We consider quantum metrology in noisy environments, where the effect of noise and decoherence limits the achievable gain in precision by quantum entanglement. We show that by using tools from quantum error correction this limitation can be overcome. This is demonstrated in two scenarios, including a many-body Hamiltonian with single-qubit dephasing or depolarizing noise and a single-body Hamiltonian with transversal noise. In both cases, we show that Heisenberg scaling, and hence a quadratic improvement over the classical case, can be retained. Moreover, for the case of frequency estimation we find that the inclusion of error correction allows, in certain instances, for a finite optimal interrogation time even in the asymptotic limit.

We establish general limits on how precise a parameter, e.g., frequency or the strength of a magnetic field, can be estimated with the aid of full and fast quantum control. We consider uncorrelated noisy evolutions of N qubits and show that fast control allows to fully restore the Heisenberg scaling (∼ 1/N 2 ) for all rank-one Pauli noise except dephasing. For all other types of noise the asymptotic quantum enhancement is unavoidably limited to a constant-factor improvement over the standard quantum limit (∼ 1/N ) even when allowing for the full power of fast control. The latter holds both in the single-shot and infinitelymany repetitions scenarios. However, even in this case allowing for fast quantum control helps to improve the asymptotic constant factor. Furthermore, for frequency estimation with finite resource we show how a parallel scheme utilizing any fixed number of entangled qubits but no fast quantum control can be outperformed by a simple, easily implementable, sequential scheme which only requires entanglement between one sensing and one auxiliary qubit.

Coherent superposition is a key feature of quantum mechanics that underlies the advantage of quantum technologies over their classical counterparts. Recently, coherence has been recast as a resource theory in an attempt to identify and quantify it in an operationally well-defined manner. Here we study how the coherence present in a state can be used to implement a quantum channel via incoherent operations and, in turn, to assess its degree of coherence. We introduce the robustness of coherence of a quantum channel-which reduces to the homonymous measure for states when computed on constant-output channels-and prove that: i) it quantifies the minimal rank of a maximally coherent state required to implement the channel; ii) its logarithm quantifies the amortized cost of implementing the channel provided some coherence is recovered at the output; iii) its logarithm also quantifies the zero-error asymptotic cost of implementation of many independent copies of a channel. We also consider the generalized problem of imperfect implementation with arbitrary resource states. Using the robustness of coherence, we find that in general a quantum channel can be implemented without employing a maximally coherent resource state. In fact, we prove that every pure coherent state in dimension larger than 2, however weakly so, turns out to be a valuable resource to implement some coherent unitary channel. We illustrate our findings for the case of single-qubit unitary channels.

We consider quantum metrology for unitary evolutions generated by parameter-dependent Hamiltonians. We focus on the unitary evolutions generated by the Ising Hamiltonian that describes the dynamics of a one-dimensional chain of spins with nearest-neighbour interactions and in the presence of a global, transverse, magnetic field. We analytically solve the problem and show that the precision with which one can estimate the magnetic field (interaction strength) given one knows the interaction strength (magnetic field) scales at the Heisenberg limit, and can be achieved by a linear superposition of the vacuum and N free fermion states. In addition, we show that GreenbergerHorne-Zeilinger-type states exhibit Heisenberg scaling in precision throughout the entire regime of parameters. Moreover, we numerically observe that the optimal precision using a product input state scales at the standard quantum limit.

We investigate different quantum parameter estimation scenarios in the presence of noise, and identify optimal probe states. For frequency estimation of local Hamiltonians with dephasing noise, we determine optimal probe states for up to 70 qubits, and determine their key properties. We find that the so-called one-axis twisted spin-squeezed states are only almost optimal, and that optimal states need not to be spin-squeezed. For different kinds of noise models, we investigate whether optimal states in the noiseless case remain superior to product states also in the presence of noise. For certain spatially and temporally correlated noise, we find that product states no longer allow one to reach the standard quantum limit in precision, while certain entangled states do. Our conclusions are based on numerical evidence using efficient numerical algorithms which we developed in order to treat permutational invariant systems.

We determine the quantum states and measurements that optimize the accessible information in a reference frame alignment protocol associated with the groups U (1), corresponding to a phase reference, and Z M , the cyclic group of M elements. Our result provides an operational interpretation of the G-asymmetry which is information-theoretic and which was thus far lacking. In particular, we show that in the limit of many copies of the bounded-size quantum reference frame, the accessible information approaches the Holevo bound. This implies that the rate of alignment of reference frames, measured by the (linearized) accessible information per system, is equal to the regularized, linearized G-asymmetry. The latter quantity is equal to the number variance in the case where G = U (1). Quite surprisingly, for the case where G = Z M and M 4, it is equal to a quantity that is not additive in general, but instead can be super-additive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analogue.where G(·) ≡ dgT (g)(·)T (g) † is the G-twirling operation, with the integral performed over the Haar measure dg, T a representation of G on the Hilbert space H d and S the von Neumann entropy. For the case of finite groups, the integral in the G-twirling operation is replaced by a sum, and the invariant Haar measure is given by 1/|G|, where |G| is the order of the group. The G-asymmetry was shown to be equal to the relative entropy of frameness [6], the latter being analogous to the relative entropy of entanglement [9].The relative entropy plays a crucial role in many quantum resource theories. Its importance comes from the fact that its asymptotic version provides the unique rate for reversible transformations [10]. This property was demonstrated with the discovery that the regularized New Journal of Physics 14 (2012) 073022 (http://www.njp.org/) 3 relative entropy of entanglement is the unique function that quantifies the rate of interconversion between states in a reversible theory of entanglement, where all types of non-entangling operations are allowed [11]. More recently, the importance of the relative entropy was demonstrated in the resource theory of thermodynamics [12,13]. However, in the resource theory of reference frames the regularized relative entropy is zero on all states [6]. We will therefore need to rescale it in order to find its operational meaning.In this work, we consider the case where G = Z M , the cyclic group of M elements, and the case where G = U (1) associated with the important case of photon number super-selection. For these cases we provide an operational interpretation of the G-asymmetry that is information theoretic, different from the interpretation in [8] of the G-asymmetry as extractable mechanical work. In particular, we find the strategy for aligning a pair of reference frames associated with G that optimizes the amount of accessible information between the true relation, g ∈ G, of the two reference frames and th...

Quantum metrology offers a quadratic advantage over classical approaches to parameter estimation problems by utilising entanglement and nonclassicality. However, the hurdle of actually implementing the necessary quantum probe states and measurements, which vary drastically for different metrological scenarios, is usually not taken into account. We show that for a wide range of tasks in metrology, 2D cluster states (a particular family of states useful for measurement-based quantum computation) can serve as flexible resources that allow one to efficiently prepare any required state for sensing, and perform appropriate (entangled) measurements using only single qubit operations. Crucially, the overhead in the number of qubits is less than quadratic, thus preserving the quantum scaling advantage. This is ensured by using a compression to a logarithmically sized space that contains all relevant information for sensing. We specifically demonstrate how our method can be used to obtain optimal scaling for phase and frequency estimation in local estimation problems, as well as for the Bayesian equivalents with Gaussian priors of varying widths. Furthermore, we show that in the paradigmatic case of local phase estimation 1D cluster states are sufficient for optimal state preparation and measurement.

We consider the usage of dynamical decoupling in quantum metrology, where the joint evolution of system plus environment is described by a Hamiltonian. We show that by ultra-fast unitary control operations acting locally only on system qubits, noise can be eliminated while the desired evolution is only reduced by at most a constant factor, leading to Heisenberg scaling. We identify all kinds of noise where such an approach is applicable. Only noise that is generated by the Hamiltonian to be estimated itself cannot be altered. However, even for such parallel noise, one can achieve an improved scaling as compared to the standard quantum limit for any local noise by means of symmetrization. Our results are also applicable in other schemes based on dynamical decoupling, e.g. the generation of highfidelity entangling gates.