Short answer: In quantum metrology with N probes, the standard quantum limit says precision improves with the square root of N, while entangled probes can in an idealized noiseless case reach Heisenberg scaling, which improves with N itself. A 2012 Nature Communications paper shows that with decoherence the maximal possible enhancement in the limit of infinite N is generically a constant factor, not a quadratic improvement. A Nature Communications paper on quantum error correction and a 2014 New Journal of Physics paper on correlated noise describe conditions under which Heisenberg scaling can be recovered. These are theory results.
Noise and the limit
The 2012 authors say precision in experiments such as gravitational wave detection and atomic-clock frequency calibration is strongly limited by the shot noise factor, which goes as one over the square root of N, with N the number of probes such as photons or atoms. Entangled probes can overcome the bound, and in an idealized scenario give Heisenberg scaling, one over N. They show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N is generically a constant factor rather than a quadratic improvement. They provide tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming, and apply them to depolarization, dephasing, spontaneous emission and photon loss.
Error correction
The error-correction paper says quantum mechanics imposes a fundamental limit on measurement precision, the Heisenberg limit, which can be achieved for noiseless quantum systems but is not achievable in general for systems subject to noise. It finds a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed that suppresses the noise without obscuring the signal, and the optimal code can be found by solving a semidefinite program.
Correlated noise
The 2014 New Journal of Physics paper says deviations from strictly unitary dynamics quickly restore the standard scaling dictated by the central limit theorem, and that product and maximally entangled states become asymptotically equivalent when the noisy evolution is both local and strictly Markovian. It notes that temporal correlations in the noise have been shown to lift this equivalence, while fully spatially correlated noise allows the identification of decoherence-free subspaces. The authors analyze noise with finite correlation length and show that there exist robust entangled state preparations that display persistent Heisenberg scaling despite environmental decoherence, even for small correlation length. They say their results emphasize the relevance of noise correlations to quantum advantage.
Limits
These are theoretical bounds and conditions. The error-correction result assumes noiseless ancillas and fast, accurate quantum processing, which are strong assumptions, and the correlated-noise result applies to the noise models analyzed. None of the papers reports a deployed sensor.
FAQ
What is the Heisenberg limit?
The fundamental limit on measurement precision set by quantum mechanics, which improves as one over N, per the error-correction paper. It is reachable for noiseless systems.
What is the standard quantum limit?
Precision improving as one over the square root of N, the scaling with independent probes or when noise prevents enhancement.
Does noise always rule out Heisenberg scaling?
The 2012 paper shows the enhancement is generically a constant factor under decoherence, and the other two papers give conditions that can restore it.
Can error correction help?
The error-correction paper gives a necessary and sufficient condition for Heisenberg scaling under Markovian noise, assuming noiseless ancillas and fast, accurate processing.
Does correlated noise matter?
The 2014 paper says yes: robust entangled preparations can keep Heisenberg scaling with finite noise correlation length.
All field notes