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Journal of Statistical Physics

A Perron–Frobenius Type of Theorem for Quantum Operations

Matthew Lagro, Wei-Shih Yang, Sheng Xiong

Published Aug 22, 2017• 4 citations
Publisher
DOI
10.1007/s10955-017-1862-3

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We introduce the concept of \mathfrak{F} -positive elements in planar algebras and establish a Perron–Frobenius theorem for such elements. We investigate the existence and uniqueness of the Perron–Frobenius eigenspace. When the eigenspace is not one dimensional, we characterize its multiplicative structure. Furthermore, we interpret the Perron–Frobenius eigenspace as the space of logical qubits for mixed states in quantum information theory. We also explore the connection between these mathematical results and quantum error correction, particularly in relation to the Knill–Laflamme theorem.
The Perron-Frobenius theorem is applied to identify the superfluid transition of a two-component Fermi gas with a zero-range s-wave interaction. According to the quantum cluster expansion method of Lee and Yang, the grand partition function is expressed by the Lee-Yang contracted 0-graphs. A singularity of an infinite series of ladder-type Lee-Yang contracted 0-graphs is analyzed. We point out that the singularity is governed by the Perron-Frobenius eigenvalue of a certain primitive matrix which is defined in terms of the two-body cluster functions and the Fermi distribution functions. As a consequence, it is found that there exists a unique fugacity at the phase transition point, which implies that there is no fragmentation of Bose-Einstein condensates of dimers and Cooper pairs at the ladder-approximation level of Lee-Yang contracted 0-graphs. An application to a Bose-Einstein condensate of strongly bounded dimers is also made.