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arXiv:1008.2510

Measurement As Spontaneous Symmetry Breaking, Non-locality and Non-Boolean Holism

Partha Ghose

Published Aug 15, 2010• 3 citations
arXivHTMLPDF

Abstract

It is shown that having degenerate ground states over the domain of the wavefunction of a system is a sufficient condition for a quantum system to act as a measuring apparatus for the system. Measurements are then instances of spontaneous symmetry breaking to one of these ground states, induced by environmental perturbations. Together with non-Boolean holism this constitutes an optimal formulation of quantum mechanics that does not imply non-locality.

DOI
10.48550/arxiv.1008.2510

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It is shown that having degenerate ground states over the domain of the wavefunction of a system is a sufficient condition for a quantum system to act as a measuring apparatus for the system. Measurements are then instances of spontaneous symmetry breaking to one of these ground states, induced by environmental perturbations. Together with non-Boolean holism this constitutes an optimal formulation of quantum mechanics that does not imply non-locality.
Abstract The extended Bose–Hubbard model for a double-well potential with pair tunneling is studied through both exact diagonalization and mean field theory (MFT). When pair tunneling is strong enough, the ground state wavefunction predicted by the MFT is complex and doubly degenerate while the quantum ground state wavefunction is always real and unique. The time reversal symmetry is spontaneously broken when the system transfers from the quantum ground state into one of the mean field ground states upon a small perturbation. As the gap between the lowest two levels decreases exponentially with particle number, the required perturbation inducing the spontaneous symmetry breaking (SSB) is infinitesimal for particle number of typical cold atom systems. The quantum ground state is further analyzed with the Penrose–Onsager criterion, and is found to be a fragmented condensate. The state also develops the pair correlation and has non-vanishing pair order parameter instead of the conventional single particle order parameter. When this model is generalized to optical lattice, a pair superfluid can be generated. The mean field ground state can be regarded as effective ground state in this simple model. The detailed computation for this model enables us to offer an in-depth discussion of the relation between SSB and effective ground state, giving a glimpse on how nonlinearity arises in the SSB of a quantum system.
In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.
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.