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Entanglement-free Heisenberg-limited phase estimation


Measurement underpins all quantitative science. A key example is the measurement of optical phase, used in length metrology and many other applications. Advances in precision measurement have consistently led to important scientific discoveries. At the fundamental level, measurement precision is limited by the number N of quantum resources (such as photons) that are used. Standard measurement schemes, using each resource independently, lead to a phase uncertainty that scales as 1/—known as the standard quantum limit. However, it has long been conjectured1,2 that it should be possible to achieve a precision limited only by the Heisenberg uncertainty principle, dramatically improving the scaling to 1/N (ref. 3). It is commonly thought that achieving this improvement requires the use of exotic quantum entangled states, such as the NOON state4,5. These states are extremely difficult to generate. Measurement schemes with counted photons or ions have been performed with N ≤ 6 (refs 6–15), but few have surpassed the standard quantum limit12,14 and none have shown Heisenberg-limited scaling. Here we demonstrate experimentally a Heisenberg-limited phase estimation procedure. We replace entangled input states with multiple applications of the phase shift on unentangled single-photon states. We generalize Kitaev’s phase estimation algorithm16 using adaptive measurement theory17,18,19,20 to achieve a standard deviation scaling at the Heisenberg limit. For the largest number of resources used (N = 378), we estimate an unknown phase with a variance more than 10 dB below the standard quantum limit; achieving this variance would require more than 4,000 resources using standard interferometry. Our results represent a drastic reduction in the complexity of achieving quantum-enhanced measurement precision.

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Figure 1: Quantum circuit diagrams of Kitaev’s phase estimation algorithm and our generalization.
Figure 2: Conceptual diagram of the algorithm’s implementation as a Mach–Zehnder interferometer.
Figure 3: Schematic of the experiment.
Figure 4: Standard deviations of distributions of phase estimates for varying numbers of resources N.


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We thank M. Mitchell, D. Bulger and S. Lo for discussions. This work was supported by the Australian Research Council and the Queensland State Government.

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Correspondence to G. J. Pryde.

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Higgins, B., Berry, D., Bartlett, S. et al. Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393–396 (2007).

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