The elusive Heisenberg limit in quantum-enhanced metrology

Abstract

Quantum precision enhancement is of fundamental importance for the development of advanced metrological optical experiments, such as gravitational wave detection and frequency calibration with atomic clocks. Precision in these experiments is strongly limited by the 1/√N shot noise factor with N being the number of probes (photons, atoms) employed in the experiment. Quantum theory provides tools to overcome the bound by using entangled probes. In an idealized scenario this gives rise to the Heisenberg scaling of precision 1/N. Here we show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N amounts generically to a constant factor rather than quadratic improvement. We provide efficient and intuitive tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: depolarization, dephasing, spontaneous emission and photon loss.

Figure 1. Quantum metrology and the CS idea.

(a) General scheme for quantum-enhanced metrology. N-probe quantum state fed into N parallel channels is sensing an unknown channel parameter ϕ. An estimator is inferred from a measurement result on the output state. (b) CS of a quantum channel. The channel Λ_ϕ is interpreted as a mixture of other channels Λ_X, where the dependence on ϕ is moved into the mixing probabilities p_ϕ(X).

Figure 2. Estimation precision in presence of decoherence.

Log–log plot of a generic dependence of quantum-enhanced parameter estimation uncertainty in the presence of decoherence as a function of the number of probes used. While for small number of probes the curve for achievable precision follows the Heisenberg scaling, it asymptotically flattens to approach dependence. The 'const' represents the quantum enhancement factor. The exemplary data corresponds to the case of phase estimation using N photons in a Mach–Zehnder interferometer with 5% losses in both arms.

Figure 3. Local classical stimulation.

Schematic representation of a local classical stimulation (CS) of a channel Λ_ϕ at ϕ₀ that lies inside the convex set of quantum channels (solid oval). The optimal CS has to be valid only in the neighbourhood of along the curve (solid arched line) and corresponds to a binary mixture of channels Λ_±, which rest on the tangent (dashed) line at the two outermost points. Then, the precision of estimation can be lower bounded just using the distances ɛ_±.

Figure 4. Graphical representation of decoherence models.

Two-level atom decoherence processes are illustrated with a corresponding shrinking of the Bloch ball (blue) for the cases of: (a) depolarization, (b) dephasing and (c) spontaneous emission. The estimated parameter ϕ represents the angle of rotation about the z axis, whereas η specifies the strength of decoherence and effectively the size of shrinkage in the x–y plane. In the example (d) of the lossy interferometer, ϕ corresponds to the additional phase acquired by photons travelling in the upper arm and η stands for the power transmission coefficient in each of the arms.