Quantum phase estimation with lossy interferometers

We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.

Figure 1

(Color online) Interferometric phase estimation scheme. An input state is fed into an interferometer consisting of two channels. Channel $a$ acquires a phase $\phi$ relative to channel $b$. Measurements are performed on the output state yielding an estimated value, ${\phi }_{est}$, of the phase $\phi$. The beam splitters symbolize photon losses.

Figure 2

(Color online) Parameters of the optimal state: $|\psi ⟩={\sum }_{i=0}^N\sqrt{x_i}|i,N-i⟩$, for phase estimation with $N=10$ photons for the case of losses in one only arm, i.e., ${\eta }_a=\eta$, ${\eta }_b=1$.

Figure 3

(Color online) The minimal phase uncertainty achieved with various $N=10$ photon states for losses in one arm, i.e., ${\eta }_a=\eta$, ${\eta }_b=1$. Red, dashed line: $N00N$ state; blue, solid line: optimal state (cf. Fig. 2); bright green, dotted line: optimal two-component state; dark green, dashed-dotted line: $N00N$ chopping strategy. The shaded area is bounded by the SIL achievable with classical states and the Heisenberg limit $1/N$.

Figure 4

(Color online) Parameters of the optimal state: $|\psi ⟩={\sum }_{i=0}^N\sqrt{x_i}|i,N-i⟩$, for phase estimation with $N=10$ photons for the case of equal losses in both arms, i.e., ${\eta }_a={\eta }_b=\eta$. Due to this symmetry we have $x_k=x_{N-k}$.

Figure 5

(Color online) The minimal phase uncertainty achieved with various $N=10$ photon states for equal losses in both arms, i.e., ${\eta }_a={\eta }_b=\eta$. Red, dashed line: $N00N$ state; blue, solid line: optimal state (cf. Fig. 4); bright green, dotted line: the state optimal for lossless phase estimation in the global approach [see Eq. (48)]; dark green, dashed-dotted line: the $N00N$ chopping strategy. The shaded area is bounded by the SIL achievable with classical states and the Heisenberg limit $1/N$.