nonnegative tensors.展开更多
If a 3-tuple (A:H_1→H_1,B:H_2→H_1,C:H_2→H_2) of operators on Hilbert spaces is given,we proved that the operator A:= on H=H_1⊕H_2 is≥0 if and only if A≥0,R(B) R(A1/2)and C≥B~* A^+ B, where A^+ is the generalize...If a 3-tuple (A:H_1→H_1,B:H_2→H_1,C:H_2→H_2) of operators on Hilbert spaces is given,we proved that the operator A:= on H=H_1⊕H_2 is≥0 if and only if A≥0,R(B) R(A1/2)and C≥B~* A^+ B, where A^+ is the generalized inverse of A. In general,A^+ is a closed operator,but since R(B) R(A1/2),B~* A^+ B is bounded yet.展开更多
The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if...The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if are compact operators, then We give inequality which is equivalent to and more general than the above inequalities, which states that if are compact operators,展开更多
We give a strategy for nonlocal unambiguous discrimination (UD) among N linearly independent nonorthogonal qudit states lying in a higher-dimensional Hilbert space. The procedure we use is a nonlocal positive operator...We give a strategy for nonlocal unambiguous discrimination (UD) among N linearly independent nonorthogonal qudit states lying in a higher-dimensional Hilbert space. The procedure we use is a nonlocal positive operator valued measurement (POVM) in a direct sum space. This scheme is designed for obtaining the conclusive nonlocal measurement results with a finite probability of success. We construct a quantum network for realizing the nonlocal UD with a set of two-level remote rotations, and thus provide a feasible physical means to realize the nonlocal UD.展开更多
We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to sev...We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No. 11071178） and the Research Foundation of the Education Department of Sichuan Province, China （Grant No. 12ZB106）.
文摘A scheme that probabilistically realizes hierarchical quantum state sharing of an arbitrary unknown qubit state with a four-qubit non-maximally entangled |χ state is presented in this paper. In the scheme, the sender Alice distributes a quantum secret with a Bell-state measurement and publishes her measurement outcomes via a classical channel to three agents who are divided into two grades. One agent is in the upper grade, while the other two agents are in the lower grade. Then by introducing an ancillary qubit, the agent of the upper grade only needs the assistance of any one of the other two agents for probabilistically obtaining the secret, while an agent of the lower grade needs the help of both the other two agents by using a controlled-NOT operation and a proper positive operator-valued measurement instead of the usual projective measurement. In other words, the agents of two different grades have different authorities to reconstruct Alice's secret in a probabilistic manner. The scheme can also be modified to implement the threshold-controlled teleportation.
基金supported by the National Natural Science Foundation of China(Grant No.10571113).
文摘The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.
文摘In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.
文摘In this paper,we consider the splitting operators into the product of two positive operators.For given operators A and T∈B(H)with A≥0 and R(A)=R(T),if T can be splitted as T=AX for some positive operator X,then the existence of X and the properties of T are studied.The related research are the solutions of operator equations T=XAX and TX=XAX,which have some particular properties and broad applications in many fields.The conditions for the existence of solutions or idempotent solutions of these kinds of equations are studied and new representations of the general solutions are given.
文摘Quantum teleportation provides a ＂bodiless＂ way of transmitting the quantum state from one object to another, at a distant location, using a classical communication channel and a previously shared entangled state. In this paper, we present a tripartite scheme for probabilistic teleportation of an arbitrary single qubit state, without losing the information of the state being teleported, via a fourqubit cluster state of the form ｜Ф） 1234 = α｜10000） ＋β｜1010） ＋ γ｜0101） - η｜1111）, as the quantum channel, where the nonzero real numbers α, β, γ, and η satisfy the relation ｜α｜^2 ＋ ｜β｜^2 ＋ ｜γ｜^2 ＋ ｜η｜^2 = 1. With the introduction of an auxiliary qubit with state ｜0}, using a suitable unitary transformation and a positive-operator valued measure （POVM）, the receiver can recreate the state of the original qubit. An important advantage of the teleportation scheme demonstrated here is that, if the teleportation fails, it can be repeated without teleporting copies of the unknown quantum state, if the concerned parties share another pair of entangled qubit. We also present a protocol for quantum information splitting of an arbitrary two-particle system via the aforementioned cluster state and a Bell-state as the quantum channel. Problems related to security attacks were examined for both the cases and it was found that this protocol is secure. This protocol is highly efficient and easy to implement.
文摘A nonlinear version of Krein Rutman Theorem is established.This paper presents aunified proof of the Krein Rutman Theorem for linear operators and for nonlinear operators,and ofthe Perron-Frobenius theorem for nonnegative matrices and for nonnegative tensors.
文摘If a 3-tuple (A:H_1→H_1,B:H_2→H_1,C:H_2→H_2) of operators on Hilbert spaces is given,we proved that the operator A:= on H=H_1⊕H_2 is≥0 if and only if A≥0,R(B) R(A1/2)and C≥B~* A^+ B, where A^+ is the generalized inverse of A. In general,A^+ is a closed operator,but since R(B) R(A1/2),B~* A^+ B is bounded yet.
文摘The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if are compact operators, then We give inequality which is equivalent to and more general than the above inequalities, which states that if are compact operators,
基金supported by the Natural Science Foundation of Guangdong Province, China (Grant No. 06029431)
文摘We give a strategy for nonlocal unambiguous discrimination (UD) among N linearly independent nonorthogonal qudit states lying in a higher-dimensional Hilbert space. The procedure we use is a nonlocal positive operator valued measurement (POVM) in a direct sum space. This scheme is designed for obtaining the conclusive nonlocal measurement results with a finite probability of success. We construct a quantum network for realizing the nonlocal UD with a set of two-level remote rotations, and thus provide a feasible physical means to realize the nonlocal UD.
基金supported by National Natural Science Foundation of China(Grant No.11301022)the State Key Laboratory of Rail Traffic Control and Safety,Beijing Jiaotong University(Grant Nos.RCS2014ZT20 and RCS2014ZZ001)+1 种基金Beijing Natural Science Foundation(Grant No.9144031)the Hong Kong Research Grant Council(Grant Nos.Poly U 501909,502510,502111 and 501212)
文摘We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.
