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arXiv:quant-ph/0203106 · Physical Review Letters

Stability of Quantum States of Finite Macroscopic Systems against Classical Noises, Perturbations from Environments, and Local Measurements

Akira Shimizu, Takayuki Miyadera

Published Dec 16, 2002• 93 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.89.270403

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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 study the stabilities of quantum states of macroscopic systems, against noises, against perturbations from environments, and against local measurements. We show that the stabilities are closely related to the cluster property, which describes the strength of spatial correlations of fluctuations of local observables, and to fluctuations of additive operators. The present theory has many applications, among which we discuss the mechanism of phase transitions in finite systems and quantum computers with a huge number of qubits.
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.
We study a general macroscopic quantum system of a finite size, which will exhibit a symmetry breaking if the system size goes to infinity, when the system interacts with an environment. We evaluate the decoherence rates of the anomalously fluctuating vacuum (AFV), which is the symmetric ground state, and the pure phase vacua (PPVs). By making full use of the locality and huge degrees of freedom, we show that there can exist an interaction with an environment which makes the decoherence rate of the AFV anomalously fast, whereas PPVs are less fragile.
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 one-dimensional systems, spontaneous symmetry breaking (SSB) states are fragile by nature, as the injection of a non-zero energy density above the ground state is expected to restore the symmetry. This instability implies that local perturbations can lead to macroscopic correlation profiles, a breaking of clustering properties and even macroscopic quantum superpositions. In this work, we investigate the effect of interaction on this phenomenology by considering an interacting model with conserved domain wall number, that possesses a ferromagnetic ground state breaking the Z2 symmetry of the Hamiltonian. We first show that a local quench in this system amplifies quantum interferences, producing a macroscopic magnetisation profile that directly reflects the scattering phase of the model. Then, we use two asymmetry measures, namely the Entanglement Asymmetry (EA) and Quantum Fisher Information (QFI), to characterise the quantum coherence associated with the fluctuations of the magnetisation. By focusing on subsystems comparable in size to the light-cone of the perturbation, we confirm the emergence of macroscopic quantum coherence throughout the whole perturbed region. Finally, we discuss the link between EA and QFI and show that the variance/EA inequality for pure state can be generalised to a QFI/EA inequality for mixed states.
We prove the existence of spontaneous symmetry breaking in suitably low-energy eigenstates of certain gapless and frustrated many-body quantum systems, namely symmetric quantum perturbations to classical models which exhibit spontaneous symmetry breaking of a finite group at some positive temperature. Additionally, the classical model need not be local in space, as long as it satisfies a quantum analogue of the Peierls condition. As an example of our technique, we establish robust ferromagnetism in random-bond Ising models in d=2d= 2d=2 dimensions with sufficiently biased random couplings, with weak transverse field. Our mathematical technique is based on establishing quantum bottlenecks, similar to a "many-body WKB" method for evaluating tunneling rates. Using these same methods, we provide new proofs of metastability and the slow decay of the false vacuum, applicable to gapless metastable states. Our work represents a first step towards a rigorous classification of stable gapless quantum phases.