量子关联及其动力学的研究
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摘要
量子信息因为具有经典信息所不具备的优势而受到了广泛关注。量子关联作为量子信息中的重要资源,在量子计算、量子相变和量子态的广播等方面有着重要的应用。然而,任何真实的量子系统不可避免地会与环境发生相互作用,进而导致退相干。因此量子关联的动力学是一个重要的研究课题。
本文研究量子关联及其动力学,主要内容如下：
第一,研究了初始关联存在时开放系统的动力学为完全正定映射的条件。当初始关联存在时,要得到完全正定映射,必须限定初始态集合；否则,开放系统的约化动力学不是一个完全正定映射。我们分别在态空间的直和分解框架和分配映射框架下给出这样的态集合的结构。在态空间的直和分解框架下,我们给出了这种初始态集合的一般结构,证明了对于任意的时间演化算符USE(t)这些态的约化动力学总是完全正定映射。我们的结果表明,不仅量子失协等于零的态集合、
一些量子失协不等于零的可分离态集合的约化动力学是完全正定映射,而且一些量子纠缠态的集合的约化动力学也是完全正定映射。以前的文献中给出的直积态集合和量子失协等于零的态集合是我们给出的初始态集合的特例。在分配映射框架下,我们证明了对于2×N维量子系统来说,本文所给出的初始态集合是使开放系统的约化动力学为完全正定映射的最大态集合,即该态集合所限定的关联是完全正定映射的充要条件。同时给出了M×N维量子系统对应的态集合的完全正定的分配映射。这说明态空间的直和分解框架和分配映射框架给出的结果是一致的。
第二,研究了玻色热库对两体量子关联的动力学的影响。分别考虑了贝尔对角态受两个独立的玻色热库和一个共同的玻色热库影响时,它的量子失协的几何度量的动力学。我们发现受两个独立的玻色热库影响时,量子失协的几何度量的动力学在次欧姆谱密度(0 第三,研究了常见信道对多体量子关联的动力学的影响。考虑了一类n体量子态分别受比特翻转信道、比特相位翻转信道、相位翻转信道和退极化信道影响时,它的q-全局量子失协和几何全局量子失协的动力学。结果表明,该类态受比特翻转信道影响后,q-全局量子失协和几何全局量子失协关于p=0.5对称,并在此处取得最小值零,同时二者发生突然转变现象的条件相同。不同的是,q-全局量子失协只有在参数q=1并且n是偶数的情况时,才有可能发生冻结现象,而几何全局量子失协发生冻结现象的情形则不受n的影响。当时间参数p保持不变时,q-全局量子失协是参数q的先增后减函数。该类态受比特相位翻转信道和相位翻转信道影响后的q-全局量子失协和几何全局量子失谐的动力学变化与受比特翻转信道影响后的动力学变化情况类似。该类态受退极化信道影响后,q-全局量子失协和几何全局量子失协是时间参数p的先减后增函数,并在p=0.75时取得最小值零。此时量子关联不存在突然转变或者冻结现象。
Quantum information has attracted much attention since it has many advan-tages over classical information. Quantum correlation as an important resource in quantum information has important applications in quantum computation, quan-tum phase transition, and broadcasting of quantum states. However, any real quan-tum system inevitably interacts to some extent with its environment, then quantum decoherence can occur if the process is irreversible. Therefore, investigation of the dynamics of quantum correlation is important and necessary.
In this thesis, we investigate quantum correlation and its dynamics. The main results are as follows:
First, we investigate completely positive maps for an open system interacting with its environment in the presence of initial correlations. In this situation, we have to restrict the set of initial states. Therefore, we consider the set of initial states that share one common completely positive map within the framework of direct sum decomposition of state space and the framework of assignment maps. In the framework of direct sum decomposition of state space, a general expression of the initial states are explicitly given, and we prove that the reduced dynamics of the open system can be described by a completely positive map for arbitrary unitary operator USE(t) as long as the initial states of the combined system are with the structure which we have given. The set of initial states includes not only separable states with vanishing or nonvanishing quantum discord but also entangled states. It significantly extends the previous results which can be taken as special cases of our results. In the framework of assignment maps, we prove that the set of initial states given in the framework of direct sum decomposition of state space is a necessary and sufficient condition for2×N quantum systems of which the reduced dynamics is a completely positive map for arbitrary unitary operator USE(t). We also give the expression of assignment maps which are completely positive for the initial states of M x N quantum systems. These results mean that the framework of direct sum decomposition of state space and the framework of assignment maps are consistent.
Second, we investigate the dynamics of quantum correlations for bipartite quan-tum systems. We investigate the dynamics of geometric measure of quantum discord of two qubits in two independent reservoirs and one common reservoir, respectively, where the initial states of the two qubits are Bell-diagonal states. When the two qubits are in two independent reservoirs, the dynamics of geometric measure of quantum discord for the reservoirs with sub-Ohmic (0 Third, we investigate the dynamics of quantum correlations for multipartite quantum systems. We consider the dynamics of q-global quantum discord and ge-ometric global quantum discord of a class of n-qubt states which undergo a local bit flip channel, a bit flip and phase flip channel, a phase flip channel, and a de-polarizing channel, respectively. Our results show that the dynamics of q-global quantum discord and geometric global quantum discord are symmetric with respec-t to p=0.5, and they also obtain the minimum value zero at p=0.5. The sudden transition phenomenon of q-global quantum discord and geometric global quantum discord occurs at the same conditions, q-global quantum discord may have a frozen phenomenon if q=1and n is even. However, geometric measure of quantum dis-cord may have a frozen phenomenon for all n. When the time parameter p remains the same value, q-global quantum discord at first is a monotonic increasing function of parameter q and then becomes a monotonic decreasing function of parameter q. The dynamics of q-global quantum discord and geometric global quantum discord for the states which undergo a local bit flip channel or a local bit flip and phase flip channel or a local phase flip channel are similar to each other. In the depolarizing channel, the dynamics of q-global quantum discord and geometric global quantum discord obtain the minimum value zero at p=0.75, and they are a monotonic de-creasing function of parameter p at first and then become a monotonic increasing function, but in this channel there is no sudden transition phenomenon or frozen phenomenon.
本文研究量子关联及其动力学,主要内容如下：
第一,研究了初始关联存在时开放系统的动力学为完全正定映射的条件。当初始关联存在时,要得到完全正定映射,必须限定初始态集合；否则,开放系统的约化动力学不是一个完全正定映射。我们分别在态空间的直和分解框架和分配映射框架下给出这样的态集合的结构。在态空间的直和分解框架下,我们给出了这种初始态集合的一般结构,证明了对于任意的时间演化算符USE(t)这些态的约化动力学总是完全正定映射。我们的结果表明,不仅量子失协等于零的态集合、
一些量子失协不等于零的可分离态集合的约化动力学是完全正定映射,而且一些量子纠缠态的集合的约化动力学也是完全正定映射。以前的文献中给出的直积态集合和量子失协等于零的态集合是我们给出的初始态集合的特例。在分配映射框架下,我们证明了对于2×N维量子系统来说,本文所给出的初始态集合是使开放系统的约化动力学为完全正定映射的最大态集合,即该态集合所限定的关联是完全正定映射的充要条件。同时给出了M×N维量子系统对应的态集合的完全正定的分配映射。这说明态空间的直和分解框架和分配映射框架给出的结果是一致的。
第二,研究了玻色热库对两体量子关联的动力学的影响。分别考虑了贝尔对角态受两个独立的玻色热库和一个共同的玻色热库影响时,它的量子失协的几何度量的动力学。我们发现受两个独立的玻色热库影响时,量子失协的几何度量的动力学在次欧姆谱密度(0
Quantum information has attracted much attention since it has many advan-tages over classical information. Quantum correlation as an important resource in quantum information has important applications in quantum computation, quan-tum phase transition, and broadcasting of quantum states. However, any real quan-tum system inevitably interacts to some extent with its environment, then quantum decoherence can occur if the process is irreversible. Therefore, investigation of the dynamics of quantum correlation is important and necessary.
In this thesis, we investigate quantum correlation and its dynamics. The main results are as follows:
First, we investigate completely positive maps for an open system interacting with its environment in the presence of initial correlations. In this situation, we have to restrict the set of initial states. Therefore, we consider the set of initial states that share one common completely positive map within the framework of direct sum decomposition of state space and the framework of assignment maps. In the framework of direct sum decomposition of state space, a general expression of the initial states are explicitly given, and we prove that the reduced dynamics of the open system can be described by a completely positive map for arbitrary unitary operator USE(t) as long as the initial states of the combined system are with the structure which we have given. The set of initial states includes not only separable states with vanishing or nonvanishing quantum discord but also entangled states. It significantly extends the previous results which can be taken as special cases of our results. In the framework of assignment maps, we prove that the set of initial states given in the framework of direct sum decomposition of state space is a necessary and sufficient condition for2×N quantum systems of which the reduced dynamics is a completely positive map for arbitrary unitary operator USE(t). We also give the expression of assignment maps which are completely positive for the initial states of M x N quantum systems. These results mean that the framework of direct sum decomposition of state space and the framework of assignment maps are consistent.
Second, we investigate the dynamics of quantum correlations for bipartite quan-tum systems. We investigate the dynamics of geometric measure of quantum discord of two qubits in two independent reservoirs and one common reservoir, respectively, where the initial states of the two qubits are Bell-diagonal states. When the two qubits are in two independent reservoirs, the dynamics of geometric measure of quantum discord for the reservoirs with sub-Ohmic (0
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