Asymptotic nonclassical-state conversion by linear optics
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Problem
What is the optimal asymptotic conversion rate between finite-energy bosonic states under passive linear-optical protocols?
Let \(\rho\) and \(\sigma\) be finite-energy density operators on finitely many bosonic modes, with \(\sigma\) nonclassical. Finite energy means finite total mean photon number. Classical states form the trace-norm closed convex hull of multimode coherent states. The allowed deterministic protocols \(\Lambda_k\) use passive linear-optical unitaries, ancillary coherent-state mixtures, destructive measurements, classical feed-forward, and discarding modes. Measurements may be non-Gaussian. The complete protocol is trace preserving; failure outcomes cannot be omitted.
Use the trace norm \(\lVert X\rVert_1:=\operatorname{Tr}\sqrt{X^\dagger X}\). A rate \(r\geq0\) is achievable if an allowed sequence \((\Lambda_k)_{k\geq1}\) satisfies
Determine \(R_{\mathrm{LO}}(\rho\to\sigma):=\sup\{r\geq0:r\text{ is achievable}\}\), with achievability defined by Eq. (1).
Source
This rate-evaluation question is formulated from Ferrari et al., Theorem 23 and Section IX. Their converse permits all classicality-preserving channels, whereas the question fixes passive optical protocols [FLTP23].
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Let \(\mathcal C\) be the trace-norm closed convex hull of coherent states. Define \(N_r(\omega):=\inf_{\tau\in\mathcal C}D(\omega\Vert\tau)\), and \(N_r^{\mathrm M}(\omega):=\inf_{\tau\in\mathcal C}\sup_M D(M(\omega)\Vert M(\tau))\). Here \(D\) is relative entropy with base-two logarithms and \(M\) ranges over measurements. When the ratio is well defined,
\begin{equation} R_{\mathrm{LO}}(\rho\to\sigma)\leq\frac{N_r(\rho)}{N_r^{\mathrm M}(\sigma)}. \tag{2} \end{equation}Theorem 23 proves Eq. (2) for the larger class of classicality-preserving channels, and therefore applies to the specified linear-optical protocols. [FLTP23]
Nonzero achievable rates are known: for the Fock state \(|j\rangle\) with \(j\) photons and \(\omega_{j,p}:=p|j\rangle\langle j|+(1-p)|0\rangle\langle0|\), where \(j\geq2\) and \(0<p\leq1\),
\begin{equation} p\leq R_{\mathrm{LO}}(\omega_{j,p}\to|j-1\rangle\langle j-1|) \leq\frac{p\log_2(j!e^j/j^j)}{\log_2((j-1)!e^{j-1}/(j-1)^{j-1})}, \tag{3} \end{equation}Proposition 47 gives Eq. (3). Its upper bound tends to \(p\) as \(j\to\infty\). [FLTP23]
Comment
Known achievable and converse bounds do not determine the rate for arbitrary state pairs. Classicality-preserving channels and passive optical protocols need not have the same optimal rates.