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arXiv:quant-ph/0507055 · Physical Review A

Ancilla-assisted enhancement of channel estimation for low-noise parameters

Masahiro Hotta, Tokishiro Karasawa, Masanao Ozawa

Published Nov 29, 2005• 33 citations
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Abstract

In order to make a unified treatment for estimation problems of a very small noise or a very weak signal in a quantum process, we introduce the notion of a low-noise quantum channel with one noise parameter. It is known in several examples that prior entanglement together with nonlocal output measurement improves the performance of the channel estimation. In this paper, we study this ``ancilla-assisted enhancement'' for estimation of the noise parameter in a general low-noise channel. For channels on two level systems we prove that the enhancement factor, the ratio of the Fisher information of the ancilla-assisted estimation to that of the original one, is always upper bounded by $3∕2$. Some conditions for the attainability are also given with illustrative examples.

DOI
10.1103/physreva.72.052334

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In order to make a unified treatment for estimation problems of a very small noise or a very weak signal in a quantum process, we introduce the notion of a low-noise quantum channel with one noise parameter. It is known in several examples that prior entanglement together with nonlocal output measurement improves the performance of the channel estimation. In this paper, we study this ``ancilla-assisted enhancement'' for estimation of the noise parameter in a general low-noise channel. For channels on two level systems we prove that the enhancement factor, the ratio of the Fisher information of the ancilla-assisted estimation to that of the original one, is always upper bounded by 3∕23∕23∕2. Some conditions for the attainability are also given with illustrative examples.
We consider the achievability problem of the Cram\'er-Rao bound for multiparameter estimations. In general, it is not achievable due to the noncommutativity of optimal measurements for the corresponding parameters. However, we show that, under certain conditions, it can always be attained up to the leading order in the parameters as long as $D\ensuremath{\leqslant}N\ensuremath{-}1$, where DDD and NNN denote the number of parameters and the dimension of the system, respectively. After proving that, we discuss the achievability in the context of channel estimation for a general channel called a low-noise channel, which is very useful for investigating parameter estimation in the leading order. This allows us to find an ancilla-assisted enhancement effect: if entangled input states with an ancilla system are utilized for the channel estimation together with collective measurements on those output states, the bound becomes achievable for $D\ensuremath{\leqslant}{N}^{2}\ensuremath{-}1$.
We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the kkk-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most kkk, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension k∗k^*k∗ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.
The notion of low-noise channels was recently proposed and analyzed in detail in order to describe noise-processes driven by environment [M. Hotta, T. Karasawa and M. Ozawa, Phys. Rev. A72, 052334 (2005)]. An estimation theory of low-noise parameters of channels has also been developed. In this report, we address the low-noise parameter estimation problem for the NNN-body extension of low-noise channels. We perturbatively calculate the Fisher information of the output states in order to evaluate the lower-bound of the mean-square error of the parameter estimation. We show that the maximum of the Fisher information over all input states can be attained by a factorized input state in the leading order of the low-noise parameter. Thus, to achieve optimal estimation, it is not necessary for there to be entanglement of the NNN subsystems, as long as the true low-noise parameter is sufficiently small.
A well-studied scenario in quantum parameter estimation theory arises when the parameter to be estimated is imprinted on the initial state by a Hamiltonian of the form $\ensuremath{\theta}G$. For such ``phase-shift Hamiltonians'' it has been shown that one cannot improve the channel quantum Fisher information by adding ancillas and letting the system interact with them. Here we investigate the general case, where the Hamiltonian is not necessarily a phase shift, and show that in this case in general it is\mathit{is}is possible to increase the quantum channel information and to reach an upper bound. This can be done by adding a term proportional to the derivative of the Hamiltonian, or by subtracting a term from the original Hamiltonian. Neither method makes use of any ancillas, which shows that, for quantum channel estimation with an arbitrary parameter-dependent Hamiltonian, entanglement with an ancillary system is not necessary to reach the best possible sensitivity. By adding an operator to the Hamiltonian we can also modify the time scaling of the channel quantum Fisher information. We illustrate our techniques with nitrogen vacancy center magnetometry and the estimation of the direction of a magnetic field in a given plane using a single spin-1 as probe.
The metrological ability of carefully designed probes can be wrecked by the presence of noise which alters their evolution, reducing the information on the parameter of interest. This spoils the signal-to-noise ratio of any possible measurement. Here we show an experiment in which the introduction of an ancilla improves the estimation of an optical phase with respect to a standard single-qubit strategy. The advantage originates from the coherent coupling of the probe and the ancilla both at the initialization and measurement stages, generating and selecting a subset of overall configurations less affected by the noise.
Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value requires a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting n ancillas, the postselection probability can be increased by a factor of n while keeping the weak value fixed (compared to n uncorrelated attempts with one ancilla), which is the optimal scaling with n that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the relevant quantum Cramér-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.