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arXiv:1107.0354 · Science China Physics Mechanics and Astronomy

Fidelity of states in infinite-dimensional quantum systems

Jinchuan Hou, Xiaofei Qi

Published Aug 9, 2012• 16 citations
arXivHTMLPDF
DOI
10.1007/s11433-012-4840-4

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Instead of unitary freedom for finite-dimensional cases, bi-contractive freedom in the operator-sum representation for quantum channels of infinite-dimensional systems is established. Specifically, if the channel sends every pure state to a finite rank state, then the isometric freedom feature holds. Then, a method of computing entanglement fidelity and a relation between quantum fidelity and entanglement fidelity for infinite-dimensional systems are obtained. In addition, upper and lower bounds of the quantum fidelity, and their connection to the trace distance, are also provided.
The Kosterlitz–Thouless transition is studied from the representation of the systems’s ground state wavefunctions in terms of matrix product states for a quantum system on an infinite-size lattice in one spatial dimension. It is found that, in the critical regime for a one-dimensional quantum lattice system with continuous symmetry, the newly developed infinite matrix product state algorithm automatically leads to infinite degenerate ground states, due to the finiteness of the truncation dimension. This results in pseudo continuous symmetry spontaneous breakdown, which allows the introduction of a pseudo-order parameter that must be scaled down to zero, in order to be consistent with the Mermin–Wegner theorem. We also show that the ground state fidelity per lattice site exhibits a catastrophe point , thus resolving a controversy regarding whether or not the ground state fidelity is able to detect the Kosterlitz–Thouless transition.