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arXiv:1702.00639 · Quantum Science and Technology

Locally optimal symplectic control of multimode Gaussian states

Uther Shackerley-Bennett, Alberto Carlini, Vittorio Giovannetti, Alessio Serafini

Published Aug 25, 2017• 3 citations
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Abstract

Abstract The relaxation of a system to a steady state is a central point of interest in many attempts to advance control over the quantum world. In this paper, we consider control through instantaneous Gaussian unitary operations on the ubiquitous lossy channel, and find locally optimal conditions for the cooling and heating of a multimode Gaussian state subject to losses and possibly thermal noise. This is done by isolating the parameters that encode entropy and temperature and by deriving an equation for their evolution. This equation is in such a form that it grants clear insight into how relaxation may be helped by instantaneous quantum control. It is thus shown that squeezing is a crucial element in optimising the rate of change of entropic properties under these channels. Exact relaxation times for heating and cooling are derived, up to an arbitrarily small distance from the fixed point of the lossy channel with locally optimal strategies.

DOI
10.1088/2058-9565/aa8883

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Abstract The relaxation of a system to a steady state is a central point of interest in many attempts to advance control over the quantum world. In this paper, we consider control through instantaneous Gaussian unitary operations on the ubiquitous lossy channel, and find locally optimal conditions for the cooling and heating of a multimode Gaussian state subject to losses and possibly thermal noise. This is done by isolating the parameters that encode entropy and temperature and by deriving an equation for their evolution. This equation is in such a form that it grants clear insight into how relaxation may be helped by instantaneous quantum control. It is thus shown that squeezing is a crucial element in optimising the rate of change of entropic properties under these channels. Exact relaxation times for heating and cooling are derived, up to an arbitrarily small distance from the fixed point of the lossy channel with locally optimal strategies.
The relaxation of a system to a steady state is a central point of interest in many \nattempts to advance control over the quantum world. In this paper, we consider control through \ninstantaneous Gaussian unitary operations on the ubiquitous lossy channel, and find locally \noptimal conditions for the cooling and heating of a multimode Gaussian state subject to losses \nand possibly thermal noise. This is done by isolating the parameters that encode entropy and \ntemperature and by deriving an equation for their evolution. This equation is in such a form that \nit grants clear insight into how relaxation may be helped by instantaneous quantum control. It \nis thus shown that squeezing is a crucial element in optimising the rate of change of entropic \nproperties under these channels. Exact relaxation times for heating and cooling are derived, up \nto an arbitrarily small distance from the fixed point of the lossy channel with locally optimal \nstrategies. Keywords: Gaussian states, coherent control, open quantum systems
We consider the problem of time-optimal control of a continuous bosonic quantum system subject to the action of a Markovian dissipation. In particular, we consider the case of a one-mode Gaussian quantum system prepared in an arbitrary initial state and which relaxes to the steady state due to the action of the dissipative channel. We assume that the unitary part of the dynamics is represented by Gaussian operations which preserve the Gaussian nature of the quantum state, i.e., arbitrary phase rotations, bounded squeezing, and unlimited displacements. In the ideal ansatz of unconstrained quantum control (i.e., when the unitary phase rotations, squeezing, and displacement of the mode can be performed instantaneously), we study how control can be optimized for speeding up the relaxation towards the fixed point of the dynamics and we analytically derive the optimal relaxation time. Our model has potential and interesting applications to the control of modes of electromagnetic radiation and of trapped levitated nanospheres.
The rise of quantum technology has put control at the centre of advancements in quantum mechanics. The union of quantum mechanics with mathematical control theory is a meeting that is leading to a much deeper insight into our interaction with the bizarre properties of quantum theory. Often, the study of discrete variable systems is the focus for making this union. Here, we look at how control theory may be applied to the continuous variable theory of Gaussian states. Special emphasis is given to control of the covariance matrix of these states, as it is here that we find the entanglement and entropic properties of the state. We begin by exploring some initial results for the geometry of Gaussian states, revealing different manifold structures dependent on symplectic eigenvalue degeneracy. In this geometrical setting a proposal for an extension of Williamson's theorem is put forward and partially completed. It is often interesting to look at restricted sets of Hamiltonians and ask what transformations can be performed with concatenations of their corresponding unitaries. Controllable systems are those for which the entire group of interest is possible to enact. We explore an uncontrollable system in a single mode and give a physical analysis as to why it behaves this way. This leads to ideas to move forwards for a necessary and sufficient condition for control on the symplectic group that has been conjectured since 1972. Later, we transfer to the question of open dynamics. We focus on a particular and ubiquitous channel known as 'lossy' or 'the attenuation channel'. An equation is derived describing the evolution for the symplectic invariants of a Gaussian state undergoing such dynamics. The equation of the former chapter is used to explore the evolution of entropy and entanglement. Optimal protocols are developed for the manipulation of these properties undergoing lossy dynamics.
We develop a method for deriving Gaussian optimality results for bosonic quantum systems from inequalities for qubit systems. Inspired by Gross's derivation of the Gaussian logarithmic Sobolev inequality from an inequality on the two-point space, we use qubits as quantum two-point spaces and lift bosonic Weyl operators to pseudo-Weyl operators on large qubit systems. By controlling weighted norms and relative entropy in the fluctuation limit, we establish two transfer theorems that convert tensorized qubit inequalities into inequalities for bosonic Gaussian channels. As a first application, we prove that the sharp 222-logarithmic Sobolev inequality for the bosonic Ornstein--Uhlenbeck semigroup tensorizes without loss in the constant. We also establish a sharp output entropy inequality for tensor powers of a class of qubit channels. This includes a quantum Mrs.~Gerber's lemma determining the minimum output entropy of tensor powers of the qubit depolarizing channel at fixed input entropy. Applying our relative entropy transfer theorem, we obtain multimode constrained minimum output entropy bounds for Gaussian additive-noise channels and thermal attenuators in the heating range.
We consider a two-level quantum system prepared in an arbitrary initial state and relaxing to a steady state due to the action of a Markovian dissipative channel. We study how optimal control can be used for speeding up or slowing down the relaxation towards the fixed point of the dynamics. We analytically derive the optimal relaxation times for different quantum channels in the ideal ansatz of unconstrained quantum control (a magnetic field of infinite strength). We also analyze the situation in which the control Hamiltonian is bounded by a finite threshold. As by-products of our analysis, we find that (i) if the qubit is initially in a thermal state hotter than the environmental bath, quantum control can not speed up its natural cooling rate; (ii) if the qubit is initially in a thermal state colder than the bath, it can reach the fixed point of the dynamics in finite time if a strong control field is applied; (iii) in the presence of unconstrained quantum control, it is possible to keep the evolved state indefinitely and arbitrarily close to special initial states which are far away from the fixed points of the dynamics.
We study the possibility of taking bosonic systems subject to quadratic Hamiltonians and a noisy thermal environment to nonclassical stationary states by feedback loops based on weak measurements and conditioned linear driving. We derive general analytical upper bounds for the single-mode squeezing and multimode entanglement at steady state, depending only on the Hamiltonian parameters and on the number of thermal excitations of the bath. Our findings show that, rather surprisingly, larger number of thermal excitations in the bath allow for larger steady-state squeezing and entanglement if the efficiency of the optimal continuous measurements conditioning the feedback loop is high enough. We also consider the performance of feedback strategies based on homodyne detection and show that, at variance with the optimal measurements, it degrades with increasing temperature.