In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\ensuremath{\theta}$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space ${H}_{w}(\ensuremath{\theta})$ increases with $\ensuremath{\theta}$ and with time, which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of bounded and unbounded DTQWs and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare ${H}_{w}(\ensuremath{\theta})$ to the full QFI ${H}_{f}(\ensuremath{\theta})$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\ensuremath{\theta}$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.
Discrete-time quantum walk (DTQW) represents a convenient mathematical framework for describing the motion of a particle on a discrete set of positions when this motion is conditioned by the values of certain internal degrees of freedom, which are usually referred to as the coin of the particle. As such, and owing to the inherent dependence of the position distribution on the coin degrees of freedom, DTQWs naturally emerge as promising candidates for quantum metrology. In this paper, we explore the use of DTQWs as quantum probes in scenarios where the parameter of interest is encoded in the internal degree of freedom of the walker and investigate the role of the topology of the walker's space on the attainable precision. In particular, we start considering the encoding of the parameter by rotations for a walker on the line and evaluate the quantum Fisher information (QFI) and the position Fisher information (FI), explicitly determining the optimal initial state in position space that maximizes the QFI across all encoding schemes. This allows us to understand the role of interference in the position space and to introduce an optimal topology, which maximizes the QFI of the coin parameter and makes the position FI equal to the QFI.
We introduce the driven discrete time quantum walk (QW), where walkers are added during the walk instead of only at the beginning. This leads to interference in walker number and very different dynamics when compared to the original QW. These dynamics have two regimes, which we illustrate using the one-dimensional line. Then, we explore a search application which has certain advantages over current search protocols, namely that it does not require a complicated initial state nor a specific measurement time to observe the marked state. Finally, we describe a potential experimental implementation using existing technology.
Abstract The enhancement of the precision of phase estimation in quantum metrology is investigated by employing parity-time ( P T ) symmetric operation. We derive the exact expressions of the quantum Fisher information (QFI) and the success probability of the phase estimation for an exactly solving model consisting of a qubit interacting with a structured reservoir. Special attention is paid to the two schemes, i.e. pre-and post- P T symmetric operations. We show that the QFI can be obviously enhanced by means of the P T symmetric operation in different regimes. In addition, we also show that the magnitude of the decoherence involved in the P T symmetric operation can be a general complex number, which extends the applicable scope of the P T symmetric operation approach.
High-precision quantum parameter estimation is fundamental to the advancement of quantum metrology. Although reservoir engineering provides a powerful approach to improve estimation by tailoring system-environment interactions, the role of the squeezing phase and correlations arising from the sequential utilization of the same squeezed reservoir remains inadequately explored. In this work, we employ a correlated squeezed-thermal reservoir to enhance the precision of estimating the phase parameter
φ and the correlation factor
μ, both individually and simultaneously. We show that the squeezing phase
Φ is crucial for achieving quantum-enhanced precision, with optimal phase-matching conditions that depend strongly on
μ. Specifically, we derive the near-optimal phase-matching relations aimed at maximizing the quantum Fisher information (QFI) for both
φ and
μ, as well as minimizing the total variance
Δsim in joint estimation. Furthermore, we show that the joint estimation variance is dominated by
Fφ, which motivates our search for the phase-matching conditions that minimize
Δsim. Through the ratio
R of variances, we demonstrate that joint estimation conserves quantum resources and maintains high precision when the squeezing phase is optimized for
Fφ, despite the inherent incompatibility of the parameters. These findings provide practical insights into reservoir engineering strategies for high-precision quantum sensing and information processing.
We address multiparameter quantum estimation for one-dimensional discrete-time quantum walks and its applications to quantum metrology. We use the quantum walker as a probe for the unknown parameters encoded in its coin degrees of freedom. We find an analytic expression of the quantum Fisher information matrix for the most general coin operator, and show that only two out of the three coin parameters can be accessed. We also prove that the resulting two-parameter coin model is asymptotically classical, i.e., the Uhlmann curvature vanishes. Finally, we apply our findings to relevant case studies, including the simultaneous estimation of charge and mass in the discretized Dirac model.