Quantum Enhanced Multiple Phase Estimation

We study the simultaneous estimation of multiple phases as a discretized model for the imaging of a phase object. We identify quantum probe states that provide an enhancement compared to the best quantum scheme for the estimation of each individual phase separately as well as improvements over classical strategies. Our strategy provides an advantage in the variance of the estimation over individual quantum estimation schemes that scales as $\mathcal{O}(d)$, where $d$ is the number of phases. Finally, we study the attainability of this limit using realistic probes and photon-number-resolving detectors. This is a problem in which an intrinsic advantage is derived from the estimation of multiple parameters simultaneously.

Figure 1

Discretized phase imaging model. We consider the simultaneous estimation of $d$ phases using a setup consisting of state preparation (green), independent phase application in each mode (blue), and state measurement (purple).

Figure 2

Strategies for multiple phase estimation using $N=16$ photons. The red line gives the total variance $|\Delta {\mathit{\theta }}_s{|}^2$ for the quantum simultaneous strategy using the states $|{\psi }_s⟩$, the blue dashed line gives the variance $|\Delta {\mathit{\theta }}_{\mathrm{ind}}{|}^2$ achievable using $N00N$ states, and the cyan dashed line gives the variance $|\Delta {\mathit{\theta }}_{\mathrm{clas}}{|}^2$ for an equivalent classical state.

Figure 3

(a) Realistic probes: The green line gives numerical calculations of the total variance from the QCRB for the simultaneous estimation of 4 phases using $\mathrm{HB}(n,4)$ states as a function of $n$. For comparison, the blue and red dashed lines give the QCRB for equivalent $N00N$ and $|{\psi }_s⟩$ states, respectively. (b) Realistic measurements: The green dots show the total variance for the simultaneous estimation of $d$ phases using a $\mathrm{HB}(1,d)$ state and a measurement apparatus consisting of a Fourier multiport followed by PNRDs. The green line gives the QCRB variance error for the same $\mathrm{HB}(1,d)$ state, while the blue and red dashed lines again give the QCRB for equivalent $N00N$ and $|{\psi }_s⟩$ states, respectively.