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IEEE Conference on Decision and Control (CDC)

Controllability properties of a continuously monitored qubit

Francesca Carlotta Chittaro, Paolo Mason

Published Dec 16, 2024• 0 citations
Publisher

Abstract

We consider a stochastic dynamics describing the evolution of a qubit controlled by an external field and subject to continuous-time measurements. Motivated by stabilization techniques recently developed in, e.g., [10], [11], we investigate the support of the corresponding solution, which is a random variable taking values on the space of two-by-two density matrices. By making use of the Strook-Varadhan support theorem and by classical geometric control arguments we compute the support for two possible choices of the measurement and Hamiltonian operators. In one case we show that, in Bloch coordinates, the support is always contained inside an ellipsoid depending on the physical parameters of the system. More precisely, a solution starting from the ellipsoid never exit it, with probability one, and every open subset of the ellipsoid is visited with nonzero probability for some choice of the control function. In the second case the support coincides with (the interior of) the Bloch ball: every open subset of the Bloch ball is visited with nonzero probability up to suitably choosing the control function.

DOI
10.1109/cdc56724.2024.10885991

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We consider a stochastic dynamics describing the evolution of a qubit controlled by an external field and subject to continuous-time measurements. Motivated by stabilization techniques recently developed in, e.g., [10], [11], we investigate the support of the corresponding solution, which is a random variable taking values on the space of two-by-two density matrices. By making use of the Strook-Varadhan support theorem and by classical geometric control arguments we compute the support for two possible choices of the measurement and Hamiltonian operators. In one case we show that, in Bloch coordinates, the support is always contained inside an ellipsoid depending on the physical parameters of the system. More precisely, a solution starting from the ellipsoid never exit it, with probability one, and every open subset of the ellipsoid is visited with nonzero probability for some choice of the control function. In the second case the support coincides with (the interior of) the Bloch ball: every open subset of the Bloch ball is visited with nonzero probability up to suitably choosing the control function.
The evolution of two-level quantum systems (qubits) in the presence of continuous-time weak measurements can be described by stochastic differential equations, which is dominated by the Hamiltonian and decoherence channels. In more concrete terms, the conventional density matrix evolves according to the stochastic master equation in the Schrödinger picture. Particularly, for a monitored qubit, the deterministic submanifolds concerning the dynamical evolution of the density matrix has been previously obtained. As a complement, We have recently developed deterministic submanifolds for the backward-evolving quantum state for a monitored qubit. In this paper, in order to better understand the behavior of a monitored qubit, by using the Bloch sphere representation and the properties of Pauli matrices, we provide a closed-form expression for probability distribution over the deterministic submanifolds of the backward evolving dynamics in different scenarios. We further analyze the backward-evolving trajectories of the quantum state by taking into account the analytic solutions and the corresponding submanifolds under heterodyne/homodyne detection associated with non-demolition or fluorescence measurements.
A continuously monitored quantum bit (qubit) exhibits competition between unitary Hamiltonian dynamics and non-unitary measurement-collapse dynamics, which for diffusive measurements form an enlarged transformation group equivalent to the Lorentz group of spacetime. We leverage this equivalence to develop a four-dimensional generalization of the three-dimensional Bloch ball to visualize the state of a monitored qubit as the four-momentum of an effective classical charge affected by a stochastic electromagnetic force field. Unitary qubit dynamics generated by Hermitian Hamiltonians correspond to elliptic spatial rotations of this effective charge while non-unitary qubit dynamics generated by non-Hermitian Hamiltonians or stochastic measurement collapse correspond to hyperbolic Lorentz boosts. Notably, to faithfully emulate the stochastic qubit dynamics arising from continuous qubit measurement, the stochastic electromagnetic fields must depend on the velocity of the charge they are acting on. Moreover, continuous qubit measurements admit a dynamical delayed choice effect where a future experimental choice can appear to retroactively determine the type of past measurement backaction, so the corresponding point charge dynamics can also exhibit delayed choice Lorentz transformations in which a future experimental choice determines whether stochastic force fields are electric or magnetic in character long after they interact with the particle.
Abstract In this work, we study controllability in the set of all density matrices for a two-level open quantum system driven by coherent and incoherent controls. In Pechen (2011 Phys. Rev. A 84 042106) an approximate controllability, i.e. controllability with some precision, was shown for generic N -level open quantum systems driven by coherent and incoherent controls. However, the explicit formulation of this property, including the behavior of this precision as a function of transition frequencies and decoherence rates of the system, was not known. The present work provides a rigorous analytical study of reachable sets for two-level open quantum systems. First, it is shown that for N = 2 the presence of incoherent control does not affect the reachable set (while incoherent control may affect the time necessary to reach particular state). Second, the reachable set in the Bloch ball is described and it is shown that already just for one coherent control any point in the Bloch ball can be achieved with precision δ ∼ γ / ω , where γ is the decoherence rate and ω is the transition frequency. Typical values are δ ≲ 10 −3 that implies high accuracy of achieving any density matrix. Moreover, we show that most points in the Bloch ball can be exactly reached, except of two lacunae of size ∼ δ . For two coherent controls, the system is shown to be completely controllable in the set of all density matrices. Third, the reachable set as a function of the final time is found and shown to exhibit a non-trivial structure.
Abstract The evolution of a quantum system subject to continuous weak measurement is described by a stochastic differential equation (SDE). It has been shown that the conventional density matrix characterized by such SDE, under experimentally relevant settings, remains within a deterministically evolving manifold of low dimension, whose associated equations can be computed explicitly. In this paper, as a complement, it is demonstrated that the effect matrix , associated to a quantum system at the time t to express its measurement during the interval , also remains confined to a deterministically evolving manifold. In particular, by using the Bloch sphere representation together with the properties of Pauli matrices (as the quantum representation of quaternions), deterministic submanifolds for the effect matrix concerning a monitored qubit are given explicitly. The qubit stays on a deterministically evolving curve in the homodyne amplitude detection scenario, and on a deterministically moving surface under heterodyne amplitude detection. This deterministic characterization should be of interest for parameter estimation and towards obtaining computationally efficient quantum filters for real‐time quantum control.
Abstract Quantum mechanics admits two distinct evolutions: deterministic unitary dynamics governed by the Schrödinger equation and the probabilistic collapse of the wave function. We show that the continuous collapse of a quantum state under measurement can, on a trajectory-by-trajectory basis, be equivalently described as unitary evolution generated by a time- and state-dependent Hermitian Hamiltonian with stochastic parameters. While the ensemble dynamics remains non-unitary, each individual trajectory thus admits a unitary representation. We derive explicit forms of such Hamiltonians for projective measurements on arbitrary n -level systems and for continuous position measurements of a harmonic oscillator, and we propose experimental schemes to test these predictions. Our framework provides a new approach to modeling and controlling continuously monitored quantum systems using only state-dependent unitary resources.
Research in the field of quantum control usually assumes complete knowledge of the state and parameters. In this paper, a driven and damped two-level quantum system under continuous observation is considered. Its time evolution is governed by an operator-valued stochastic master equation and conditioned on the recorded measurement data. The system is transformed to state space representation and its observability is proven analytically. An extended Kalman filter is employed in order to both reconstruct the density operator describing the full state of the system, and to dynamically identify key parameters. Convincing results are obtained in numerical simulations. Overall, a practical engineering approach to quantum control is presented.