マクロな量子多体系が熱平衡化するか、という問題が、一般的な形では決定不能命題であることを証明しました。
言ってることがイントロからほとんど理解できない。まず決定不能とは
決定可能集合の概念と同様、決定可能な理論や論理体系の定義は、「実効的方法 (effective method)」や「計算可能関数 (computable function)」によって与えられる。これらは一般にチャーチ＝チューリングのテーゼと等しいと見なされている。実際、論理体系や理論が決定不能であるという証明は、計算可能性の形式定義を使い、ある適当な集合が決定可能集合ではないことを示し、チャーチのテーゼを使って、その理論や論理体系が実効的方法では決定可能でないことを示す。帰納的関数（きのうてきかんすう）は、数理論理学と計算機科学において、直観的に「計算可能」な自然数から自然数への部分関数のクラスである。計算可能性理論では、μ再帰関数はチューリングマシンで計算可能な関数と正確に一致することが示されている。
Thermalization in quantum many-body systems have attracted interest of researchers in var- ious research fields. In these fields, we investigate what determines the presence or absence of thermalization and how to understand thermalizing and non-thermalizing phenomena. This problem is an old longstanding problem that it has already been discussed by Boltz- mann [1] and von Neumann [2]. Recent development of experimental techniques on cold atoms pushes this old problem toward a new step, and thermalization of isolated quantum many-body systems has been minutely examined in laboratory [3–7]. Elaborated experi- ments have revealed some unexpected behaviors including quantum many-body scars [8], which has recently been studied intensively [9–13]. In the theoretical approach, various suggestive notions including typicality of states [14–19] and dynamics [20–23], eigenstate thermalization hypothesis [24–29] and its violation [30–34], weak-version of eigenstate ther- malization hypothesis [29,35], the effective dimension [19,36–39], the quantum many-body localization [40–46], generalized Gibbs ensemble [47–50] and many other important find- ings [51–54] have been raised and some aspects of thermalization have been well clarified (see a review paper [55]). However, the full understanding of thermalization is still far from our present position. Particularly, one of the central problem, determining whether a given system thermalizes or not, has still been left unsolved.
In this paper, we consider this problem by a completely different approach from con- ventional ones. We employ the viewpoint from theoretical computer science and quantum information, and prove that the above problem of thermalization is undecidable. More pre- cisely, we prove that the expectation value of an observable A after relaxation with a given
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Hamiltonian H of a one-dimensional system is undecidable. Our result of undecidability is still valid if we restrict the class of the observable A as a shift sum of a given one-body observable, the Hamiltonian H as a shift-invariant nearest-neighbor interaction, and the initial state as a product state of identical states on a single site (except the first site). This result directly implies that there is no general procedure to decide the presence or absence of thermalization, obviously no general theorem.
Undecidability sometimes appears in physics. Examples are dynamical systems [56], re- peated quantum measurement [57], and spectral gap in quantum many-body systems [58], and our finding tells that quantum thermalization stands in this line. Our proof is inspired by the following two tools: the reduction to the halting problem of Turing machine [59,60] and Feynman-Kitaev type quantum emulation of classical machines [61,62]. With several technical ideas, we successfully construct a quantum many-body systems where the desti- nation after relaxation is determined by the halting problem of Turing machine.
This paper is organized as follows: In Sec. 2, we provide a pedagogical review of the motivation and the problem of quantum thermalization. In Sec. 3, we state two main theorems and a technical lemma. The latter lemma is the most important result in the theoretical aspect. In Sec. 4, we derive two main theorems from the technical lemma. The remainder of this paper is devoted to prove this lemma. In Sec. 5, we briefly sketch the proof strategy. In Sec. 6, we introduce a classical universal reversible Turing machine, which is responsible for the halting problem. In Sec. 7, we construct the Hamiltonian of the quantum system emulating the classical dynamics of the Turing machine. Since the details of dynamics of our quantum system is a little complicated, in Sec. 8 we introduce an analogous setting to our original one but easier to treat, and solve this dynamics. In Sec. 9, we go back to the original setting and construct the initial state with which the expectation value after relaxation is indeed undecidable.
