理解Lindblad方程局限性的半定规划方法
Semidefinite programming for understanding the limitations of Lindblad equations
AI总结：
通过半定规划方法，研究Lindblad主方程在弱耦合下同时准确描述布居和相干性的可行性，发现大多数参数下无法实现，但某些情况下可牺牲相干性满足局域守恒律。
摘要：
Lindblad量子主方程（LEs）是描述弱耦合到热浴的量子系统最流行的工具。但近期工作表明，在许多情况下这种马尔可夫描述存在根本性局限：即使到系统-浴耦合的领头阶，它们也无法同时捕捉布居和相干性。这可能导致违反热化等基本性质以及局域守恒律相关的连续性方程，即使在实际设置中这些性质是预期的。这引出一个问题：给定一个物理情境，我们如何知道是否存在一个LE能以所需精度描述它？这里我们证明，对于平衡态和非平衡稳态（NESS），该问题可以简洁地表述为一个半定规划（SDP），一种凸优化技术。如果SDP的解能以所需精度找到，那么对于所选设置，LE描述是可能的。如果不能，则根本上无法获得任何LE描述，表明在该特定设置下，即使在弱系统-浴耦合下，一致的马尔可夫处理也是不可能的。考虑耦合到多个热浴的少量子比特各向同性XXZ型模型，我们发现大多数参数区间内，无法获得能准确给出领头阶布居和相干性的LE描述，导致严格的不可行结果。然而，在某些情况下，在部分参数区间内，可以获得具有正确布居但不准确相干性且满足局域守恒律的LE描述。我们的工作凸显了半定规划在分析物理一致的LEs中的威力，从而有助于理解弱系统-浴耦合下马尔可夫描述的极限。
英文摘要：
Lindbladian quantum master equations (LEs) are the most popular descriptions for quantum systems weakly coupled to baths. But, recent works have established that in many situations such Markovian descriptions are fundamentally limited: they cannot simultaneously capture populations and coherences even to the leading-order in system-bath couplings. This can cause violation of fundamental properties like thermalization and continuity equations associated with local conservation laws, even when such properties are expected in the actual setting. This begs the question: given a physical situation, how do we know if there exists an LE that describes it to a desired accuracy? Here we show that, for both equilibrium and non-equilibrium steady states (NESS), this question can be succinctly formulated as a semidefinite program (SDP), a convex optimization technique. If a solution to the SDP can be found to a desired accuracy, then an LE description is possible for the chosen setting. If not, no LE description is fundamentally attainable, showing that a consistent Markovian treatment is impossible even at weak system-bath coupling for that particular setting. Considering few qubit isotropic XXZ-type models coupled to multiple baths, we find that in most parameter regimes, LE description giving accurate populations and coherences to leading-order is unattainable, leading to rigorous no-go results. However, in some cases, LE description having correct populations but inaccurate coherences, and satisfying local conservation laws, is possible over some of the parameter regimes. Our work highlights the power of semidefinite programming in the analysis of physically consistent LEs, thereby, in understanding the limits of Markovian descriptions at weak system-bath couplings.
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