Continuous measurements play a pivotal role in the study of dynamical open quantum systems. ``Dyne'' detections are among the most widespread and efficient measurement schemes, and give rise to quantum diffusion of the conditioned state. In this work we study under what conditions the detector dependency of the conditional state of a quantum system subject to diffusive monitoring can be demonstrated experimentally, in the sense of ruling out any detector-independent pure-state dynamical model for the system. We consider an arbitrary number $L$ of environments to which the system is coupled, and an arbitrary number $K$ of different types of dyne detections. We prove that nontrivial necessary conditions for such a demonstration can be determined efficiently by semidefinite programming. To determine sufficient conditions, different physical environmental couplings and Hamiltonians for a qubit and different sets of diffusive monitorings are scrutinized. We compare the threshold efficiencies that are sufficient in the various cases, as well as cases previously considered in the literature, to suggest the most feasible experimental options.
Continuous measurements play a pivotal role in the study of dynamical open quantum systems. ``Dyne'' detections are among the most widespread and efficient measurement schemes, and give rise to quantum diffusion of the conditioned state. In this work we study under what conditions the detector dependency of the conditional state of a quantum system subject to diffusive monitoring can be demonstrated experimentally, in the sense of ruling out any detector-independent pure-state dynamical model for the system. We consider an arbitrary number L of environments to which the system is coupled, and an arbitrary number K of different types of dyne detections. We prove that nontrivial necessary conditions for such a demonstration can be determined efficiently by semidefinite programming. To determine sufficient conditions, different physical environmental couplings and Hamiltonians for a qubit and different sets of diffusive monitorings are scrutinized. We compare the threshold efficiencies that are sufficient in the various cases, as well as cases previously considered in the literature, to suggest the most feasible experimental options.
It was recently argued (Wiseman and Gambetta 2012 Phys. Rev. Lett. 108 220402 ) that the stochastic dynamics (jumps or diffusion) of an open quantum system are not inherent to the system, but rather depend on the existence and nature of a distant detector. The proposed experimental tests involved homodyne detection, giving rise to quantum diffusion, and required efficiencies of well over 50%. Here we prove that this requirement ( ) is universal for diffusive-type detection, even if the system is coupled to multiple baths. However, this no-go theorem does not apply to quantum jumps, and we propose a test involving a qubit with jump-type detectors, with a threshold efficiency of only 37%. That is, quantum jumps are 'more quantum', and open the way to practical experimental tests. Our scheme involves a novel sort of adaptive monitoring scheme on a system coupled to two baths.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
Dynamical quantum jumps were initially conceived by Bohr as objective events associated with the emission of a light quantum by an atom. Since the early 1990s they have come to be understood as being associated rather with the detection of a photon by a measurement device, and that different detection schemes result in different types of jumps (or diffusion). Here we propose experimental tests to rigorously prove the detector dependence of the stochastic evolution of an individual atom. The tests involve no special preparation of the atom or field, and the required efficiency can be as low as η≈58%.
We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing among a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time within any desired accuracy. Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.
Decoherence is an unavoidable effect of the quantum system coupled to an environment, which leads to the system evolving from a pure state to a mixed one. A critical issue here is that the reduction of purity is quantum or classical, which corresponds to whether or not entanglement is generated between the system and environment. We report a proof-of-principle experimental investigation of such entanglement in a simple but paradigmatic case, in which a qubit system interacts with a pure dephasing channel. The pure dephasing process is realized in a linear photonic system, and the experiment is a validation in a very controlled setting. By adopting previous methods to detect qubit-environment entanglement, we prove that such entanglement can be detected by performing local measurements on only one subsystem, even when the other subsystem in the pair cannot be detected. Our experiment paves the way for investigating features of open quantum system dynamics.
We revisit aspects of monitoring observables with continuous spectrum in a quantum system subject to dissipative (Lindbladian) or conservative (Hamiltonian) evolutions. After recalling some of the salient features of the case of pure monitoring, we deal with the case when monitoring is in competition with a Lindbladian evolution. We show that the strong measurement limit leads to a diffusion on the spectrum of the observable. For the case with competition between observation and Hamiltonian dynamics, we exhibit a scaling limit in which the crossover between the classical regime and a diffusive regime can be analyzed in details.