Controllability of the periodic quantum Ising spin chain and the Onsager algebra

Abstract The Ising spin model is a fundamental theoretical model in statistical physics which has received large attention in the context of quantum phase transitions (Sachdev S 2011 Quantum Phase Transitions (Cambridge University) Press) and, more recently, in quantum information (King et al 2022 Nat. Phys. 18 1324–8). In this paper, we present a controllability analysis of the quantum Ising periodic chain (1-dimensional) of n spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> particles (Pfeuty 1970 Ann. Phys 57 79–90) where the interpolating parameter between the two Hamiltonians plays the role of the control. The controllability is characterized by the dynamical Lie algebra generated by the two Hamiltonians (D’Alessandro 2021 Introduction to Quantum Control and Dynamics (CRC Press Taylor and Francis)). It is known (see, e.g. (Minami 2021 Nucl. Phys. B 973 115599; Roan 1991 Max-Plank-. Math. ; Von Gehlen and Rittenberg 1985 Nucl. Phys. B 257 351–70) that such a Lie algebra is an Onsager algebra . We present an alternative derivation of this result which is based on the algebra of Pauli matrices and Pauli strings . This derivation displays the explicit basis of the Lie algebra and of its simple ideals and center in the relevant representation of the space of n qubits <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>⊗</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . We discuss how these results match the ones of the seminal paper by Onsager (Onsager 1944 Phys. Rev. 65 3–4) which were obtained in the context of classical statistical mechanics. Applications to quantum information and control are also discussed.