Nonclassicality of Mixed States with Photon Number Coherence

TL;DR

This work provides exact analytical results for the operational resource theory (ORT) nonclassicality measure and metrological power for a broad class of rank-two bosonic mixed states, including those with photon-number coherence, and develops numerical methods to treat higher-rank states. It reveals that photon-number coherence generally enhances nonclassicality and metrological usefulness, but the relationship is nuanced, with piecewise regimes and entanglement-sudden-death–like plateaus under dephasing. The study unifies purity, coherence, and metrological advantages within a convex-roof framework and demonstrates regimes where the metrological power saturates or falls short of the ORT bound. These insights advance understanding of nonclassical resources in realistic, mixed quantum states and have implications for designing quantum optical sensors with coherence-managed resources.

Abstract

The operational resource theory (ORT) measure is a nonclassicality measure for bosonic states, notable for its resource-theoretic properties and connection to metrology. However, it can be difficult to evaluate, being linked to an optimization problem for mixed states. Here, we present the first ORT measure calculations for mixed states with photon number coherence. We give exact formulas governing the ORT measure of a broad class of rank-two mixed states, and numerical solutions for some higher-rank states. We also compare the nonclassicality of these states to their metrological power, thus showing in what regimes the metrological power manages to saturate the ORT bound. Throughout, we consider the role of coherence. In particular, we show that nonclassicality and metrological power never increase under bosonic dephasing, but may plateau in a manner similar to entanglement sudden death. Nevertheless, lowering photon number coherence more freely can sometimes yield more nonclassical and metrologically useful states.

Figures (13)

• Figure 1: Effect of dephasing $\ket{n+m}\bra{n}\rightarrow e^{-im\omega_0t}e^{-|m|\gamma t}\ket{n+m}\bra{n}$ (as due to a Lorentzian frequency profile) on the ORT measure $\mathcal{N}(\hat{\rho})$ for states that are, at $t=0$, pure superpositions $\sqrt{p_{n+m}}\ket{n+m}+\sqrt{1-p_{n+m}}e^{i\chi}\ket{n}$. Particular values for the $m=1$ curve are: $n=9$, $p_{10}=0.5$. For the $m=2$ curve: $n=6$, $p_8=0.75$. For the $m>2$ curve: $m=3$, $n=3$, and $p_6=0.5$.
• Figure 2: Six scenarios given the inequalities controlling $\mathcal{N}$ and $\mathcal{M}$ in section \ref{['sec:Rank2NoCoherence']}. The solid (blue) curve represents $p(1-p)(r_{12}+r_{21})^2$, the dashed (red) curve represents $4p(1-p)r_{12}r_{21}$, and the dotted (gray) lines represent possible behaviors for $|\braket{\hat{a}^2}|$. Note that if $r_{12}=r_{21},$ the solid (blue) and dashed (red) curves coincide.
• Figure 3: The ORT measure $\mathcal{N}$ (solid blue) and the metrological power $\mathcal{M}$ (dashed black) of mixed even and odd cat states, as a function of the population $p$ of the even cat state for (a) $\alpha=0.5$ and for (b) $\alpha=700$. $\mathcal{N}_{+}=|\alpha|^2(\tanh{|\alpha|^2}+1)$ and $\mathcal{N}_{-}=|\alpha|^2(\coth{|\alpha|^2}+1)$ are the nonclassicalities of the pure even and odd cat states, which approach one another as $|\alpha|\rightarrow\infty$.
• Figure 4: ORT measure $\mathcal{N}$ and metrological power $\mathcal{M}$ of mixtures of squeezed vacuum and photon-subtracted squeezed vacuum for a squeezing parameter below ($\gamma=0.1$) and above ($\gamma=0.3$) $\gamma_{\text{crit}}\approx0.19193$, as a function of the population $p$ of the squeezed vacuum state. To clarify, the ORT measure for states with $\gamma=0.1$ does not reach zero for any $p$.
• Figure 5: Fig. \ref{['fig:rank2ORT']} shows the ORT measure $\mathcal{N}$ (solid, blue) and metrological power $\mathcal{M}$ (dashed, black) of mixtures of level-skipped states with $n=1$ and $|\beta_2|=\sqrt{0.75}$, as a function of the population $p$ of $\ket{\phi_1}$. Fig. \ref{['fig:rank2ORT_bvar']} shows the variation of $(p_L,p_R)$ with $|\beta_2|$ when $n=1$. The black dotted line represents $(p_L,p_R)\approx(0.234823,0.890667)$ for the case shown in \ref{['fig:rank2ORT']}.