We study the stability of quantum states of macroscopic systems of finite volume V. By using both the locality and huge degrees of freedom, we show the following: (i) If square fluctuation of every additive operator is O(V) or less for a pure state, then it is not fragile for any weak classical noises or weak perturbations from environments. (ii) If square fluctuation of some additive operator is O(V2) for a pure state, then it is fragile for some of these. (iii) If a state, pure or mixed, has the "cluster property," then it is stable against local measurements, and vice versa. Among many applications, we discuss the mechanism of symmetry-breaking in finite systems.
We consider finite macroscopic systems, i.e. systems of large but finite degrees of freedom, which we believe are poorly understood as compared with small systems and infinite systems. We focus on pure states that do not have the "cluster property". Such a pure state is entangled macroscopically, and is quite anomalous in view of many-body physics because it does not approach any pure state in the infinite-size limit. However, we often encounter such anomalous states when studying finite macroscopic systems, such as quantum computers with many qubits and finite systems that will exhibit symmetry breaking in the infinite-size limit. We study stabilities of such anomalous states in general systems. In contrast to the previous works, we obtain general and universal results, by making full use of the locality of the theory. Using the general results, we discuss roles of anomalous states in quantum computers, and the mechanism of emergence of a classical world from quantum theory.
In quantum systems of a macroscopic size V, such as interacting many particles and quantum computers with many qubits, there exist pure states such that fluctuations of some intensive operator A is anomalously large, = O(V^0), which is much larger than that assumed in thermodynamics, = O(1/V). By making full use of the locality, we show, starting from Hamiltonians of macroscopic degrees of freedom, that such states decohere at anomalously fast rates when they are weakly perturbed from environments.
In quantum many-body systems subjected to generic perturbations, eigenstates are typically fragile and prone to strong mixing, leading to rapid thermalization in dynamics to an equilibrium state described by statistical mechanics. However, there exist robust counterexamples that defy this typical expectation, exhibiting novel non-equilibrium phenomena and offering potential applications in (quantum) information processing. This dissertation explores such systems and establishes their stability against perturbations with mathematical rigor. First, by generalizing the exponentially slow tunneling effect in single-particle quantum mechanics, we develop a theory of metastability in quantum many-body systems, demonstrating their nonperturbatively long lifetimes. As a canonical application, we provide the first general and tight bounds on the slow decay of false vacua. Second, it is known that ground states in finite spatial dimensions can resist arbitrary local perturbations through topologically ordered phases of matter. We extend this notion of stability to quantum error-correcting codes in infinite dimensions, specifically quantum low-density parity-check (LDPC) codes. These codes are of significant interest due to their promise of lower fault-tolerance overhead compared to finite-dimensional codes. Lastly, we analyze classical counterparts of LDPC codes and prove that their eigenstates remain localized in the many-body Hilbert space under generic perturbations. This finding presents the first unambiguous example of a robust violation of the eigenstate thermalization hypothesis. To achieve these results, we introduce advanced techniques to control operator locality, leveraging and generalizing methods such as Lieb-Robinson bounds, cluster expansions, and Schrieffer-Wolff transformations. These tools not only underpin our specific findings but may also prove valuable for broader challenges in mathematical physics of quantum many-body systems.
The stability of quantum systems to perturbations of the Hamiltonian is studied. This stability is quantified by the fidelity. Dependence of fidelity on the initial state as well as on the dynamical properties of the system is considered. In particular, systems having a chaotic or regular classical limit are analysed. The fidelity decay rate is given by an integral of the correlation function of the perturbation and is thus smaller the faster correlation function decays. If the perturbation can be written as a time derivative of another operator, meaning that the time averaged perturbation vanishes, fidelity freezes at a constant value and starts to decay only after a long time inversely proportional to the perturbation strength. In composite systems stability of entanglement to perturbations of the Hamiltonian is analysed in terms of purity. For regular systems purity decay is shown to be independent of Planck's constant for coherent initial states in the semiclassical limit. The accelerated decoherence of macroscopic superpositions is also explained. The theory of fidelity decay is applied to the stability of quantum computation and an improved quantum Fourier transform algorithm is designed.
We study macroscopic entanglement of various pure states of a one-dimensional
N-spin system with
N⪢1. Here, a quantum state is said to be macroscopically entangled if it is a superposition of macroscopically distinct states. To judge whether such superposition is hidden in a general state, we use an essentially unique index
p: A pure state is macroscopically entangled if
p=2, whereas it may be entangled but not macroscopically if
p<2. This index is directly related to fundamental stabilities of many-body states. We calculate the index
p for various states in which magnons are excited with various densities and wave numbers. We find macroscopically entangled states
(p=2) as well as states with
p=1. The former states are unstable in the sense that they are unstable against some local measurements. On the other hand, the latter states are stable in the senses that they are stable against any local measurements and that their decoherence rates never exceed
O(N) in any weak classical noises. For comparison, we also calculate the von Neumann entropy
SN∕2(N) of a subsystem composed of
N∕2 spins as a measure of bipartite entanglement. We find that
SN∕2(N) of some states with
p=1 is of the same order of magnitude as the maximum value
N∕2. On the other hand,
SN∕2(N) of the macroscopically entangled states with
p=2 is as small as
O(logN)⪡N∕2. Therefore larger
SN∕2(N) does not mean more instability. We also point out that these results are partly analogous to those for interacting many bosons. Furthermore, the origin of the huge entanglement, as measured either by
p or
SN∕2(N), is discussed to be due to spatial propagation of magnons.