复杂性约束的量子热力学
Complexity-constrained quantum thermodynamics
AI总结：
本文提出复杂性约束下的量子热力学框架，证明复杂性熵和复杂性相对熵分别量化重置状态的最小功成本及非平凡哈密顿量下的最优功成本，并揭示其在数据压缩与解耦等信息论任务中的应用，将时间复杂性概念整合入热力学资源理论。
摘要：
量子复杂性衡量实现量子过程的难度，例如制备一个状态或实现一个酉操作。我们提出了一种方法，用于量化在过程的复杂性受到限制时实现该过程所需的热力学资源。我们关注信息擦除或 Landauer 擦除这一典型任务，其中 n 量子比特存储器被重置为全零状态。我们证明，通过复杂性约束过程重置任意状态所需的最小热力学功，由该状态的复杂性熵来量化。因此，复杂性熵量化了重置状态时在功成本与复杂性成本之间的权衡。如果量子比特具有非平凡（但为直积）哈密顿量，则最优功成本由复杂性相对熵确定。复杂性熵量化了系统在计算能力有限的观察者看来所具有的随机性程度。类似地，复杂性相对熵量化了此类观察者区分两个状态的能力。我们证明了复杂性（相对）熵的基本性质，并确定了复杂性熵在随机电路下的行为。此外，我们确定了复杂性熵的信息论应用。如果压缩算法必须使用受限数量的门，复杂性熵可量化数据压缩所需的资源。我们进一步引入了复杂性条件熵，它自然地出现在信息论解耦的复杂性约束变体中。假设该熵遵循所猜测的链式法则，我们证明该熵界定了在计算能力有限的裁判判定下，人们能从参考系统中解耦的量子比特数量。总体而言，我们的框架扩展了热力学的资源理论方法，以整合由复杂性量化的时间概念。
英文摘要：
Quantum complexity measures the difficulty of realizing a quantum process, such as preparing a state or implementing a unitary. We present an approach to quantifying the thermodynamic resources required to implement a process if the process's complexity is restricted. We focus on the prototypical task of information erasure, or Landauer erasure, wherein an n-qubit memory is reset to the all-zero state. We show that the minimum thermodynamic work required to reset an arbitrary state, via a complexity-constrained process, is quantified by the state's complexity entropy. The complexity entropy therefore quantifies a trade-off between the work cost and complexity cost of resetting a state. If the qubits have a nontrivial (but product) Hamiltonian, the optimal work cost is determined by the complexity relative entropy. The complexity entropy quantifies the amount of randomness a system appears to have to a computationally limited observer. Similarly, the complexity relative entropy quantifies such an observer's ability to distinguish two states. We prove elementary properties of the complexity (relative) entropy and determine the complexity entropy's behavior under random circuits. Also, we identify information-theoretic applications of the complexity entropy. The complexity entropy quantifies the resources required for data compression if the compression algorithm must use a restricted number of gates. We further introduce a complexity conditional entropy, which arises naturally in a complexity-constrained variant of information-theoretic decoupling. Assuming that this entropy obeys a conjectured chain rule, we show that the entropy bounds the number of qubits that one can decouple from a reference system, as judged by a computationally bounded referee. Overall, our framework extends the resource-theoretic approach to thermodynamics to integrate a notion of time, as quantified by complexity.
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