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Sum rule for non-adiabatic geometric phases

Adam Fredriksson, Erik Sjöqvist

Abstract

Berry monopoles always cancel when summing over a complete set of energy eigenstates. We demonstrate that analogous sum rules exist for geometric phases and their underlying 2-forms in non-adiabatic evolution. Our result has implications for qudit computation as it limits the types of gates that can be implemented by purely geometric means.

Sum rule for non-adiabatic geometric phases

Abstract

Berry monopoles always cancel when summing over a complete set of energy eigenstates. We demonstrate that analogous sum rules exist for geometric phases and their underlying 2-forms in non-adiabatic evolution. Our result has implications for qudit computation as it limits the types of gates that can be implemented by purely geometric means.

Paper Structure

This paper contains 2 theorems, 12 equations.

Key Result

Theorem 1

Given an orthonormal set $\{|\phi_{j}\rangle\}_{j = 1}^{d}$ of local sections spanning $\mathscr{H}_d$, in terms of which we define the curvature 2-forms $F^{(j)} = i\langle d\phi_{j}| \wedge |d\phi_{j}\rangle$, it holds that

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof