Data-processing inequalities for quantum metrology

Abstract

We apply the classical data-processing inequality to quantum metrology to show that manipulating the classical information from a quantum measurement cannot aid in the estimation of parameters encoded in quantum states. We further derive a quantum data-processing inequality to show that coherent manipulation of quantum data also cannot improve the precision in estimation. In addition, we comment on the assumptions necessary to arrive at these inequalities and how they might be avoided, providing insights into enhancement procedures which are not provably wrong. Parameter estimation is an integral part of physics. Quantum metrology refers to the study of the ultimate limits in the accuracy of estimates given the structure imposed by quantum theory [1,2]. Estimation at or near this limit is important for practical objectives such as improving time and frequency standards [3,4] as well as fundamental physics, such as the detection of gravitational waves [5].Researchers have found many novel approaches to quantum metrology using, for example, multipass interferometers [6], machine learning techniques [7], and computational Bayesian statistics [8] as well as new bounds [9,10] in increasingly more general scenarios. Here we supplement these results with one of a different flavor. We provide a very general bound on the estimation accuracy in quantum metrology when noise or data processing (either classical or coherent) is present. Precisely, we give a classical and quantum data processing inequality which shows that estimators based on the raw data are optimal. In other words, processing the data cannot improve quantum metrology. We conclude by showing how to avoid the inequalities with more exotic procedures which can be classed into three conceptually intuitive categories: (1) processing data in a way dependent on the parameter; (2) circumventing an imposed operational restriction; or (3) modifying the dynamics which impart the parameter. Consider a statistical model defining a likelihood function Pr(x|θ ; C). In words, there is an experimental context C whose outcomes are labeled by the random variable x and θ is an unknown parameter to be estimated. For quantum metrology, the goal is estimate a parameter which defines a quantum dynamical process:

Optimal Gaussian measurements for phase estimation in single-mode Gaussian metrology

“…Note that the probability distribution of the measurement outcome forΠ 0 being a squeezed thermal state can be decomposed into a mixture of those forΠ 0 being squeezed vacuum states. We thus assumê Π 0 to be only the squeezed vacuum state without loss of generality according to the data processing inequality [28,29]. Typical types of Gaussian measurement are the homodyne measurement [shown in Fig.…”

Improving quantum parameter estimation by monitoring quantum trajectories

“…They further conjectured that F is a non-decreasing function of the measurement efficiency. Here our general formalism allows a simple generalization: since an imperfect measurement can be regarded as a perfect measurement followed by a non-unitary quantum operation, while any θ-independent quantum operation cannot increase the QFI [84], F [ρ ext|{E l } ] and hence F is a nondecreasing function of the measurement efficiency on the environment for any environment.…”

“…follows from the simple fact that ρ ext|{E l } is obtained from |Ψ ext by a non-unitary operation, which cannot increase the QFI [84]. Alternatively, we rewrite their difference as…”

Quantum Markov monogamy inequalities

“…With rising interest in quantum information theory, classical data processing inequalities were extended to the regime of quantum processes [12], setting an appropriate approach to define new constraints on quantum Markovian processes. Indeed, intense research has been undertaken in this direction since then [13][14][15][16][17][18][19][20]. Naturally, further information-theoretic constraints imposed by quantum Markovian processes is of great interest and it is the main problem addressed in this paper.…”