Fidelity is a sub-martingale for discrete-time quantum filters

Fidelity is known to increase through any Kraus map: the fidelity between two density matrices is less than the fidelity between their images via a Kraus map. We prove here that, in average, fidelity is also increasing for any discrete-time quantum filter: fidelity between the density matrix of the underlying Markov chain and the density matrix of its associated quantum filter is a sub-martingale. This result is not restricted to pure states. It also holds true for mixed states

Models and Feedback Stabilization of Open Quantum Systems

At the quantum level, feedback-loops have to take into account measurement back-action. We present here the structure of the Markovian models including such back-action and sketch two stabilization methods: measurement-based feedback where an open quantum system is stabilized by a classical controller; coherent or autonomous feedback where a quantum system is stabilized by a quantum controller with decoherence (reservoir engineering). We begin to explain these models and methods for the photon box experiments realized in the group of Serge Haroche (Nobel Prize 2012). We present then these models and methods for general open quantum systems.Comment: Extended version of the paper attached to an invited conference for the International Congress of Mathematicians in Seoul, August 13 - 21, 201

Asymptotic expansions of Laplace integrals for quantum state tomography

Bayesian estimation of a mixed quantum state can be approximated via maximum likelihood (MaxLike) estimation when the likelihood function is sharp around its maximum. Such approximations rely on asymptotic expansions of multi-dimensional Laplace integrals. When this maximum is on the boundary of the integration domain, as it is the case when the MaxLike quantum state is not full rank, such expansions are not standard. We provide here such expansions, even when this maximum does not belong to the smooth part of the boundary, as it is the case when the rank deficiency exceeds two. These expansions provide, aside the MaxLike estimate of the quantum state, confidence intervals for any observable. They confirm the formula proposed and used without precise mathematical justifications by the authors in an article recently published in Physical Review A.Comment: 17 pages, submitte

Deterministic submanifolds and analytic solution of the stochastic differential master equation describing a qubit

This paper studies the stochastic differential equation (SDE) associated to a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves

Robust open-loop stabilization of Fock states by time-varying quantum interactions

A quantum harmonic oscillator (spring subsystem) is stabilized towards a target Fock state by reservoir engineering. This passive and open-loop stabilization works by consecutive and identical Hamiltonian interactions with auxiliary systems, here three-level atoms (the auxiliary ladder subsystem), followed by a partial trace over these auxiliary atoms. A scalar control input governs the interaction, defining which atomic transition in the ladder subsystem is in resonance with the spring subsystem. We use it to build a time-varying interaction with individual atoms, that combines three non-commuting steps. We show that the resulting reservoir robustly stabilizes any initial spring state distributed between 0 and 4n+3 quanta of vibrations towards a pure target Fock state of vibration number n. The convergence proof relies on the construction of a strict Lyapunov function for the Kraus map induced by this reservoir setting on the spring subsystem. Simulations with realistic parameters corresponding to the quantum electrodynamics setup at Ecole Normale Superieure further illustrate the robustness of the method

Singular perturbations and Lindblad-Kossakowski differential equations

We consider an ensemble of quantum systems whose average evolution is described by a density matrix, solution of a Lindblad-Kossakowski differential equation. We focus on the special case where the decoherence is only due to a highly unstable excited state and where the spontaneously emitted photons are measured by a photo-detector. We propose a systematic method to eliminate the fast and asymptotically stable dynamics associated to the excited state in order to obtain another differential equation for the slow part. We show that this slow differential equation is still of Lindblad-Kossakowski type, that the decoherence terms and the measured output depend explicitly on the amplitudes of quasi-resonant applied field, i.e., the control. Beside a rigorous proof of the slow/fast (adiabatic) reduction based on singular perturbation theory, we also provide a physical interpretation of the result in the context of coherence population trapping via dark states and decoherence-free subspaces. Numerical simulations illustrate the accuracy of the proposed approximation for a 5-level systems.Comment: 6 pages, 2 figure

Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations

For discrete-time systems, governed by Kraus maps, the work of D. Petz has characterized the set of universal contraction metrics. In the present paper, we use this characterization to derive a set of quadratic Lyapunov functions for continuous-time systems, governed by Lindblad differential equations, that have a steady-state with full rank. An extremity of this set is given by the Bures metric, for which the quadratic Lyapunov function is obtained by inverting a Sylvester equation. We illustrate the method by providing a strict Lyapunov function for a Lindblad equation designed to stabilize a quantum electrodynamic "cat" state by reservoir engineering. In fact we prove that any Lindblad equation on the Hilbert space of the (truncated) harmonic oscillator, which has a full-rank equilibrium and which has, among its decoherence channels, a channel corresponding to the photon loss operator, globally converges to that equilibrium.Comment: Submitted (10 pages, 1 figure

Controllability of the 1D Schrodinger equation by the flatness approach

We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control input achieving the exact controllability in the energy space. As an application, we derive an explicit pair of control inputs achieving the exact steering to zero for a simply-supported beam