Indistinguishability-enabled coherence for quantum metrology

Quantum coherence plays a fundamental and operational role in different areas of physics. A resource theory has been developed to characterize the coherence of distinguishable particles systems. Here we show that indistinguishability of identical particles is a source of coherence, even when they are independently prepared. In particular, under spatially local operations, states that are incoherent for distinguishable particles, can be coherent for indistinguishable particles under the same procedure. We present a phase discrimination protocol, in which we demonstrate the operational advantage of using two indistinguishable particles rather than distinguishable ones. The coherence due to the quantum indistinguishability significantly reduces the error probability of guessing the phase, using the most general measurements. The role played by particle statistics in the protocol is also investigated.

Figure 1

CNOT gate for identical particles in the sLOCC framework.

Figure 2

Phase discrimination game.

Figure 3

(a) Probability of error in function of the phase difference ${\varphi }_{12}$, with $p_1=1/3$, ${\left|l\right|}^2={\left|r^{\prime }\right|}^2=\frac{1}{2}$, and ${\omega }_{\downarrow \uparrow }-{\omega }_{\uparrow \downarrow }=1$. The orange dashed line corresponds to $P_{\mathrm{err}}$ [Eq. (12)] when there is no overlap ($l^{\prime }r=0$) or, analogously, to the case with distinguishable particles. The blue one corresponds to ${\tilde{P}}_{\mathrm{err}}$ [Eq. (13)], i.e., to the case of spatial overlap with $|l^{\prime }{|}^2={\left|r\right|}^2=\frac{1}{2}$. (b) Contour plot of $P_{\mathrm{err}}$ [Eq. (12)] in function of $r$ and $l^{\prime }$, with ${\varphi }_{12}=\pi$.

Figure 4

Probability of error in function of ${\varphi }_{12}$ with ${\left|l\right|}^2={\left|r^{\prime }\right|}^2=1/2$ for identical particles initially prepared in the state of Eq. (14): distinguishable (blue solid line) and indistinguishable with $|l^{\prime }{|}^2={\left|r\right|}^2=1/2$ (orange-dashed line for bosons and green dotted line for fermions), with $a=b$, ${\omega }_{\downarrow \uparrow }=3$, ${\omega }_{\uparrow \downarrow }=2$ and ${\omega }_{\downarrow \downarrow }=1$.

Figure 5

Three-dimensional (3D) plot of the probability of error with ${\left|l\right|}^2={\left|r^{\prime }\right|}^2=1/2$ for identical particles initially prepared in the state of Eq. (14): distinguishable (blue) and indistinguishable with $|l^{\prime }{|}^2={\left|r\right|}^2=1/2$ (orange for bosons and green for fermions) in function of ${\varphi }_{12}$ and of ${\omega }_{\downarrow \downarrow }$, with ${\left|l\right|}^2={\left|r^{\prime }\right|}^2=1/2$, $a=b$, ${\omega }_{\downarrow \uparrow }=3$, ${\omega }_{\uparrow \downarrow }=2$.