arXiv subjects
Publications and source records attributed to Jinchuan Hou.
Linear maps preserving pure states of a quantum system of any dimension are characterized. This is then used to establish a structure theorem for linear maps that preserve separable pure states in multipartite systems. As an application, a characterization of separable pure state preserving affine maps is obtained.
We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New classes of positive maps are constructed. Some open questions are answered.
Let $Φ_{t,π}: M_3({\mathbb C}) \rightarrow M_3({\mathbb C})$ be a linear map defined by $Φ_{t,π}(A)=(3-t)\sum_{i=1}^3E_{ii}AE_{ii}+t\sum_{i=1}^3E_{i,π(i)}AE_{i,π(i)}^†-A$, where $0\leq t\leq 3$ and $π$ is a permutation of $(1,2,3)$. We show that the Hermitian matrix $W_{Φ_{t,π}}$ induced by $Φ_{t,π}$ is an optimal entanglement witness if and only if $t=1$ and $π$ is cyclic.
Maps $Φ$ which do not increase the spectrum on complex matrices in a sense that $\Sp(Φ(A)-Φ(B))\subseteq\Sp(A-B)$ are classified.
We show that a local channel cannot create quantum discord (QD) for zero QD states of size $d\geq3$ if and only if either it is a completely decohering channel or it is a nontrivial isotropic channel. For the qubit case this propertiy is additionally characteristic to the completely decohering channel or the commutativity-preserving unital channel. In particular, the exact forms of the completely decohering channel and the commutativity-preserving unital qubit channel are proposed. Consequently, our results confirm and improve the conjecture proposed by X. Hu et al. for the case of $d\geq3$ and improve the result proposed by A. Streltsov et al. for the qubit case. Furthermore, it is shown that a local channel nullifies QD in any state if and only if it is a completely decohering channel. Based on our results, some protocols of quantum information processing issues associated with QD, especially for the qubit case, would be experimentally accessible.
In this paper, in terms of the relation between the state and the reduced states of it, we obtain two inequalities which are valid for all separable states in infinite-dimensional bipartite quantum systems. One of them provides an entanglement criterion which is strictly stronger than the computable cross-norm or realignment (CCNR) criterion.
Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$ and $ξ\in{\mathbb C}$ a scalar. It is shown that an additive map $L$ on $\mathcal M$ satisfies $L(AB-ξBA)=L(A)B-ξBL(A)+L(B)A-ξAL(B)$ whenever $A,B\in{\mathcal M}$ with $AB=0$ if and only if one of the following statements holds: (1) $ξ=1$, $L=φ+f$, where $φ$ is an additive derivation on $\mathcal M$ and $f$ is an additive map from $\mathcal M$ into its center vanishing on $[A,B]$ with $AB=0$; (2) $ξ=0$, $L(I)\in{\mathcal Z}({\mathcal M})$ and there exists an additive derivation $φ$ such that $L(A)=φ(A)+L(I)A$ for all $A$; (3) $ξ=-1$, $L$ is a Jordan derivation; (4) $ξ$ is rational and $ξ\not=0, \pm1$, $L$ is an additive derivation; (5) $ξ$ is not rational, there exists an additive derivation $φ$ satisfying $φ(ξI)=ξL(I)$ such that $L(A)=φ(A) + L(I)A$ for all $A \in{\mathcal M}$. A linear map $L$ on $\mathcal M$ satisfies $L(AB-ξBA)=L(A)B-ξBL(A)+L(B)A-ξAL(B)$ whenever $A,B\in{\mathcal M}$ with $AB=0$ if and only if there exists a $T\in\mathcal M$ and a linear map $f:{\mathcal M}\rightarrow{\mathcal Z}({\mathcal M})$ vanishing on $[A,B]$ with $AB=0$ such that (i) $ξ=1$, $L(A)=AT-TA+f(A)$ for all $A\in\mathcal M$; (ii) $ξ=0$, $L(I)\in{\mathcal Z}({\mathcal M})$ and $L(A)=AT-(T-L(I))A$ for all $A\in{\mathcal M}$; (iii) $ξ\not=0,1$, $L(A)=AT-TA$ for all $A \in{\mathcal M}$.
Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $Φ:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective map. It is shown that $Φ$ is strong skew commutativity preserving (that is, satisfies $Φ(A)Φ(B)-Φ(B)Φ(A)^*=AB-BA^*$ for all $A,B\in{\mathcal M}$) if and only if there exists some self-adjoint element $Z$ in the center of ${\mathcal M}$ with $Z^2=I$ such that $Φ(A)=ZA$ for all $A\in{\mathcal M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.
Quantum correlation is a key to our understanding of quantum physics. In particular, it is essential for the powerful applications to quantum information and quantum computation. There exist quantum correlations beyond entanglement, such as quantum discord (QD) and measurement-induced nonlocality (MiN) [Phys. Rev. Lett. \textbf{106}, 120401(2011)]. In [Phys. Rev. A \textbf{77}, 022113(2008)], a subclass of PPT states so-called strong positive partial transposition (SPPT) states was introduced and it was conjectured there that SPPT states are separable. However, it was illustrated with examples in [Phys. Rev. A \textbf{81}, 064101(2010)] that this conjecture is not true. Viewing the original SPPT as SPPT up to part B, in the present paper, we define SPPT state up to part A and B respectively and present a separable class of SPPT states, that is the super SPPT (SSPPT) states, in terms of local commutativity. In addition, classical-quantum (CQ) states and nullity of MiN are characterized via local commutativity. Based on CQ states, the geometric measure of quantum discord (GMQD) for infinite-dimensional case is proposed. Consequently, we highlight the relation among MiN, QD(GMQD), SSPPT and separability through a unified approach for both finite- and infinite-dimensional systems: zero MiN implies zero QD(GMQD), zero QD(GMQD) signals SSPPT and SSPPT guarantees separability, but the converses are not valid.
Concurrence is an important entanglement measure for states in finite-dimensional quantum systems that was explored intensively in the last decade. In this paper, we extend the concept of concurrence to infinite-dimensional bipartite systems and show that it is continuous and does not increase under local operation and classical communication (LOCC). Moreover, based on the partial Hermitian conjugate (PHC) criterion proposed in [Chin. Phys. Lett. \textbf{26}, 060305(2009); Chin. Sci. Bull. \textbf{56}(9), 840--846(2011)], we introduce a concept of the PHC measure and show that it coincides with the concurrence, which provides another perspective on the concurrence.
In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain new indecomposable optimal witnesses which have no spanning property. These also provide new examples which support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).
In this paper we discuss the fidelity of states in infinite dimensional systems, give an elementary proof of the infinite dimensional version of Uhlmann's theorem, and then, apply it to generalize several properties of the fidelity from finite dimensional case to infinite dimensional case. Some of them are somewhat different from those for finite dimensional case.
For any bipartite systems, a universal entanglement witness of rank-4 for pure states is obtained and a class of finite rank entanglement witnesses is constructed. In addition, a method of detecting entanglement of a state only by entries of its density matrix with respect to some product basis is obtained.
In this paper, the realignment criterion and the RCCN criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let $H_A$ and $H_B$ be complex Hilbert spaces with $\dim H_A\otimes H_B=+\infty$. Let $ρ$ be a state on $H_A\otimes H_B$ and $\{δ_k\}$ be the Schmidt coefficients of $ρ$ as a vector in the Hilbert space ${\mathcal C}_2(H_A)\otimes{\mathcal C}_2(H_B)$. We introduce the realignment operation $ρ^R$ and the computable cross norm $\|ρ\|_{\rm CCN}$ of $ρ$ and show that, if $ρ$ is separable, then $\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k\leq1.$ In particular, if $ρ$ is a pure state, then $ρ$ is separable if and only if $\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k=1$.
In this paper, a class of indecomposable positive finite rank elementary operators of order $(n,n)$ are constructed. This allows us to give a simple necessary and sufficient criterion for separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order $(2,2)$ and get some new mixed entangled states that can not be detected by the positive partial transpose (PPT) criterion and the realignment criterion.
Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state $ρ$ on $H\otimes K$ is separable if and only if $(Φ\otimes I)ρ\geq 0$ for all positive finite rank elementary operators $Φ$. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.
It is shown that, every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $αI+T$ with $α$ a nonnegative number and $T$ a finite rank self-adjoint operator. We also provide two methods of constructing entanglement witness and apply them to obtain some entangled states that cannot be detected by the PPT criterion and the realignment criterion.
The question under what conditions different witnesses may detect some entangled states simultaneously is answered for both finite- and infinite-dimensional bipartite systems. Finite many different witnesses can detect some entangled states simultaneously if and only if their convex combinations are still witnesses; they can not detect any entangled state simultaneously if and only if the set of their convex combinations contains a positive operator. For two witnesses $W_1$ and $W_2$, some more can be said: (1) $W_1$ and $W_2$ can detect the same set of entangled states if and only if they are linearly dependent; (2) $W_2$ can detect more entangled states than that $W_1$ can if and only if $W_1$ is a linear combination of $W_2$ and a positive operator. As an application, some characterizations of the optimal witnesses are given and some structure properties of the decomposable optimal witnesses are presented.