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arXiv:quant-ph/0311176 · International Journal of Quantum Information

ANOMALOUS QUANTUM STATES IN FINITE MACROSCOPIC SYSTEMS

Akira Shimizu, Takayuki Miyadera, Akihisa Ukena

Published Dec 1, 2003• 0 citations
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Abstract

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.

DOI
10.1142/s0219749903000425

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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.
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 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 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.
We study macroscopic observables defined as the total value of a physical quantity over a collection of quantum systems. We show that previous results obtained for an infinite ensemble of identically prepared systems lead to incorrect conclusions for finite ensembles. In particular, exact measurement of a macroscopic observable significantly disturbs the state of any finite ensemble. However, we show how this disturbance can be made arbitrarily small when the measurements are of finite accuracy. We demonstrate a general trade-off between state disturbance and measurement coarseness as a function of the size of the ensemble. Using this trade-off, we show that the histories generated by any sequence of finite accuracy macroscopic measurements always generate a consistent family in the absence of large-scale entanglement for sufficiently large ensembles. Hence, macroscopic observables behave ``classically'' provided that their accuracy is coarser than the quantum correlation length scale of the system. The role of these observable is also discussed in the context of NMR quantum information processing and bulk ensemble quantum state tomography.
By considering (non‐relativistic) quantum mechanics as it is done in practice in particular in condensed‐matter physics, it is argued that a deterministic, unitary time evolution within a chosen Hilbert space always has a limited scope, leaving a lot of room for embedding the quantum‐classical transition into our current theories without recurring to difficult‐to‐accept interpretations of quantum mechanics. Nonunitary projections to initial and final states, the breaking of time‐reversal symmetry, a change of Hilbert space, and the introduction of classical concepts such as external potentials or localized atomic nuclei are widespread in quantum mechanical calculations. Furthermore, quantum systems require classical environments that enable the symmetry breaking that is necessary for creating the atomic configurations of molecules and crystals. This paper argues that such classical environments are provided by finite‐temperature macroscopic systems in which the range of quantum correlations and entanglement is limited. This leads to classical behavior on larger scales, and to collapse‐like events in all dynamical processes that become coupled to the thermalized degrees of freedom. image