Quantum metrology with imperfect states and detectors

Quantum enhancements of precision in metrology can be compromised by system imperfections. These may be mitigated by appropriate optimization of the input state to render it robust, at the expense of making the state difficult to prepare. In this paper, we identify the major sources of imperfection of an optical sensor: input state preparation inefficiency, sensor losses, and detector inefficiency. The second of these has received much attention; we show that it is the least damaging to surpassing the standard quantum limit in a optical interferometric sensor. Further, we show that photonic states that can be prepared in the laboratory using feasible resources allow a measurement strategy using photon-number-resolving detectors that not only attain the Heisenberg limit for phase estimation in the absence of losses, but also deliver close to the maximum possible precision in realistic scenarios including losses and inefficiencies. In particular, we give bounds for the tradeoff between the three sources of imperfection that will allow true quantum-enhanced optical metrology

Figure 1

A schematic interferometer involving HB($N$) states. BS1 and BS2 are 50/50 beam splitters, $\varphi$ denotes the phase shift of mode $c$, and PNRD is a photon-number-resolving detector. $\eta$ is the loss in the interferometer arm, while ${\eta }_p$ and ${\eta }_d$ are the preparation and detection imperfections. $\eta ={\eta }_p={\eta }_d=1$ denotes a perfect setup.

Figure 2

QFI for phase estimation as a function of the transmissivity $\eta$ for 20 input photons. Blue (dotted line): standard quantum limit. Red (dashed line): HB(10) states. Black (solid line): $N00N$ states. Green (dotted-dashed line): optimal states [7]. Inset: QFI for phase estimation as a function of the photon number $N$ for $\eta =0.9$ (top) and $\eta =0.6$ (bottom).

Figure 3

Plot of the feasibility region for beating the standard quantum limit using HB(2) states in the parameter space of preparation inefficiency, interferometer loss, and detector inefficiency ${\eta }_p,\hspace{0.5em}\eta ,\hspace{0.5em}\text{and}\hspace{0.5em}{\eta }_d$, respectively. The bottleneck in beating the standard quantum limit is the detector imperfection, followed by the preparation imperfection, and, lastly, losses in the interferometer.

Figure 4

Plot of the feasibility region for beating the standard quantum limit using HB$(1)$ states (left) in the parameter space of preparation inefficiency, interferometer loss, and detector inefficiency ${\eta }_p,\hspace{0.5em}\eta ,\hspace{0.5em}\text{and}\hspace{0.5em}{\eta }_d$, respectively. The bottleneck in beating the standard quantum limit is the detector imperfection, followed by the preparation imperfection, and, lastly, losses in the interferometer. Same for HB$(3)$ states (right).