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arXiv:1212.4581 · Physical Review A

Probe readout and quantum-limited measurements

Salini Jose, Noufal Jaseem, Anil Shaji

Published Feb 20, 2013• 8 citations
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Abstract

We consider the readout process on a probe made of $N$ qubits (two-level systems) in a quantum single-parameter estimation scheme. The parameter-dependent evolution of the probe as well as the measurements that are done for readout are assumed to be fixed. We find the optimal initial states of the probe that will saturate the quantum Cram\'er-Rao bound under these conditions. The relation between the optimal state of the $N$-qubit probe and that of the one-qubit probe is obtained for both entangling and nonentangling dynamics for the probe qubits. We study the limitations placed on the optimal initial state of the probe and on the achievable measurement uncertainty by the restrictions on the readout procedure.

DOI
10.1103/physreva.87.022330

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We consider the readout process on a probe made of NNN qubits (two-level systems) in a quantum single-parameter estimation scheme. The parameter-dependent evolution of the probe as well as the measurements that are done for readout are assumed to be fixed. We find the optimal initial states of the probe that will saturate the quantum Cram'er-Rao bound under these conditions. The relation between the optimal state of the NNN-qubit probe and that of the one-qubit probe is obtained for both entangling and nonentangling dynamics for the probe qubits. We study the limitations placed on the optimal initial state of the probe and on the achievable measurement uncertainty by the restrictions on the readout procedure.
Quantum resources, such as entanglement, can decrease the uncertainty of a parameter-estimation procedure beyond what is classically possible. This phenomenon is well described for noiseless systems with asymptotically many measurement resources by the Quantum Cramer-Rao Bound, but no general description exists for the regime of limited measurement resources. We address this problem by defining a Bayesian quantifier for uncertainty suitable for the regime of limited resources, and by developing a mathematical description for a parameter-estimation procedure which uses qubit probes to estimate a rotation angle induced on them. We simulate the qubit estimation scheme in the regime of limited resources using a single class of probe states. We find that, in noiseless systems, entanglement between qubits always decreases the uncertainty of the estimation; however, the quantum advantage decreases as fewer qubits are used in the estimation. We also find that the presence of strong dephasing noise removes the quantum advantage completely, regardless of the number of qubit probes used.
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We present a new proof of the quantum Cramer-Rao bound for precision parameter estimation [1-3] and extend it to a more general class of measurement procedures. We analyze a generalized framework for parameter estimation that covers most experimentally accessible situations, where multiple rounds of measurements, auxiliary systems or external control of the evolution are available. The proof presented demonstrates the equivalence of these more general metrology procedures to the simplest optimal strategy for which the bound is proven: a single measurement of a two-level system interacting with a time-independent Hamiltonian.
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We address the joint estimation of the two defining parameters of a displacement operation in phase space. In a measurement scheme based on a Gaussian probe field and two homodyne detectors, it is shown that both conjugated parameters can be measured below the standard quantum limit when the probe field is entangled. We derive the most informative Cram\'er-Rao bound, providing the theoretical benchmark on the estimation, and observe that our scheme is nearly optimal for a wide parameter range characterizing the probe field. We discuss the role of the entanglement as well as the relation between our measurement strategy and the generalized uncertainty relations.
This work compares the performance of single- and two-qubit probes for estimating several phase rotations simultaneously under the action of different noisy channels. We compute the quantum limits for this simultaneous estimation using collective and individual measurements by evaluating the Holevo and Nagaoka-Hayashi Cramér-Rao bounds, respectively. Several quantum noise channels are considered, namely the decohering channel, the amplitude damping channel, and the phase damping channel. For each channel, we find the optimal single- and two-qubit probes. Where possible we demonstrate an explicit measurement strategy that saturates the appropriate bound and we investigate how closely the Holevo bound can be approached through collective measurements on multiple copies of the same probe. We find that under the action of the considered channels, two-qubit probes show enhanced parameter estimation capabilities over single-qubit probes for almost all non-identity channels, i.e., the achievable precision with a single-qubit probe degrades faster with increasing exposure to the noisy environment than that of the two-qubit probe. However, in sufficiently noisy channels, we show that it is possible for single-qubit probes to outperform maximally entangled two-qubit probes. This work shows that, in order to reach the ultimate precision limits allowed by quantum mechanics, entanglement is required in both the state preparation and state measurement stages. It is hoped the tutorial-esque nature of this paper will make it easily accessible.