Gaussian quantum estimation of the loss parameter in a thermal environment

In the task of estimating the parameter η, we consider the case where an experimentalist prepares M i.i.d. copies ρ_η of an idler-signal system. Quantum metrology aims to answer the question: how precise does quantum mechanics allow to estimate the value of η? For this aim, it is useful to introduce the concept of estimator. This is an algorithm that, given the measurement outcomes {x} = {x₁, ..., x_M } as input, provides an output $\hat{\eta }(\left\{\mathbf{x}\right\})$ approximating the value of η. Examples of estimators range from sample mean calculations to more complex post-processing analysis, such as maximum likelihood or non-trivial machine-learning based estimations. Given the stochastic nature of a measurement, {x} is a random variable and $\hat{\eta }$ shall be considered as a random variable as well. In the following, we focus on unbiased estimators, defined by $\mathbb{E}[\hat{\eta }]=\eta$. Moreover, we use the 'mean square error' (MSE), i.e., ${V}_{\hat{\eta }}(\eta ,\left\{\mathbf{x}\right\})=\mathbb{E}[{(\hat{\eta }(\left\{\mathbf{x}\right\})-\eta )}^{2}]$, as uncertainty measure of the estimator $\hat{\eta }$. Notice that other uncertainty measures can be considered, each one having different interpretations [41].

In quantum mechanics, a generic measurement can be represented by a positive operator-valued measure (POVM), i.e., a set of positive semi-definite self-adjoint operators {Π_x } that sum to the identity operator, that is ${\sum }_{x}{{\Pi}}_{x}=\mathbb{I}$. The set can be either discrete, as in the case of a photon-resolving measurement, or continuous, as in the case of homodyne or heterodyne measurement. The Born rule gives us the probability densities for the measurement outcome, i.e., p(x|η) = Tr (Π_x ρ_η ). Let us introduce the Fisher information (FI) ${H}_{\eta }(\eta ,\left\{{{\Pi}}_{x}\right\})=\int p(x\vert \eta ){({\partial }_{\eta }\mathrm{ln}\,p(x\vert \eta ))}^{2}\,\mathrm{d}x$. The Cramér–Rao bound sets a limit on the minimal MSE. Given a POVM {Π_x } and its corresponding measurement outcomes {x}, we have

$$\begin{equation}{V}_{\hat{\eta }}(\eta ,\left\{\mathbf{x}\right\})\geqslant {(M{H}_{\eta }(\eta ,\left\{{{\Pi}}_{x}\right\}))}^{-1}\end{equation} \tag{ 5 }$$

for any estimator $\hat{\eta }$. This bound is always attained by a maximum likelihood estimator in the M → ∞ limit [42]. We shall mention that the Cramér–Rao bound holds strictly for unbiased estimators. There are cases where asymptotically unbiased estimators, for which $\mathbb{E}[\hat{\eta }]\to \eta$ only for infinite M, can achieve lower MSE than any unbiased estimator.

Let us define the symmetric logarithmic derivative (SLD) L_η as the self-adjoint operator satisfying the equation ∂_η ρ_η = (L_η ρ_η + ρ_η L_η )/2. Notice that the SLD is unique in supp(ρ_η ) (i.e., the support of ρ_η ), while the projector of L_η on supp${({\rho }_{\eta })}^{\perp }$ can be defined arbitrarily [20]. This non-uniqueness property does not affect the discussion that follows, as the important feature is that ρ_η L_η and L_η ρ_η are unique. One can prove the inequality [19] ${H}_{\eta }(\eta ,\left\{{{\Pi}}_{x}\right\})\leqslant \mathrm{Tr}\,({L}_{\eta }^{2}{\rho }_{\eta })\equiv {I}_{\eta }(\eta )$, holding for any {Π_x }. By using the latter inequality and the Cramér–Rao bound in equation (5), we get a bound indicating the ultimate precision allowed by quantum mechanics for a generic unbiased estimator:

$$\begin{equation}{V}_{\hat{\eta }}(\eta ,\left\{\mathbf{x}\right\})\geqslant {(M{I}_{\eta }(\eta ))}^{-1}\end{equation} \tag{ 6 }$$

for any $\hat{\eta }$ and {x}. The quantity I_η (η) is the QFI, and equation (6) is called quantum Cramér–Rao bound. This bound can be saturated by choosing a POVM $\left\{{\tilde{{\Pi}}}_{x}\right\}$ projecting on the eigenbasis of L_η [19], for which the equality ${I}_{\eta }(\eta )={H}_{\eta }(\eta ,\left\{{\tilde{{\Pi}}}_{x}\right\})$ holds. Indeed, one can see the QFI as the maximal achievable FI:

$$\begin{equation}{I}_{\eta }(\eta )=\underset{\left\{{{\Pi}}_{x}\right\}}{\mathrm{max}}{H}_{\eta }(\eta ,\left\{{{\Pi}}_{x}\right\})=\underset{\text{proj.}\;\left\{{{\Pi}}_{x}\right\}}{\mathrm{max}}{H}_{\eta }(\eta ,\left\{{{\Pi}}_{x}\right\})={H}_{\eta }(\eta ,\left\{{\tilde{{\Pi}}}_{x}\right\}).\end{equation} \tag{ 7 }$$

The optimization can be done with respect to projective POVMs, given that the maximal FI is achieved by a projective POVM. Notice that a maximum likelihood estimator applied on the measurement outcomes of the projective POVM $\left\{{\tilde{{\Pi}}}_{x}\right\}$ saturates the quantum Cramér–Rao bound in the limit of large M. However, it is not excluded the existence of different POVMs, even non-projective ones, and estimators attaining this bound.

We shall now mention that the POVMs optimizing equation (7) generally depend on the value of η. This creates a conundrum, as it may seem that knowing the optimal POVM for estimating a parameter requires a prior knowledge of the parameter value itself. This problem is solved as follows. Let us assume that η ∈ (η₀ − δη₀, η + δη₀) with high probability, where both η₀ and δη₀ are known and δη₀ is small. Then, an optimal estimation procedure consists in the iteration of the following steps:

• (a)  
  Measure a large number of times the POVM maximizing H_η (η₀, {Π_x }). This saturates the QFI I_η (η₀).
• (b)  
  Estimate the parameter η with an estimator saturating the Cramér–Rao bound, such as the maximum likelihood estimator. This provides a new estimated value η ∈ (η₁ − δη₁, η₁ + δη₁), where η₁ ∈ (η₀ − δη₀, η₀ + δη₀) and δη₁ ≪ δη₀.

This procedure requires an initial accurate enough knowledge of the η value. This knowledge can be obtained by measuring a non-optimal POVM that provides a rough estimation of η, i.e., η₀, within an uncertainty δη₀.

In the following, we drop the subscript η in the QFI, and denote I, I^IF and I^EA as the QFIs for a generic multi-mode, single-mode and two-mode states, respectively. We denote the zero temperature case (N_B = 0) with the superscript '(0)'. For instance, I^{IF, (0)} is the generic idler-free (or single-mode) QFI for N_B = 0, and with the subscript 'norm' the QFI of the normalized environment. Moreover, we call the total QFI $\mathcal{I}$ the QFI of the M-fold input state, and use the same superscript and subscript notations as for the QFI. In the case of M i.i.d. copies, the total QFI reduces to $\mathcal{I}=MI$. Similarly, we call ${\mathcal{N}}_{S}$ the total input power, which in the i.i.d. case can be expressed as ${\mathcal{N}}_{S}=M{N}_{S}$.

Regarding the estimation of the loss parameter, the following results are known in the literature.

Lemma 1 [ 34 ]. The total QFI of a generic multi-mode probe in the zero-temperature environment case is bounded as ${\mathcal{I}}^{(\mathrm{0})}\leqslant \frac{4{\mathcal{N}}_{S}}{1-{\eta }^{2}}$ for any η ∈ [0, 1) and total power ${\mathcal{N}}_{S}\geqslant 0$.

Lemma 1 can be extended to the normalized environment case.

Lemma 2 [ 40 ]. The total QFI of a generic multi-mode probe in the normalized environment case is bounded as ${\mathcal{I}}_{\mathrm{norm}}\leqslant \frac{4{\mathcal{N}}_{S}}{{N}_{B}+1-{\eta }^{2}}$ for any η and total power ${\mathcal{N}}_{S}\geqslant 0$.

Lemmas 1 and 2 are proven in the respective references for generic multi-mode probe, i.e., not necessarily i.i.d. Considering i.i.d. probe states is justified both experimentally and theoretically: i.i.d. probes are easier to generate in a lab and, as we will see, are a sufficient resource to saturate the bounds in lemmas 1 and 2. We notice that by setting M = 1, one can derive viable bounds for the QFI, i.e., ${I}^{(\mathrm{0})}\leqslant \frac{4{N}_{S}}{1-{\eta }^{2}}$ and ${I}_{\mathrm{norm}}\leqslant \frac{4{N}_{S}}{{N}_{B}+1-{\eta }^{2}}$. This means that the standard quantum limit, for which the asymptotic scaling I ∼ N_S holds, cannot be overcome. These bounds can be used to test the optimality of Gaussian probes for the QFI. Lastly, we notice that a similar bound for the unnormalized thermal environment is not available, as explained in section 4.

As mentioned in section 2.1, Gaussian states can be faithfully represented by the covariance matrix $\tilde{\mathbf{\Sigma }}$ and the first-moment vector $\tilde{\mathbf{d}}$ ⁴ . If the channel is Gaussian-preserving and the input state is Gaussian, then the manifold of output state parameterized by any of the dynamics parameters (in our case η) defines a Gaussian manifold. The QFI on this Gaussian manifold for the estimation of the parameter η is given by [25, 26]

$$\begin{equation}I=\mathrm{Tr}\left\{{\mathbf{L}}_{2}{\partial }_{\eta }\tilde{\mathbf{\Sigma }}\right\}+{({\partial }_{\eta }\tilde{\mathbf{d}})}^{\top }{\tilde{\mathbf{\Sigma }}}^{-1}({\partial }_{\eta }\tilde{\mathbf{d}}),\end{equation} \tag{ 8 }$$

where L₂ is the quadratic form of the SLD, and ${\tilde{\mathbf{\Sigma }}}^{-1}$ is the pseudoinverse of $\tilde{\mathbf{\Sigma }}$. The SLD quadratic form is the solution to the equation $4\tilde{\mathbf{\Sigma }}{\mathbf{L}}_{2}\tilde{\mathbf{\Sigma }}+\mathbf{\Omega }{\mathbf{L}}_{2}\mathbf{\Omega }=2\,{\partial }_{\eta }\tilde{\mathbf{\Sigma }}$. Since the idler-free protocol involves only single-mode states, in this case the QFI can alternatively be expressed as [26]

$$\begin{equation}{I}^{\text{IF}}=\frac{\mathrm{Tr}\left\{{\left({\tilde{\mathbf{\Sigma }}}^{-1}{\partial }_{\eta }\tilde{\mathbf{\Sigma }}\right)}^{2}\right\}}{2(1+{\mu }^{2})}\ +\frac{2{({\partial }_{\eta }\mu )}^{2}}{1-{\mu }^{4}}+{({\partial }_{\eta }\tilde{\mathbf{d}})}^{\top }{\tilde{\mathbf{\Sigma }}}^{-1}({\partial }_{\eta }\tilde{\mathbf{d}}),\end{equation} \tag{ 9 }$$

where $\mu ={\left(4\mathrm{det}\tilde{\mathbf{\Sigma }}\right)}^{-1/2}$ is the purity of the single-mode quantum state.

Since we are considering the estimation of a parameter embedded in a completely positive and trace preserving map, the QFI is convex [43], and therefore maximized by a pure-state input. We then consider pure-state probes for both the idler-free and entanglement-assisted strategies. Finally, we notice that the QFI for the estimation of η can be used to compute the ultimate precision limit for the estimation of γ via the relation ${I}_{\gamma }(\gamma )=\frac{{t}^{2}}{4}{\mathrm{e}}^{-\gamma t}{I}_{\eta }(\eta ={\mathrm{e}}^{-\gamma t/2})$.

This case has been studied in references [30, 32] in the Gaussian case. Here, we derive novel analytical results for the optimal states in the power-constrained case. In this case, the QFI takes a relatively simple form:

$$\begin{align}\hfill {I}^{\mathrm{IF,}\;(\text{0})}& =4{N}_{S}\left\{\frac{1-\xi }{1-2{\eta }^{2}\left(\sqrt{\xi {N}_{S}(1+\xi {N}_{S})}-\xi {N}_{S}\right)}\right.\hfill \\ \hfill & \quad \left.+\frac{\xi \left[{(1-{\eta }^{2})}^{2}+{\eta }^{4}\right]}{(1-{\eta }^{2})(1+2\xi {N}_{S}{\eta }^{2}(1-{\eta }^{2}))}\right\}.\hfill \end{align} \tag{ 13 }$$

Our task consists in finding ξ that optimizes I^{IF, (0)} for given values of N_S and η. This problem can be solved numerically for arbitrary parameter values, see figure 2. In the following, we seek to find the analytical behaviour of the optimal probe, as this aspect has not been studied in previous related works, i.e., references [30, 32].

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Generally speaking, both displacement and squeezing are essential for achieving optimality. However, it is interesting to look for the regimes where squeezing or displacement alone are the optimal probes. In figure 2 we can see a transition between ξ^opt = 1 and ξ^opt < 1. The following proposition characterizes this transition.

Proposition 1 [ Squeezed-vacuum state as optimal probe (N_B =0)]. ξ^opt = 1 if and only if ${N}_{S}\leqslant {\bar{N}}_{S}(\eta )$. Here,

$$\begin{equation}{\bar{N}}_{S}(\eta )=\begin{cases}0\quad \hfill & \quad \eta \leqslant 1/\sqrt{2}\hfill \\ {\bar{N}}_{S}^{(0)}(\eta )\quad \hfill & \quad \eta > 1/\sqrt{2},\hfill \end{cases}\end{equation} \tag{ 14 }$$

where ${\bar{N}}_{S}^{(0)}(\eta )$ is the only zero of ${f\!}_{1}(\eta ,{N}_{S})=\frac{1}{4{N}_{S}}{\left({\partial }_{\xi }{I}^{\mathrm{IF,}\;(\text{0})}\right)}_{\vert \,\xi =1}$.

Proof. In appendix A.2 we show that I^{IF, (0)} is concave in ξ, provided that η ≠ 0, see appendix A.2. This means that ξ = 1 is a maximum point if and only if ${f\!}_{1}(\eta ,{N}_{S}){:=}\frac{1}{4{N}_{S}}{\left({\partial }_{\xi }{I}^{\mathrm{IF,}\;(\text{0})}\right)}_{\vert \,\xi =1}\geqslant 0$. The function f₁ has at most one zero, as its derivative in N_S is negative everywhere, see appendix A.2. Since ${f\!}_{1}\to -\frac{1}{1-{\eta }^{2}}< 0$ for N_S → ∞, the zero ${\bar{N}}_{S}^{(0)}(\eta )$ is positive only if ${f\!}_{1}(\eta ,{N}_{S}=0)=\frac{2{\eta }^{2}-1}{1-{\eta }^{2}}$ is positive, which is the case for $\eta \in \left(\frac{1}{\sqrt{2}},1\right)$. □

Proposition 1 implies that the squeezed-vacuum state is never optimal for $\eta \leqslant \frac{1}{\sqrt{2}}$, or if the input power N_S is large enough. More precisely, the squeezed-vacuum state is optimal only for $\eta \geqslant \bar{\eta }({N}_{S})$, where $\bar{\eta }({N}_{S})$ is the inverse of ${\bar{N}}_{S}^{(0)}(\eta )$. The curve defined by f₁ = 0 can be computed numerically, and an analytical expansion can be derived using perturbation theory. For instance, a perturbation expansion to the first order gives us $\bar{\eta }\simeq 1-\frac{1}{c{N}_{S}}$, with c ≃ 8.86, for N_S ≫ 1, and $\bar{\eta }\simeq \frac{1}{\sqrt{2}}\left(1+\frac{\sqrt{{N}_{S}}}{2}\right)$ for N_S ≪ 1, see appendix A.3.

Understanding whether coherent states performs optimally in certain regimes is important, as these states are a close representation of a classical signal. Due to this property, many sensing protocols are compared with respect to coherent states in order to claim a quantum advantage, see references [6, 7] among others. The following is a no-go result for the coherent state as optimal probe.

Proposition 2 [No-go theorem for the coherent state as optimal probe ( N_B =0)]. The coherent state (ξ = 0) cannot be the optimal probe for any η > 0.

Proof. Due to the concavity of I^{IF, (0)} for η ≠ 0, the coherent state is optimal if and only if $\frac{1}{4{N}_{S}}{\left({\partial }_{\xi }{I}^{\mathrm{IF,}\;(\text{0})}\right)}_{\vert \,\xi =0}\leqslant 0$. However, we have $\frac{1}{4{N}_{S}}{\left({\partial }_{\xi }{I}^{\mathrm{IF,}\;(\text{0})}\right)}_{\vert \,\xi =0}=\frac{\sqrt{{N}_{S}}{\eta }^{2}}{\sqrt{\xi }}+\mathcal{O}(1)$ for ξ → 0, which is strictly positive for any η > 0. □

Let us now investigate the η → 0 limit, and show that there are power regimes where coherent state is not optimal even in this limit. We have

$$\begin{equation}{I}^{\mathrm{IF,}\;(\text{0})}=4{N}_{S}\left\{1+{g}_{1}(\xi ,{N}_{S}){\eta }^{2}\right\}+\mathcal{O}({\eta }^{4}),\quad \mathrm{a}\mathrm{s}\enspace \eta \to 0,\end{equation} \tag{ 15 }$$

where ${g}_{1}(\xi ,{N}_{S})=2(1-\xi )\sqrt{\xi {N}_{S}(1+\xi {N}_{S})}-\xi (1+2{N}_{S})$. In the N_S ≪ 1 and N_S ≫ 1 regimes, the function g₁(ξ, N_S ) is always negative, and decreasing with respect to ξ. This implies that the coherent state, corresponding to ξ = 0, is optimal in these limits. However, for intermediate values of N_S , the function g₁(ξ, N_S ) is positive for some finite ξ, meaning that the QFI is maximized for a displaced squeezed state. This behaviour of the QFI is clearly visible in figure 2.

We now move the discussion to the regimes where non-trivial displaced squeezed states optimize the QFI. In particular, we are interested in the high- and low-power regimes, where some interesting properties emerge. In the large power regime, we have

$$\begin{equation}{I}^{\mathrm{IF,}\;(\text{0})}=4{N}_{S}\frac{(1-\xi )}{1-{\eta }^{2}}+\mathcal{O}(1),\quad \mathrm{a}\mathrm{s}\enspace \xi \enspace {N}_{S}\to \infty .\end{equation} \tag{ 16 }$$

In this limit the optimal squeezing is infinitesimal, i.e., ξ^opt → 0. However, ξ^opt cannot be exactly zero, otherwise the ${(1-{\eta }^{2})}^{-1}$-scaling of the QFI disappears, as one can see using equation (13). By expanding equation (16) to the next order in ξN_S , we derive the asymptotic value ${\xi }^{\mathrm{opt}}\sim \eta /{[4{N}_{S}(1-{\eta }^{2})]}^{1/2}$, see appendix A.4. This asymptotic expansion holds for N_S ≫ η²/(1 − η²). By expanding ξ^opt around η = 1, we obtain ${\xi }^{\mathrm{opt}}\sim {[8{N}_{S}(1-\eta )]}^{-1/2}$ as in reference [30]. Interestingly, this means that in the N_S ≫ 1 regime, an infinitesimal amount of squeezing ensures the optimality of the QFI. Notice also that equation (16) virtually saturates the bound in lemma 1. Therefore, the single-mode state is asymptotically an optimum among generic multi-mode probes.

In the low-power regime, we have

$$\begin{equation}{I}^{\mathrm{IF,}\;(\text{0})}=4{N}_{S}\left\{(1-\xi )+\frac{\xi \left[{(1-{\eta }^{2})}^{2}+{\eta }^{4}\right]}{(1-{\eta }^{2})}\right\}+\mathcal{O}({N}_{S}^{3/2}),\quad \mathrm{a}\mathrm{s}\enspace {N}_{S}\to 0.\end{equation} \tag{ 17 }$$

This is a linear quantity in ξ, meaning that in this limit there is an abrupt change in the optimal ξ: ξ^opt = 0 for $\eta < 1/\sqrt{2}$, and ξ^opt = 1 otherwise. We will see that this transition is even more evident in the finite temperature case, corresponding to N_B > 0.

Finally, in the intermediate power regime, a finite squeezing is always a resource in the quantum estimation task. This is the case even for small η, as shown in equation (15). In figure 2 we see that this happens especially in the 10⁻¹ ≲ N_S ≲ 10 regime. However, figure 3 tells us that the advantage is minimal for $\eta \lesssim 1/\sqrt{2}$, and it becomes increasingly relevant only for η approaching one.

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The finite temperature case presents a passive signature, consisting in the vacuum having metrological power [44]:

$$\begin{equation}I({N}_{S}=0)=\frac{4{\eta }^{2}{N}_{B}}{\left(1-{\eta }^{2}\right)\left[1+{N}_{B}\left(1-{\eta }^{2}\right)\right]}\equiv {I}_{\text{shad}}.\end{equation} \tag{ 18 }$$

This is an effect appearing for η, N_B > 0, and it is present for a generic multi-mode state. We call this 'shadow-effect' [45], and denote its contribution to the QFI as I_shad, as in equation (18). The following results complement the analysis done in reference [32], by providing the optimal QFI and input state in the asymptotically high-brightness regime (N_S ≫ 1). In addition, we recognize an abrupt transition between ξ^opt = 0 and ξ^opt = 1 in the low-brightness regime (N_S ≪ 1).

In order to gain an intuition on the optimal probe, let us first discuss the QFI of two topical states: coherent and squeezed-vacuum states. For a coherent state as input, the QFI can be written in a closed form as

$$\begin{equation}{I}^{\text{IF}}(\xi =0)={I}_{\text{shad}}+\frac{4{N}_{S}}{1+2{N}_{B}\left(1-{\eta }^{2}\right)}\equiv {I}^{\text{coh}}.\end{equation} \tag{ 19 }$$

For a squeezed-vacuum state probe, we have a lengthy expression for the QFI, that we denote as I^IF(ξ = 1) ≡ I^sq, see appendix A.1. In figure 2, we see the presence of a clear region where ξ^opt = 1. This feature is similar to what proved in proposition 1 in the zero temperature case. In the large squeezing regime, the QFI saturates to a η-dependent value:

$$\begin{equation}{I}^{\text{sq}}=\frac{2{(1-{\eta }^{2})}^{2}+2{\eta }^{4}}{{\eta }^{2}{(1-{\eta }^{2})}^{2}}+\mathcal{O}({N}_{S}^{-1}),\quad \mathrm{a}\mathrm{s}\enspace {N}_{S}\to \infty .\end{equation} \tag{ 20 }$$

This limit holds for N_S ≫ (1 + N_B )/[η²(1 − η²)]. Let us investigate I^sq at the diverging points of equation (20). The analysis of the different regimes is complicated by the fact that the order of different limits do not commute. However, one can rely on Taylor analysis to understand which limit order corresponds to which regime of parameters, see appendix A.6 for a discussion on this. The limits η → 0 and N_S → ∞ do not commute, as ${I}^{\text{sq}}=\mathcal{O}({\eta }^{2})$ while η = 0 is a diverging point of equation (20). This is due to the fact that equation (20) holds for N_S η² ≫ 1, while ${I}^{\text{sq}}=\mathcal{O}({\eta }^{2})$ holds for N_S η² ≪ 1. Instead, at η = 1 we have

$$\begin{equation}{I}^{\text{sq}}=\frac{2{N}_{S}(1+2{N}_{B})+2{N}_{B}}{1-\eta }+\mathcal{O}(1),\quad \mathrm{a}\mathrm{s}\enspace \eta \to 1.\end{equation} \tag{ 21 }$$

At first glance, this may seem in contrast with equation (20), as equation (21) is unbounded with respect to N_S . Indeed, as a Taylor analysis reveals, equation (20) is valid for N_S (1 − η) ≫ 1 while equation (21) holds for N_S (1 − η) ≪ 1. This means that, alike the coherent states, squeezed-vacuum states do not asymptotically reach the standard quantum limit, as their QFI saturates for large enough N_S for any fixed value of η < 1, i.e., I^sq/N_S → 0 for N_S → ∞.

Let us now consider the general case of a displaced squeezed state probe. In figure 2, we see that squeezing can be a resource even when η is far from being one. In the low-power regime there is an abrupt transition from ξ^opt = 0 to ξ^opt = 1 at a certain value of η. This can be seen more clearly by expanding I^IF for small N_S :

$$\begin{equation}{I}^{\text{IF}}={I}_{\text{shad}}+4{N}_{S}\left\{\frac{1-\xi }{1+2{N}_{B}(1-{\eta }^{2})}+\xi {g}_{2}(\eta ,{N}_{B})\right\}+\mathcal{O}({N}_{S}^{3/2}),\quad \mathrm{a}\mathrm{s}\enspace {N}_{S}\to 0,\end{equation} \tag{ 22 }$$

where g₂(η, N_B ) is given in appendix A.5. It is clear that ξ^opt = 1 if ${g}_{2}(\eta ,{N}_{B}) > \frac{1}{1+2{N}_{B}(1-{\eta }^{2})}$, otherwise ξ^opt = 0. For instance, for large N_B , this abrupt change happens at $\eta \simeq 1-\frac{3}{2{N}_{B}}$, see appendix A.5.

In the large power regime, ξ^opt behaves similarly as in the N_B = 0 case, as shown in figure 2. More precisely, we have the following result for the asymptotic QFI, which generalizes (and includes) the N_B = 0 case.

Proposition 3 [ Optimal asymptotic QFI ( N_B =0)]. The optimal QFI in the large power regime is given by

$$\begin{equation}{I}^{\text{IF}}=\frac{4{N}_{S}(1-\xi )}{(1-{\eta }^{2})(1+2{N}_{B})}+\mathcal{O}(1),\quad \mathrm{a}\mathrm{s}\enspace \xi {N}_{S}\to \infty .\end{equation} \tag{ 23 }$$

Here, the optimal squeezing is given by ${\xi }^{\mathrm{opt}}\sim \eta /{[4{N}_{S}(1-{\eta }^{2})(1+2{N}_{B})]}^{1/2}$ for N_S ≫ η²/[(1 − η²)(1 + 2N_B )] and N_S ≫ N_B .

Proof. The Taylor expansion for large ξN_S is

$$\begin{align}\hfill {I}^{\text{IF}}& =\frac{4{N}_{S}(1-\xi )}{(1-{\eta }^{2})(1+2{N}_{B})}\left[1-\frac{{\eta }^{2}}{4{N}_{S}\xi (1-{\eta }^{2})(1+2{N}_{B})}\right]\hfill \\ \hfill & \quad +\frac{2(1-2{\eta }^{2}+2{\eta }^{4})}{{\eta }^{2}{(1-{\eta }^{2})}^{2}}+\mathcal{O}(1),\hfill \end{align} \tag{ 24 }$$

which holds for ξN_S ≫ η²/[(1 − η²)(1 + 2N_B )] and N_S ≫ N_B . By setting the derivative with respect to ξ to zero and solving for ξ, we obtain ${\xi }^{\mathrm{opt}}\sim \eta /{[4{N}_{S}(1-{\eta }^{2})(1+2{N}_{B})]}^{1/2}$. □

We calculate the two-mode QFI for the probe state with covariance matrix as in equation (26) and displacement $\mathbf{d}=\sqrt{2{N}_{\text{coh}}}{(\mathrm{cos}\,\theta ,\mathrm{sin}\,\theta ,0,0)}^{\top }$. By simplification with symbolic software, we verify that the resulting QFI is independent of the rotation by ϕ. See appendix B.2 for the full expression. Moreover, we have the following lemma on the optimal displacement angle, mirroring the result of lemma 3.

Lemma 5. The optimal displacement angle for the two-mode QFI is along the direction of squeezing, i.e., θ^opt = nπ, with $n\in \mathbb{N}$.

Proof. Displacement appears only in the second term of equation (8), which is computed, for the covariance matrix probe of equation (26) and dynamics as in equations (3) and (4), as

$$\begin{equation}{({\partial }_{\eta }\tilde{\mathbf{d}})}^{\top }{\tilde{\mathbf{\Sigma }}}^{-1}({\partial }_{\eta }\tilde{\mathbf{d}})=8{N}_{\text{coh}}a\left(\frac{{\mathrm{cos}}^{2}\theta }{4ay+r{\eta }^{2}}+\frac{{\mathrm{sin}}^{2}\theta }{4ay+{\eta }^{2}/r}\right),\end{equation} \tag{ 27 }$$

where $y=\left(1-{\eta }^{2}\right)\left({N}_{B}+\frac{1}{2}\right)$. If r = 1, θ is degenerate. Otherwise, if r < 1, equation (27) is maximised for θ = nπ, for $n\in \mathbb{N}$. □

With the optimal displacement along θ = nπ, the task of QFI optimization is reduced from five to three parameters, since the QFI does not depend on φ. Equivalently to the single-mode optimization, we restrict the total number of photons per mode as N_S = N_coh + N_sq.th.. We introduce the free parameter ζ² ∈ [0, 1] as the fraction of photons allocated to the covariance. In particular, ${N}_{\text{coh}}={N}_{S}\left(1-{\zeta }^{2}\right)$ and N_sq.th. = N_S ζ².

The number of photons of a squeezed thermal state with covariance matrix Σ_S as in equation (26) is ${N}_{\text{sq.th.}}=\frac{a}{2}\left(r+{r}^{-1}\right)-\frac{1}{2}$. Notice that if we for the moment fix ζ, we have fixed also the photons allocated to the covariance as N_sq.th. = N_S ζ². We use this to eliminate the parameter a, as $a=\frac{2{N}_{S}{\zeta }^{2}+1}{r+{r}^{-1}}$, and retain the free parameter r which represents the trade-off between local squeezing and correlations. Since the number of photons allocated to the covariance matrix depends on ζ, so does also the range of possible squeezing, as $r\in [2{N}_{S}{\zeta }^{2}+1-2\sqrt{{N}_{S}{\zeta }^{2}\left({N}_{S}{\zeta }^{2}+1\right)},1]$. In summary, the power-constrained two-mode QFI is parameterized on the two-dimensional space (ζ, r).

We have run exhaustive searches on the two-dimensional parameter space (ζ, r) to find the point maximizing the two-mode QFI. For each scenario in $\left\{{N}_{S}\in \left[1{0}^{-3},1{0}^{3}\right],{N}_{B}\in \left[1{0}^{-3},1{0}^{3}\right],\eta \in \left[1{0}^{-3},0.999\right]\right\}$, the point (ζ = 1, r = 1) always results to be the global maximum. That is, the optimal strategy always consists of allocating all photons to maximize correlations in the covariance matrix. Indeed, the state corresponding to (1, 1) is the TMSV. See also figure 4 for three samples of this verification with varying amounts of background noise.

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We support the numerical results analytically by showing that the point (ζ = 1, r = 1) corresponds to a local maximum of the QFI.

Proposition 4 [ TMSV as local maximum of the QFI ( N_B )]. On the parameter space of (ζ, r), the two-mode QFI is maximized at the point (1, 1).

Proof. The proof consists of evaluating the gradients at the point of interest. Assume a non-zero signal N_S > 0. We have that $\left({\partial }_{r}{I}^{\text{EA}}\right){\vert }_{\zeta =1,r=1}=0$, i.e., $\left(1,1\right)$ is a stationary point with respect to r. Furthermore

$$\begin{align}\hfill \left({\partial }_{r}^{2}{I}^{\text{EA}}\right){\vert }_{\zeta =1,r=1}& =-\frac{2{\left(1+2{N}_{S}\right)}^{2}}{g\left({N}_{S},{N}_{B}\right){\left(1+\left(1-{\eta }^{2}\right)g\left({N}_{S},{N}_{B}\right)\right)}^{2}}\hfill \\ \hfill & \quad \cdot \frac{{f\!}_{1,\mathrm{n}\mathrm{u}\mathrm{m}}\left({N}_{S},{N}_{B},\eta \right)}{{f\!}_{1,\mathrm{d}\mathrm{e}\mathrm{n}}\left({N}_{S},{N}_{B},\eta \right)}< 0.\hfill \end{align} \tag{ 28 }$$

Here,

$$\begin{align}\hfill {f\!}_{1,\mathrm{n}\mathrm{u}\mathrm{m}}& =2\left(1-{\eta }^{2}\right)\left[{N}_{S}^{3}\left(1+2{N}_{B}\right){\eta }^{2}\left(1-{\eta }^{2}\right)\left(2{N}_{B}^{2}+2{N}_{B}+1\right)\right.\hfill \\ \hfill & \quad \left.+{\left(1+{\eta }^{2}\right)}^{2}{N}_{B}^{4}\left(1+2{N}_{S}\right)\right]\hfill \\ \hfill & \quad +2{N}_{S}^{2}\left(1-{\eta }^{2}\right)\left[2{\left(1+{\eta }^{2}\right)}^{2}{N}_{B}^{4}+2\left(2+7{\eta }^{2}-{\eta }^{4}\right){N}_{B}^{3}\right.\hfill \\ \hfill & \quad \left.+\left(3+16{\eta }^{2}-{\eta }^{4}\right){N}_{B}^{2}+\left(1+9{\eta }^{2}\right){N}_{B}+2{\eta }^{2}\right]\hfill \\ \hfill & \quad +2{N}_{S}\left[2\left(2-{\eta }^{2}\right)\left(2-{\left(1-{\eta }^{2}\right)}^{2}\right){N}_{B}^{3}+\left(3+7{\eta }^{2}-5{\eta }^{4}+{\eta }^{6}\right){N}_{B}^{2}\right.\hfill \\ \hfill & \quad \left.+\left(1+4{\eta }^{2}-{\eta }^{4}\right){N}_{B}+{\eta }^{2}\right]\hfill \\ \hfill & \quad +{N}_{B}\left[4\left(1+{\eta }^{2}-{\eta }^{4}\right){N}_{B}^{2}+\left(3+3{\eta }^{2}-2{\eta }^{4}\right){N}_{B}+{\eta }^{2}+1\right] > 0,\hfill \end{align} \tag{ 29 }$$

$$\begin{align}\hfill {f\!}_{1,\mathrm{d}\mathrm{e}\mathrm{n}}=\,\,& 2{\left(1-{\eta }^{2}\right)}^{2}\left(2{N}_{S}^{2}{N}_{B}^{2}+2{N}_{S}^{2}{N}_{B}+{N}_{S}^{2}+2{N}_{S}{N}_{B}^{2}+{N}_{B}^{2}\right)\hfill \\ \hfill & \!+2g({N}_{S},{N}_{B})\left(1-{\eta }^{2}\right)+1 > 0,\hfill \end{align} \tag{ 30 }$$

where g(x, y) = x + 2xy + y. That is, the second order derivative is strictly negative. Therefore, the point (1, 1) is a maximum with respect to r for any configuration of {N_S , N_B , η}. Regarding the parameter ζ, we have

$$\begin{align}\hfill & \quad \left({\partial }_{\zeta }{I}^{\text{EA}}\right){\vert }_{r=1,\zeta =1}=\frac{2{N}_{S}{f}_{2}\left({N}_{S},{N}_{B},\eta \right)}{\left(1-{\eta }^{2}\right)\left(1+2\left(1-{\eta }^{2}\right)g\left({N}_{S},{N}_{B}\right)\right){\left(1+\left(1-{\eta }^{2}\right)g\left({N}_{S},{N}_{B}\right)\right)}^{2}} > 0\hfill \end{align} \tag{ 31 }$$

because

$$\begin{align}\hfill {f}_{2}\left({N}_{S},{N}_{B},\eta \right)& =4{N}_{S}^{2}\left(1+2{N}_{B}\right){\eta }^{2}{\left(1-{\eta }^{2}\right)}^{2}+8{N}_{S}\left(1-{\eta }^{2}\right)\left({N}_{B}^{2}{\left(1+{\eta }^{2}\right)}^{2}\right.\hfill \\ \hfill & \quad \left.+{N}_{B}\left(1+3{\eta }^{2}\right)+{\eta }^{2}\right)\hfill \\ \hfill & \quad +4{N}_{B}^{2}\left(1+{\eta }^{2}\right)\left(1-{\eta }^{4}\right)+4{N}_{B}\left(1+{\eta }^{2}\left(2-{\eta }^{2}\right)\right)+4{\eta }^{2} > 0.\hfill \end{align} \tag{ 32 }$$

That is, the QFI is locally an increasing function of ζ. The line of ζ = 1 is at the boundary of the parameter space. Therefore, the point (1, 1) is a maximum also with respect to ζ. □

We strengthen proposition 4 and show that the maximum at (ζ = 1, r = 1) is indeed the global maximum in the η → 0 and η → 1 limits. In the η → 0 case, the QFI is monotone with respect to r. This simplifies the optimization with respect to ζ. Indeed, we have

$$\begin{align}\hfill & \quad \underset{\eta \to \,0}{\mathrm{lim}}\left({\partial }_{r}{I}^{\text{EA}}\right)=\frac{128g\left({N}_{S}{\zeta }^{2},{N}_{B}\right){N}_{B}\left({N}_{B}+1\right){\left(2{N}_{S}{\zeta }^{2}+1\right)}^{2}r\left(1-{r}^{4}\right)}{{\left[4{r}^{2}{\left(1+2g\left({N}_{S}{\zeta }^{2},{N}_{B}\right)\right)}^{2}-{\left(1+{r}^{2}\right)}^{2}\right]}^{2}}\geqslant 0.\hfill \end{align} \tag{ 33 }$$

This implies that, for N_B > 0, I is an increasing function of r, with r = 0 and r = 1 the only stationary points, where r = 0 implies infinite squeezing. Because the gradient is strictly positive on r ∈ (0, 1), r = 1 is the optimal choice for any ζ. We now study the gradient with respect to ζ and evaluate it along the line of r = 1, as

$$\begin{equation}\underset{\eta \to \,0}{\mathrm{lim}}{\left({\partial }_{\zeta }{I}^{\text{EA}}\right)}_{\vert r=1}=\frac{8{N}_{S}\zeta {N}_{B}\left({N}_{B}+1\right)}{\left(1+2{N}_{B}\right){\left(1+g\left({N}_{S}{\zeta }^{2},{N}_{B}\right)\right)}^{2}}\geqslant 0.\end{equation} \tag{ 34 }$$

The only stationary point is at ζ = 0, which is a minimum. Therefore, if N_B > 0, ζ = 1 is optimal. Furthermore, there are globally no other stationary points, so (1, 1) is the global maximum as η = 0.

In the η → 1 case, the asymptotic behaviour is

$$\begin{equation}{I}^{\text{EA}}\sim \frac{2({N}_{S}{\zeta }^{2}+{N}_{B}+2{N}_{B}{N}_{S}{\zeta }^{2})}{1-\eta },\quad \mathrm{a}\mathrm{s}\enspace \eta \to 1.\end{equation} \tag{ 35 }$$

This expression is independent of r and a growing function of ζ. This implies the optimal strategy consists of allocating all photons to covariance. However, local squeezing, correlations, and any combination of the two perform equivalently. In fact, the behaviour of equation (35) at ζ = 1 is identical to that of the single-mode squeezed-vacuum, see equation (21).

For a coherent state probe, the number of probes M is irrelevant for the performance in terms of total QFI, given that ${\mathcal{I}}^{\text{IF}\;(\text{0})}(\xi =0)=4{\mathcal{N}}_{S}$. The situation changes when squeezing enters into the game. For instance, by setting ξ = 1, we get

$$\begin{equation}{\mathcal{I}}^{\mathrm{IF,}\;(\text{0})}(\xi =1)=\frac{4{\mathcal{N}}_{S}[{(1-{\eta }^{2})}^{2}+{\eta }^{4}]}{(1-{\eta }^{2})\left[1+2\frac{{\mathcal{N}}_{S}}{M}{\eta }^{2}(1-{\eta }^{2})\right]}.\end{equation} \tag{ 39 }$$

Indeed, we have ${\mathcal{I}}^{(\mathrm{0})}(\xi =1) > {\mathcal{I}}^{(\mathrm{0})}(\xi =0)$ provided that $\frac{{\mathcal{N}}_{S}}{M}< \frac{2{\eta }^{2}-1}{2{(1-{\eta }^{2})}^{2}}$. There are a couple of striking facts. First, if ${\mathcal{N}}_{S}\leqslant {\bar{N}}_{S}(\eta )$ (defined in proposition 1), then M = ∞ optimizes the total QFI, and the squeezed-vacuum is an optimal probe. This is a direct consequence of proposition 1. Second, if $\eta > \frac{1}{\sqrt{2}}$, for any total power ${\mathcal{N}}_{S}$ we can choose a sufficiently large M such that squeezed-vacuum does better than a coherent state. However, by using equation (16), we find that applying an infinitesimal squeezing to a largely displaced mode virtually saturates the bound in lemma 1:

$$\begin{equation}{\mathcal{I}}^{\mathrm{IF,}(\text{0})}=\frac{4{\mathcal{N}}_{S}(1-\xi )}{1-{\eta }^{2}}+\mathcal{O}(1),\quad \mathrm{a}\mathrm{s}\enspace \xi {\mathcal{N}}_{S}\to \infty .\end{equation} \tag{ 40 }$$

Therefore, as in the single copy QFI case, an optimized displaced squeezed state is still the optimal in general. We now show an interesting result for the optimal bandwidth given a certain amount of power at disposal.

Proposition 5. Assuming M ⩽ M_max, the total QFI ${\mathcal{I}}^{\mathrm{IF,}\;(\mathrm{0})}$ is optimized either for M = 1 or M = M_max.

Proof. Let us denote ${N}_{S}={\mathcal{N}}_{S}/M$, and extend, for simplicity, the optimization problem to the continuum. Indeed, we consider ${N}_{S}\in [{\mathcal{N}}_{S}/{M}_{\mathrm{max}},{\mathcal{N}}_{S}]$. We are interested in the N_S value that solves the optimization problem

$$\begin{align}\hfill & \quad \underset{{N}_{S}\in \left[\frac{{\mathcal{N}}_{S}}{{M}_{\mathrm{max}}},{\mathcal{N}}_{S}\right],\xi \in [0,1]}{\mathrm{max}}\frac{{\mathcal{I}}^{\mathrm{IF,}\;(\text{0})}}{4{\mathcal{N}}_{S}}\hfill \\ \hfill & =\underset{\xi \in [0,1]}{\mathrm{max}}\left\{\underset{{N}_{S}\in \left[\frac{{\mathcal{N}}_{S}}{{M}_{\mathrm{max}}},{\mathcal{N}}_{S}\right]}{\mathrm{max}}\frac{{\mathcal{I}}^{\mathrm{IF,}(\text{0})}}{4{\mathcal{N}}_{S}}\right\}\hfill \\ \hfill & =\underset{0\leqslant x\leqslant {N}_{S}^{\mathrm{opt}}(x)}{\mathrm{max}}\left\{\underset{{N}_{S}\in \left[\frac{{\mathcal{N}}_{S}}{{M}_{\mathrm{max}}},{\mathcal{N}}_{S}\right]}{\mathrm{max}}{h}_{\eta }(x,{N}_{S})\right\},\hfill \end{align} \tag{ 41 }$$

where ${h}_{\eta }(x,{N}_{S})=\frac{1-x{N}_{S}^{-1}}{1-2{\eta }^{2}\left(\sqrt{x(1+x)}-x\right)}+\frac{\left[{(1-{\eta }^{2})}^{2}+{\eta }^{4}\right]x{N}_{S}^{-1}}{(1-{\eta }^{2})(1+2x{\eta }^{2}(1-{\eta }^{2}))}$. In equation (41), we have performed the change of variable ξN_S = x, so that ${N}_{S}^{\mathrm{opt}}(x)$ is the argmax of the optimization with respect to N_S . The function h_η (x, N_S ) is linear in ${N}_{S}^{-1}$, meaning that the maximum is in one of the extreme point, i.e., ${N}_{S}^{\mathrm{opt}}(x)$ is either ${\mathcal{N}}_{S}/{M}_{\mathrm{max}}\enspace$ or ${\mathcal{N}}_{S}$ ⁵ . It follows that either M = 1 or M = M_max is the optimal choice. □

Notice that in the limit of large total power, the total QFI is optimized virtually for any M. This is clear from equation (40), where the ${\mathcal{I}}^{\mathrm{IF,}(\text{0})}$ asymptotic expression does not depend on M. The next question is whether squeezed-vacuum states perform better than any state for fixed total power and large bandwidth. This turns out to depend on the available total power, as shown the following proposition.

Proposition 6. There exists $\bar{K}(\eta )$ such that M = 1 optimizes ${\mathcal{I}}^{\mathrm{IF,}(\text{0})}$ for any ${\mathcal{N}}_{S} > \bar{K}(\eta )$. We have that $\bar{K}(\eta )=0$ for $0< \eta \leqslant \frac{1}{\sqrt{2}}$ and $\bar{K}(\eta )\geqslant {\bar{N}}_{S}(\eta )$ for $\eta > \frac{1}{\sqrt{2}}$.

Proof. Let us consider $0< \eta \leqslant \frac{1}{\sqrt{2}}$. For M = ∞, the total QFI ${\mathcal{I}}^{\mathrm{IF,}\;(\text{0})}=4{\mathcal{N}}_{S}\left(1-\xi +\xi \frac{{(1-{\eta }^{2})}^{2}+{\eta }^{4}}{1-{\eta }^{2}}\right)$ is optimized for ξ = 0, i.e., for a coherent state probe. Notice that the performance of a coherent state probe is the same for any M, i.e., ${\mathcal{I}}^{\mathrm{IF,}\;(\text{0})}(M=\infty ,\xi =0)={\mathcal{I}}^{\mathrm{IF,}\;(\text{0})}(M,\xi =0)$ for any finite M. However, due to proposition 2, for any finite M there is a squeezed coherent state that performs better than a coherent state probe, which is an absurd. It follows that M = ∞ cannot optimize the total QFI. In this case, M = 1 is optimal for any ${\mathcal{N}}_{S} > 0$.

Let us now consider $\eta > \frac{1}{\sqrt{2}}$. Let us extend the optimization domain to N_S ∈ [0, ∞]. The quantity $\frac{{\mathcal{I}}^{\mathrm{IF,}(\text{0})}}{4{\mathcal{N}}_{S}}$ is maximal for N_S = ∞, as for this value the bound in lemma 1 is saturated. This means that, referring to the optimization problem in equation (41), there exists $\bar{K}(\eta )$ such that h_η (x, N_S ) > h_η (x, 0) for any ${N}_{S}\geqslant \bar{K}(\eta )$. It follows that if ${\mathcal{N}}_{S} > \bar{K}(\eta )$, then M = 1 is optimal. In addition, $\bar{K}(\eta )\geqslant {\bar{N}}_{S}(\eta )$, where ${\bar{N}}_{S}(\eta )$ is defined in proposition 1. In fact, if ${\mathcal{N}}_{S}\leqslant {\bar{N}}_{S}(\eta )$, then M = ∞ is the optimal choice, as shown below equation (39). □

Proposition 6 consists in a worst case scenario, where the number of copies can be infinite. In the case where M is finite, then M = 1 is the optimal choice for a larger range of power values. In figure 6 we numerically show that $\tilde{K}(\eta )$ is strictly larger than ${\tilde{N}}_{S}(\eta )$. This is because when jointly optimizing the total QFI with respect to M and ξ, the squeezed-vacuum state results to be the optimal choice on a larger range of parameter values. In this case we numerically see that ξ^opt = 1 if and only if ${\mathcal{N}}_{S}\leqslant \tilde{K}(\eta )$, and the optimal value is achieved in the limits N_S → 0 and M → ∞, with the constraint $M{N}_{S}={\mathcal{N}}_{S}$.

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In section 4, we have proved that the TMSV is optimal for any system parameter choice. Therefore, we can restrict the entanglement-assisted analysis for the optimal total QFI to the TMSV case. The total QFI for a TMSV probe is independent on M, i.e., ${\mathcal{I}}^{\text{TMSV,}\;(\text{0})}=\frac{4{\mathcal{N}}_{S}}{1-{\eta }^{2}}$. No advantage with respect an optimized single-mode transmitter can be observed in the ${\mathcal{N}}_{S}\gg 1$ regime, as ${\mathcal{I}}^{\text{TMSV,}\;(\text{0})}$ approaches the optimal total QFI achieved in the idler-free case, see equation (40). However, one shall keep in mind that reaching the performance of equation (40) needs squeezing, albeit an infinitesimal amount. In other words, also the idler-free case needs non-classical resources to reach optimality. Indeed, the TMSV still shows an advantage with respect to a coherent state transmitter for any η ≠ 0. In addition, due to lemma 1, the TMSV state is indeed an optimal probe for any value of η.

In figure 6, it is shown that a factor of 2 advantage is reached for a large range of values of η, if ${\mathcal{N}}_{S}\lesssim 1$. This advantage decreases with increasing ${\mathcal{N}}_{S}$. For instance, for ${\mathcal{N}}_{S}\simeq 1$ and $\eta \simeq \frac{1}{\sqrt{2}}$ we have ${\mathcal{I}}^{\text{TMSV,}(\text{0})}\simeq 10$, which is enough to realize a sensitivity up to ${\Delta}{\hat{\eta }}^{2}\lesssim 0.1$. To achieve larger sensitivity values, the optimal displaced squeezed state shall be a better choice for an experimentalist, as it realizes similar performances for larger power as the TMSV probe, while being less experimentally demanding to generate.

As previously discussed, M = ∞ is the optimal choice for any value of ${\mathcal{N}}_{S}$, due to the presence of the shadow-effect. Let us discuss a limit where the shadow-effect is not present, and where the TMSV is expected to show a relevant advantage with respect to the single-mode case. In the finite N_B case, we expect to have an advantage of the TMSV state over the idler-free strategy for low (albeit finite) values of ${\mathcal{N}}_{S}$, similarly as it happens in the N_B = 0 case. Let us focus on the ${N}_{B}\gg {(1-{\eta }^{2})}^{-1}$ regime. The presence of the shadow-effect makes the quantum advantage disappear for finite values of η. Therefore, we set also ${N}_{B}{\eta }^{2}\ll {\mathcal{N}}_{S}/M$. In this regime, we have ${\mathcal{I}}^{\text{TMSV}}\simeq \frac{4{\mathcal{N}}_{S}(M+{\mathcal{N}}_{S})}{{N}_{B}(M+2{\mathcal{N}}_{S})}$, while for the coherent state we get ${\mathcal{I}}^{\text{coh}}=\frac{2{\mathcal{N}}_{S}}{{N}_{B}(1-{\eta }^{2})}$. It is clear that ${\mathcal{I}}^{\text{TMSV}}$ is optimized for $M\gg {\mathcal{N}}_{S}$, which also implies that η² N_B must be much smaller than 1. This agrees with the analysis done after equation (38). In this regime, the TMSV state shows a quantum advantage of 2 for arbitrarily large ${\mathcal{N}}_{S}$. In figure 5, the case of M = 1 is drawn. It is visible that the quantum advantage is present for η² N_B ≪ 1, and it disappears already for η² N_B ∼ 1.

This normalization has been used for discussing remote sensing protocols such as quantum illumination and quantum reading under the no passive signature assumption [44]. It erases the shadow-effect, and, with that, any metrological power of the vacuum state. Here, lemma 2 is relevant. In appendix C, we have computed the QFI with the normalized environment, for both the single-mode and the TMSV state case. In the idler-free case, we have

$$\begin{equation}{\mathcal{I}}_{\mathrm{norm}}^{\text{IF}}=\frac{4{\mathcal{N}}_{S}(1-\xi )}{2{N}_{B}+1-{\eta }^{2}}+\mathcal{O}(1),\quad \mathrm{a}\mathrm{s}\enspace \xi {\mathcal{N}}_{S}\to \infty \end{equation} \tag{ 42 }$$

for any M. For a coherent state input, i.e., for ξ = 0, we get that ${\mathcal{I}}_{\mathrm{norm}}^{\text{coh}}=\frac{4{\mathcal{N}}_{S}}{1+2{N}_{B}}$, meaning that an infinitesimal amount of squeezing allows us to reduce the QFI as in equation (42). Comparing this result with equation (23), we see that the ${(1-{\eta }^{2})}^{-1}$ divergence disappears. Indeed, in the N_B ≫ 1 regime, the un-squeezed coherent state is virtually the optimal probe in the idler-free setting, for any value of η.

Proposition 7 [ TMSV state as optimal probe for the normalized channel ] [47]. The infinite bandwidth (M = ∞) TMSV state is an optimal probe for the quantum estimation of η with the normalization N_B → N_B /(1 − η²), for any parameter values.

Proof. The QFI of the TMSV state can be written as

$$\begin{equation}{\mathcal{I}}_{\mathrm{norm}}^{\text{TMSV}}=\frac{4{\mathcal{N}}_{S}\left[{N}_{B}+1+{\mathcal{N}}_{S}\left({N}_{B}+1-{\eta }^{2}\right){M}^{-1}\right]}{\left({N}_{B}+1-{\eta }^{2}\right)\left[{N}_{B}+1+{\mathcal{N}}_{S}\left(2{N}_{B}+1-{\eta }^{2}\right){M}^{-1}\right]}.\end{equation} \tag{ 43 }$$

In the infinite bandwidth limit we get

$$\begin{equation}{\mathcal{I}}_{\mathrm{norm}}^{\text{TMSV}}=\frac{4{\mathcal{N}}_{S}}{{N}_{B}+1-{\eta }^{2}}+\mathcal{O}({M}^{-1}),\quad \mathrm{a}\mathrm{s}\enspace M\to \infty .\end{equation} \tag{ 44 }$$

Equation (44) saturates the ultimate bound in lemma 2 for any value of η. □

This result was first reported in reference [47]. In reference [40], the authors claim that the bound in lemma 1 is not necessarily attained. Here, we show that it is actually saturated by a TMSV probe for any value of η. The quantum advantage is limited to a factor of 2 in the QFI, and is obtained in the limit of large N_B . We notice that there is a clear qualitative distinction between in the normalized and the unnormalized models. In the unnormalized model, the shadow-effect washed out the quantum advantage for low-enough N_S . A quantum advantage is reached by the TMSV state only when the bandwidth of the classical probe is limited, and for large enough power per mode. Instead, in the normalized model, the TMSV state shows a quantum advantage for any parameter value, unless N_B = 0.

We have

$$\begin{align}\hfill \frac{1}{2{N}_{S}^{2}{\eta }^{2}}{\partial }_{\xi }^{2}{I}^{\mathrm{IF,}\;(\text{0})}& =-\frac{8(1-2{\eta }^{2}+2{\eta }^{4})}{{(1+2{N}_{S}\xi {\eta }^{2}(1-{\eta }^{2}))}^{3}}\hfill \\ \hfill & \quad +\frac{{l}_{1}({N}_{S},\eta ,\xi )}{\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}{[1-2{\eta }^{2}(\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}-{N}_{S}\xi )]}^{2}},\hfill \end{align} \tag{ 55 }$$

with

$$\begin{align}\hfill {l}_{1}({N}_{S},\eta ,\xi )& =\frac{2{N}_{S}(1-\xi ){\eta }^{2}{[1-2(\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}-{N}_{S}\xi )]}^{2}}{\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}[1-2{\eta }^{2}(\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}-{N}_{S}\xi )]}\hfill \\ \hfill & \quad -\frac{1-\xi }{\xi (1+{N}_{S}\xi )}-4[1-2(\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}-{N}_{S}\xi )].\hfill \end{align} \tag{ 56 }$$

We notice that the maximum value of l₁ is reached for η = 1 and N_S ξ → ∞, for which l₁ ∼ −(1 − ξ)/(2N_S ξ²). Therefore l₁ < 0, and ${\partial }_{\xi }^{2}{I}^{\mathrm{IF,}\;(\text{0})}$ is negative for any parameter values unless η = 0.

Let us compute $f(\eta ,{N}_{S},\xi )=\frac{1}{4{N}_{S}}{\partial }_{\xi }{I}^{\mathrm{IF,}\;(\text{0})}$:

$$\begin{align}\hfill f(\eta ,{N}_{S},\xi )& =\frac{2{N}_{S}(1-\xi ){\eta }^{2}\left(\frac{1+2{N}_{S}\xi }{2\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}}-1\right)}{{\left[1-2{\eta }^{2}\left(\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}-{N}_{S}\xi \right)\right]}^{2}}\hfill \\ \hfill & \quad -\frac{2{N}_{S}\xi {\eta }^{2}(1-2{\eta }^{2}+2{\eta }^{4})}{{\left[1+2{N}_{S}\xi {\eta }^{2}(1-{\eta }^{2})\right]}^{2}}\hfill \\ \hfill & \quad +\frac{1-2{\eta }^{2}+2{\eta }^{4}}{(1-{\eta }^{2})(1+2{N}_{S}\xi {\eta }^{2}(1-{\eta }^{2}))}\hfill \\ \hfill & \quad -\frac{1}{1-2{\eta }^{2}\left(\sqrt{{N}_{S}\xi (1+{N}_{S}\xi )}-{N}_{S}\xi \right)}.\hfill \end{align} \tag{ 57 }$$

Let us now study the derivative in ξ = 1, i.e., ${f\!}_{1}(\eta ,{N}_{S})=\frac{1}{4{N}_{S}}{\left({\partial }_{\xi }{I}^{\mathrm{IF,}\;(\text{0})}\right)}_{\vert \xi =1}$ it and its derivative with respect to N_S :

$$\begin{align}\hfill {f\!}_{1}(\eta ,{N}_{S})& =\frac{{(1-{\eta }^{2})}^{2}+{\eta }^{4}}{(1-{\eta }^{2}){[1+2{N}_{S}{\eta }^{2}(1-{\eta }^{2})]}^{2}}\hfill \\ \hfill & \quad -\frac{1}{1-2{\eta }^{2}(\sqrt{{N}_{S}(1+{N}_{S})}-{N}_{S})},\hfill \end{align} \tag{ 58 }$$

$$\begin{align}\hfill \frac{1}{{\eta }^{2}}{\partial }_{{N}_{S}}{f\!}_{1}(\eta ,{N}_{S})& =-\frac{2\left(\frac{1+2{N}_{S}}{2\sqrt{{N}_{S}(1+{N}_{S})}}-1\right)}{{\left[1+2{\eta }^{2}\left({N}_{S}-\sqrt{{N}_{S}(1+{N}_{S})}\right)\right]}^{2}}\hfill \\ \hfill & \quad -\frac{4[{(1-{\eta }^{2})}^{2}+{\eta }^{4}]}{{\left[1+2{N}_{S}{\eta }^{2}(1-{\eta }^{2})\right]}^{3}}< 0.\hfill \end{align} \tag{ 59 }$$

This means that f₁ is always decreasing in N_S . Notice that ${f\!}_{1}\to -\frac{1}{1-{\eta }^{2}}$ for N_S → ∞, and that ${f\!}_{1}(\eta ,{N}_{S}=0)=\frac{2{\eta }^{2}-1}{1-{\eta }^{2}}$ is positive for some $\eta > 1/\sqrt{2}$.

The function f is singular in ξ = 0. Its expansion is

$$\begin{equation}f(\eta ,{N}_{S},\xi )=\frac{\sqrt{{N}_{S}}{\eta }^{2}}{\sqrt{\xi }}+\mathcal{O}(1),\quad \xi \to 0.\end{equation} \tag{ 60 }$$