Thermodynamical formulation of entanglement theory and quantum simulations of many-body systems

纠缠既是量子理论基础中的一个重要概念，又是量子信息科学中的一个重要资源。因此，考虑到纠缠的几种表现形式，发展纠缠的定量理论是量子信息理论的中心目标。已经确定，在任意大量相同状态副本的限制下，纠缠变换与热力学具有显著的相似之处。然而，迄今为止，是否存在一种形式上等同于热力学的纠缠操纵设置的问题仍然难以捉摸。在这方面，我的研究将解决以下几点：可逆转换：我将分析几种量子操作，其中纠缠资源的可逆转换，因此与热力学的正式联系可以保持。我会试着（不）证明一些类的可逆性。-热力学类比：根据第一项的发现，我将确定热力学中先进概念（如温度和热量）的纠缠理论的对应物，并探索这些将为理解量子相关性带来的新见解。-含义：最后，我将研究这种连接对纠缠理论和热力学的含义。特别是，我将分析从热力学中获得的新见解如何有助于解决纠缠理论中的开放问题，例如可加性问题和渐近极限下纠缠测度的等价性。我研究的第二部分将涉及使用良好控制的量子系统作为复杂量子多体动力学的量子模拟器。我将分析耦合微腔阵列提供的丰富可能性，这是最近被提出的一种有前途的新型量子模拟器，具有允许单个站点可寻址的显著优势。此外，从更基本的角度来看，我将研究在量子计算机和经典资源上模拟多体哈密顿量的复杂性问题。我计划进行的研究将涉及以下几点：-自旋哈密顿量：我将研究耦合微腔阵列中工程各向异性自旋哈密顿量的实验可行方法，考虑到有前途的系统的特殊性，如耦合到rf腔的库珀和平盒，环形微腔中的原子和光子晶体微腔中的量子点。-拓扑保护量子存储器：基于上一项，我将分析使用这些多体哈密顿量来创建拓扑保护量子存储器的可行性。耦合腔阵列的全局部可寻址性也将被探讨，以实现受保护量子比特的主动纠错、初始化和量子处理。-局部哈密顿量与谱间隙的复杂性：我将解决在一个和多个维度的局部哈密顿量基态下计算局部观测值期望值的计算复杂性。我们将特别分析这种复杂性与系统谱隙的关系。

Entanglement is a key concept both in the foundations of quantum theory and, as a resource, in quantum information science. It is thus a central goal of quantum information theory to developed a quantitative theory of entanglement, considering its several manifestations. It has been established that entanglement transformations, in the limit of an arbitrarily large number of identical copies of the state, shares remarkable similarities with thermodynamics. However, the question whether there is a setting for entanglement manipulation which is formally equivalent to thermodynamics has remained elusive to date. In this respect, my research will address the following points: - Reversible transformations: I will analyse several classes of quantum operations for which a reversible conversion of entangled resources and hence a formally connection with thermodynamics could hold. I will try to (dis)prove reversibility for some of these classes.- Thermodynamics analogies: Following the findings in the first item, I will identify counterparts in entanglement theory of advanced concepts in thermodynamics, such as temperature and heat, and explore the new insights that these would bring to the understanding of quantum correlations. - Implications: Finally, I will investigate the implications of such a connection both to entanglement theory and to thermodynamics. Particularly, I will analyse how the new insights gained from thermodynamics can help in the solution of open problems in entanglement theory, e.g. additivities questions and equivalence of entanglement measures in the asymptotic limit. The second strand of my research will be concerned with the use of well controlled quantum systems as quantum simulators of complex quantum many-body dynamics. I will analyse the rich possibilities offered by arrays of coupled micro-cavities, which have been recently proposed as a promising new type of quantum simulator with the distinguishing advantage of allowing the addressability of single sites. Moreover, from a more fundamental perspective, I will investigate issues concerning the complexity of simulating many-body Hamiltonians both on a quantum computer and by classical resources. The research I plan to conduct will address the following points: - Spin Hamiltonians: I will investigate experimentally feasible ways of engineering anisotropic spin Hamiltonians in arrays of coupled micro-cavities, considering the particulars of promising systems such as Cooper paix boxes coupled to rf cavities, atoms in toroidal microcavities, and quantum dots in photonic crystal micro-cavities. - Topologically protected quantum memories: Based on the previous item, I will analyse the feasibility of using these many-body Hamiltonians for the creation of topologically protected quantum memories. The full local addressability of arrays of coupled cavities will also be explored to the realization of active error correction, initialization and quantum processing of the protected qubit. - Complexity of local Hamiltonians versus spectral gap: I will address the computational complexity of calculating the expectation value of local observables in the ground state of local Hamiltonians in one and more dimensions. It will be analysed in particular the dependence of this complexity with the spectral gap of the system.