We study quantum metrology for unitary dynamics. Analytic solutions are given for both the optimal unitary state preparation starting from an arbitrary mixed state and the corresponding optimal measurement precision. This represents a rigorous generalization of known results for optimal initial states and upper bounds on measurement precision which can only be saturated if pure states are available. In particular, we provide a generalization to mixed states of an upper bound on measurement precision for time-dependent Hamiltonians that can be saturated with optimal Hamiltonian control. These results make precise and reveal the full potential of mixed states for quantum metrology.The standard paradigm of quantum metrology involves the preparation of an initial state, a parameterdependent dynamics, and a consecutive quantum measurement of the evolved state. From the measurement outcomes the parameter can be estimated [1][2][3]. Naturally, it is the goal to estimate the parameter as precisely as possible, i.e., to reduce the uncertainty ∆α = Var(α) 1 2 of the estimatorα of the parameter α that we want to estimate. Quantum coherence and nonclassical correlations in quantum sensors help to reduce this uncertainty compared to what is possible with comparable classical resources [4,5]. The ultimate precision limit is given by the quantum Cramér-Rao bound ∆α ≥ (M I α ) −1 2 which depends on the number of measurements M and the quantum Fisher information (QFI) I α which is a function of the state [6,7]. When the number of measurements is fixed, as they correspond to a limited resource, precision is optimal when the QFI is maximal which involves an optimization with respect to the state.In this Letter, we consider a freely available state ρ, unitary freedom to prepare an initial state from ρ, and unitary parameter-dependent dynamics of the quantum system (see Fig. 1). The parameter-dependent dynamics will be called sensor dynamics in the following in order to distinguish it from the state preparation dynamics. For instance, in a spin system the unitary freedom can be used to squeeze the spin before it is subjected to the sensor dynamics, as it is the case in many quantum-enhanced measurements [8][9][10][11]. In the worst case scenario, only the maximally mixed state is available, which does not change under unitary state preparation or unitary sensor dynamics and, thus, no information about the parameter can be gained. In the best-case scenario the available state is pure, when the maximal QFI as well as the optimal state to be prepared are well-known [12,13].The appeal and advantage of the theoretical study of unitary sensor dynamics lies in the analytic solutions that can be found that allow fundamental insights in the limits of quantum metrology and the role of resources such as measurement time and system size. The QFI maximized available state unitary state preparation initial state readout unitary sensor dynamics available state Figure 1. Schematic representation of the metrology protocol.with respect to initial stat...
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 θG. 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 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 to the original Hamiltonian. Neither method makes use of any ancillas which shows that for quantum channel estimation with 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 NV-center magnetometry and the estimation of the direction of a magnetic field in a given plane using a single spin-1 as probe.
We investigate in detail a recently introduced “coherent averaging scheme” in terms of its usefulness for achieving Heisenberg limited sensitivity in the measurement of different parameters. In the scheme, N quantum probes in a product state interact with a quantum bus. Instead of measuring the probes directly and then averaging as in classical averaging, one measures the quantum bus or the entire system and tries to estimate the parameters from these measurement results. Combining analytical results from perturbation theory and an exactly solvable dephasing model with numerical simulations, we draw a detailed picture of the scaling of the best achievable sensitivity with N, the dependence on the initial state, the interaction strength, the part of the system measured, and the parameter under investigation. In particular, we identify the situations allowing one to reach Heisenberg‐limited scaling of the sensitivity.
We consider quantum channel-estimation for depolarizing channels and phase-flip channels extended by ancilla qubits and fed with a GHZ or W state. After application of the channel one or several qubits can be lost, and we calculate the impact of the loss on the quantum Fisher information that determines the smallest uncertainty with which the parameters of these channels can be estimated
Abstract:We study very generally to what extent the uncertainty with which a phase shift can be estimated in quantum metrology can be reduced by extending the Hamiltonian that generates the phase shift to an ancilla system with a Hilbert space of arbitrary dimension, and allowing arbitrary interactions between the original system and the ancilla. Such Hamiltonian extensions provide a general framework for open quantum systems, as well as for "non-linear metrology schemes" that have been investigated over the last few years. We prove that such Hamiltonian extensions cannot improve the sensitivity of the phase shift measurement when considering the quantum Fisher information optimized over input states.
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