Florian Fröwis

Improved Quantum Metrology Using Quantum Error Correction

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.

Measures of macroscopicity for quantum spin systems

We investigate the notion of “macroscopicity” in the case of quantum spin systems and provide two main results. First, we motivate the quantum Fisher information as a measure for the macroscopicity of quantum states. Second, we compare the existing literature of this topic. We report on a hierarchy among the measures and we conclude that one should carefully distinguish between “macroscopic quantum states” and “macroscopic superpositions”, which is a strict subclass of the former. PACS numbers: 03.65.-w,03.67.Mn,03.65.UdWe investigate the notion of ‘macroscopicity’ in the case of quantum spin systems and provide two main results. Firstly, we motivate the quantum Fisher information as a measure of the macroscopicity of quantum states. Secondly, we make a comparison with the existing literature on this topic. We report on a hierarchy among the measures and we conclude that one should carefully distinguish between ‘macroscopic quantum states’ and ‘macroscopic superpositions’, which is a strict subclass of the former.

Kind of entanglement that speeds up quantum evolution

The “speed” of unitary quantum evolution was recently shown to be connected to entanglement in multipartite quantum systems. Here, we discuss a tighter version of the Mandelstam-Tamm uncertainty relation that depends on the Fisher information. The passage time is estimated by a lower bound that depends inversely proportional to the square root of the Fisher information. This leads to a better understanding of the origin of a fast quantum time evolution of entangled states.

Detecting Large Quantum Fisher Information with Finite Measurement Precision.

We propose an experimentally accessible scheme to determine the lower bounds on the quantum Fisher information (QFI), which ascertains multipartite entanglement or usefulness for quantum metrology. The scheme is based on comparing the measurement statistics of a state before and after a small unitary rotation. We argue that, in general, the limited resolution of collective observables prevents the detection of large QFI. This can be overcome by performing an additional operation prior to the measurement. We illustrate the power of this protocol for present-day spin-squeezing experiments, where the same operation used for the preparation of the initial spin-squeezed state improves also the measurement precision and hence the lower bound on the QFI by 2 orders of magnitude. We also establish a connection to the Leggett-Garg inequalities. We show how to simulate a variant of the inequalities with our protocol and demonstrate that large QFI is necessary for their violation with coarse-grained detectors.

Stability of encoded macroscopic quantum superpositions

The multipartite Greenberger-Horne-Zeilinger (GHZ) state is a paradigmatic example of a highly entangled multipartite state with distinct quantum features. However, the GHZ state is very sensitive to generic decoherence processes, where its quantum features and in particular its entanglement diminish rapidly, thereby hindering possible practical applications. In this paper, we discuss GHZlike quantum states with a block-local structure and show that they exhibit a drastically increased stability against noise for certain choices of block-encoding, thereby extending results of Ref. [Phys. Rev. Lett. 106, 110402 (2011)]. We analyze in detail the decay of the interference terms, the entanglement in terms of distillable entanglement, and negativity as well as the notion of macroscopicity as measured by the so-called index q, and provide general bounds on these quantities. We focus on an encoding where logical qubits are themselves encoded as GHZ states, which leads to so-called concatenated GHZ (C-GHZ) states. We compare the stability of C-GHZ states with other types of encodings, thereby showing the superior stability of the C-GHZ states. Analytic results are complemented by numerical studies, where tensor network techniques are used to investigate the quantum properties of multipartite entangled states under the influence of decoherence.

Effective noise channels for encoded quantum systems

We investigate effective noise channels for encoded quantum systems with and without active error correction. Noise acting on physical qubits forming a logical qubit is thereby described as a logical noise channel acting on the logical qubits, which leads to a significant decrease of the effective system dimension. This provides us with a powerful tool to study entanglement features of encoded quantum systems. We demonstrate this framework by calculating lower bounds on the lifetime of distillable entanglement and the negativity for encoded multipartite qubit states with different encodings. At the same time, this approach leads to a simple understanding of the functioning of (concatenated) error correction codes.

Tensor operators: Constructions and applications for long-range interaction systems

We consider the representation of operators in terms of tensor networks and their application to the ground-state approximation and time evolution of systems with long-range interactions. We provide an explicit construction to represent an arbitrary many-body Hamilton operator in terms of a one-dimensional tensor network (i.e., as a matrix product operator). For pairwise interactions, we show that such a representation is always efficient and requires a tensor dimension growing only linearly with the number of particles. For systems obeying certain symmetries or restrictions we find optimal representations with minimal tensor dimension. We discuss the analytic and numerical approximation of operators in terms of low-dimensional tensor operators. We demonstrate applications for time evolution and the ground-state approximation, in particular for long-range interaction with inhomogeneous couplings. The operator representations are also generalized to other geometries such as trees and two-dimensional lattices, where we show how to obtain and use efficient tensor network representations respecting a given geometry.

Optimal quantum states for frequency estimation

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.

Linking measures for macroscopic quantum states via photon-spin mapping

We review and compare several measures that identify quantum states that are “macroscopically quantum”. These measures were initially formulated either for photonic systems or for spin ensembles. Here, we compare them through a simple model which maps photonic states to spin ensembles. On one hand, we reveal problems for some spin measures to handle correctly photonic states that typically are considered to be macroscopically quantum. On the other hand, we find significant similarities between other measures even though they were differently motivated.

Certifiability criterion for large-scale quantum systems

Can one certify the preparation of a coherent, many-body quantum state by measurements with bounded accuracy in the presence of noise and decoherence? Here, we introduce a criterion to assess the fragility of large-scale quantum states, which is based on the distinguishability of orthogonal states after the action of very small amounts of noise. States which do not pass this criterion are called asymptotically incertifiable. We show that if a coherent quantum state is asymptotically incertifiable, there exists an incoherent mixture (with entropy at least log2) which is experimentally indistinguishable from the initial state. The Greenberger-Horne-Zeilinger states are examples of such asymptotically incertifiable states. More generally, we prove that any so-called macroscopic superposition state is asymptotically incertifiable. We also provide examples of quantum states that are experimentally indistinguishable from highly incoherent mixtures, i.e. with an almost-linear entropy in the number of qubits. Finally, we show that all unique ground states of local gapped Hamiltonians (in any dimension) are certifiable.