Gaussian Quantum Illumination via Monotone Metrics

Quantum illumination (QI) seeks to discern the presence or absence of a low reflectivity target by utilizing an additional idler mode of light to enhance the detection probability [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Coherent states are the benchmark to compare the performance of QI to classical illumination, where one uses only one mode of light to detect a target. It was first shown in Ref. [2] that two-mode squeezed vacuum (TMSV) states can outperform coherent states. The performance is quantified by the ability to distinguish two different quantum states $\rho_{0}$ and $\rho_{1}$ each corresponding to the state of target absent and present.

The Helstrom bound gives the minimum total uncertainty when distinguishing $\rho_{0}$ and $\rho_{1}$ [15]. The quantum Chernoff bound indicates the ability to distinguish $\rho_{0}$ and $\rho_{1}$ in the asymptotic limit of symmetric hypothesis testing, when we have $N$ copies of the same state and optimal collective measurement on all $N$ copies is possible. More precisely, it states that the Helstrom bound of the tensor product states $\rho_{0,1}^{\otimes N}$ decays exponentially in $N$ [16]:

	$\displaystyle\gamma_{\text{col}}$	$\displaystyle=\lim_{N\rightarrow\infty}-\frac{1}{N}\text{Log }P_{err}\left(\rho_{0}^{\otimes N},\rho_{1}^{\otimes,N}\right)$		(1)
		$\displaystyle=-\min_{0\leq s\leq 1}\text{Log Tr }\rho_{0}^{s}\rho_{1}^{1-s},$		(2)

where $P_{err}$ denotes the Helstrom bound. We refer to $\gamma_{\text{col}}$ as the collective decay constant. TMSV states can show a maximum of 6 dB enhancement in this decay constant compared to a coherent state with same signal power in the regime of low signal power, low target reflectivity and high background noise [2].

As performing collective measurements on all copies at once is typically hard, one would seek the behavior of error probabilities when only local measurements combining each incoming signal and idler mode are possible. Given a local measurement $\hat{O}$, we have two probability distributions $\mu_{0}$ and $\mu_{1}$ corresponding to $\rho_{0}$ and $\rho_{1}$, and the problem is to distinguish between the two probability distributions given $N$ samples drawn from either distribution. The error probability again decays exponentially, and when the two states are close, the decay constant is given by the signal-to-noise ratio $(m_{0}-m_{1})^{2}/(2(\sigma_{0}+\sigma_{1})^{2})$ [17]. Here, $m_{i}$ and $\sigma_{i}^{2}$ are the mean and variance of the probability distribution $\mu_{i}$ $(i=0,1)$. We define the local decay constant $\gamma_{\text{loc}}$ to be the decay constant maximized over all possible local measurements $\hat{O}$. However, no simple formula like Eq. (2) seems to be known for $\gamma_{\text{loc}}$ in general.

Computation of the decay constants is complicated in general, even when considering only Gaussian states [18, 19, 20]. Gaussian states, which include coherent states, TMSV states, and thermal states, have been considered in various works of QI for their ease in both experimental and theoretical aspects [2, 3, 4, 5, 6, 7, 8, 9]. For Gaussian states, the main obstacle is the necessity of symplectic diagonalization of the two output states [21, 2, 6] or computation of the symmetric logarithmic derivative [24, 22, 23]. These procedures make analytic formulae of decay constants highly complicated, if exists.

In QI, the low reflectivity condition implies that the difference between the two states of interest is infinitesimal. In this case, the decay constants also become infinitesimal quantities and give infinitesimal distances between quantum states [16, 18]. These distances are examples of monotone metrics, infinitesimal distances between quantum states which decrease under completely positive trace preserving (CPTP) maps [25]. Taking advantage of this, we compute $\gamma_{\text{col}}$ and $\gamma_{\text{loc}}$ directly in terms of first-order moments and covariance matrix of the input two-mode Gaussian state. Our results only need to compute the symplectic diagonalization of the idler mode, which are in contrast to previous methods requiring symplectic diagonalization of the two-mode Gaussian states $\rho_{0}$ and $\rho_{1}$. We also show that in the large background noise limit, no symplectic diagonalization is needed to compute the decay constants. We apply our results to analyze the effect of single-mode Gaussian unitaries on both signal and idler modes on the performance of QI.

This paper is organized as follows. In Sec. II, we provide a short review on Gaussian states and quantum illumination. In Sec. III, the results of Petz on monotone metrics are summarized and its restriction to Gaussian states is computed in terms of first-order moments and covariance matrix. As $\gamma_{\text{col}}$ and $\gamma_{\text{loc}}$ are described by monotone metrics, we provide in Sec. IV analytic formulae for the decay constants in terms of first-order moments and covariance matrix. These are used to show that TMSV states are indeed optimal for QI among pure Gaussian states with same signal power. Also we investigate the effect of single-mode Gaussian operations on the signal mode to enhance QI performance. In Sec. V, we consider the effect of idler loss on the performance of QI. We show that single-mode Gaussian operations on the idler mode are not advantageous to overcome idler loss and identify the region of idler loss and noise where observing quantum advantage over coherent states is possible. In Sec. VI, we identify the region of correlations between the signal and idler modes of QI needed to see quantum advantage. Sec. VII gives a summary of the results.

Infinitesimal distances on a space are described by a metric, which is an inner product on the tangent space at each point of the space. We are interested in the space of invertible density matrices or the space of invertible Gaussian states. For the former case, the tangent space at each point is the collection of Hermitian traceless matrices. We denote the inner product as $g_{\rho}$ for each invertible density matrix $\rho$. If the metric decreases under all completely positive trace preserving (CPTP) maps, we call the metric to be monotone:

$$g_{\Phi(\rho)}(\Phi(A),\Phi(A))\leq g_{\rho}(A,A)$$

(9)

where $\Phi$ is a CPTP map and $A$ is a tangent vector at $\rho$. When $\Phi$ is a unitary evolution, hence invertible, applying the monotone property twice with $\Phi$ and $\Phi^{-1}$ implies that the inner product is invariant under unitaries:

$$g_{\Phi(\rho)}(\Phi(A),\Phi(A))=g_{\rho}(A,A).$$

(10)

Monotone metrics arise naturally from general quantum divergences. Let $D(\rho||\sigma)$ be a quantum divergence monotone under CPTP maps, for example, the relative entropy or Rényi entropies. Then the divergence between states with infinitesimal difference gives rise to a monotone metric as $D(\rho||\rho+\epsilon\;d\rho)=\epsilon^{2}g_{\rho}(d\rho,d\rho)+o(\epsilon^{2})$ [27].

Petz classified monotone metrics on finite dimensional quantum spaces [25]. For each monotone metric $g$, there is an operator monotone function $f$ satisfying $f(t)=tf(t^{-1})$ such that $g_{\rho}$ is given by

	$\displaystyle g_{\rho}(A,B)$	$\displaystyle=\text{Tr }A^{\dagger}c_{f}(\mathbf{L}_{\rho},\mathbf{R}_{\rho})(B),$		(11)
	$\displaystyle c_{f}(x,y)$	$\displaystyle=1/(yf(x/y)),$		(12)

where $A,B$ are tangent vectors at $\rho$. The definition of an operator monotone function, which is immaterial in this paper, can be found in Ref. [28]. Here, $\mathbf{L}_{\rho}$ and $\mathbf{R}_{\rho}$ are left and right multiplication by $\rho$ respectively. If $\rho$ is diagonalized as $\rho=\sum\lambda_{n}\ket{\psi_{n}}\bra{\psi_{n}}$ with $\braket{\psi_{n}}{\psi_{m}}=\delta_{nm}$, then the metric is

$$g_{\rho}(A,B)=\sum_{n,m}c_{f}(\lambda_{n},\lambda_{m})A_{nm}^{*}B_{nm}$$

(13)

where $A_{nm}=\braket{\psi_{n}}{A}{\psi_{m}}$ are matrix elements of $A$ and $B_{nm}$ are analogously defined. Infinitesimal values of the collective and local decay constants $\gamma_{\text{col(loc)}}$ are related to monotone metrics and their associated functions are given by $f_{\text{col}}=2(\sqrt{t}+1)^{2}$ and $f_{\text{loc}}=4(t+1)$ respectively [18, 27].

In the following, we compute the monotone metric associated to an operator monotone function $f$ restricted to Gaussian states. We first determine the metric in terms of infinitesimal changes in symplectic eigenvalues and infinitesimal unitaries, and then write the expression in terms of first-order moments and covariance matrix. From unitary invariance, it suffices to compute the metric at diagonal states, which are thermal states. Let $\rho$ be an $n$-mode thermal state with symplectic eigenvalues $\nu_{1},\cdots,\nu_{n}>1$. The tangent vectors at $\rho$ can be classified into two types, ones which commute with $\rho$ and ones which do not. The tangent vectors which commute with $\rho$ arise from infinitesimal changes in $\nu_{i}$:

$$\frac{\partial\rho}{\partial\nu_{i}}=\left(\frac{2\hat{a}_{i}^{\dagger}\hat{a}_{i}}{\nu_{i}^{2}-1}-\frac{1}{\nu_{i}+1}\right)\rho.$$

(14)

The tangent vectors which do not commute with $\rho$ arise from infinitesimal unitaries:

$$i[\hat{H},\rho],$$

(15)

where $\hat{H}$ is a Hamiltonian generator describing displacement, squeezing or beam splitting action.

Unitary invariance combined with the polarization identity is a powerful tool to show certain tangent vectors are orthogonal to each other. Given a symmetric real bilinear form $g$, the polarization identity implies

$$g(u+v,u+v)=g(u-v,u-v)\Rightarrow g(u,v)=0$$

(16)

for vectors $u,v$. Thermal states and tangent vectors $\tfrac{\partial\rho}{\partial\nu_{i}}$ are invariant under local phase shifts, while the tangent vectors $i[\hat{H},\rho]$ are not. This immediately gives

$$g_{\rho}\left(\frac{\partial\rho}{\partial\nu_{i}},i[\hat{H},\rho]\right)=0.$$

(17)

Furthermore, the $2n(n+1)$ tangent vectors $i[\hat{H},\rho]$ with

	$\displaystyle\hat{H}$	$\displaystyle=\hat{x}_{i},\;\hat{p}_{i},\;\frac{1}{2}\left(\hat{a}_{i}^{2}+\hat{a}_{i}^{\dagger 2}\right),\;\frac{i}{2}\left(\hat{a}_{i}^{2}-\hat{a}_{i}^{\dagger 2}\right),\;\hat{a}_{i}\hat{a}_{j}+\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger},\;$
		$\displaystyle i\left(\hat{a}_{i}\hat{a}_{j}-\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\right),\;\hat{a}_{i}^{\dagger}\hat{a}_{j}+\hat{a}_{i}\hat{a}_{j}^{\dagger},\;i\left(\hat{a}_{i}^{\dagger}\hat{a}_{j}-\hat{a}_{i}\hat{a}_{j}^{\dagger}\right),$		(18)

are all mutually orthogonal. For example, a local $\pi/2$ phase shift on mode $i$ maps $\hat{x}_{i}\mapsto\hat{p}_{i}$, $\hat{p}_{i}\mapsto-\hat{x}_{i}$, so unitary invariance and the polarization identity imply

$\displaystyle\begin{split}g_{\rho}&(i[\hat{x}_{i}+\hat{p}_{i},\rho],i[\hat{x}_{i}+\hat{p}_{i},\rho])\\ &\hskip 56.9055pt=g_{\rho}(i[\hat{p}_{i}-\hat{x}_{i},\rho],i[\hat{p}_{i}-\hat{x}_{i},\rho])\\ &\Rightarrow\;g_{\rho}(i[\hat{x}_{i},\rho],i[\hat{p}_{i},\rho])=0.\end{split}$

(19)

Therefore, it suffices to determine the metric on tangent vectors $\tfrac{\partial\rho}{\partial\nu_{i}}$ and the values of $g_{\rho}(i[\hat{H},\rho],i[\hat{H},\rho])$ for each $\hat{H}$ from Eq. (18).

We determine the metric on tangent vectors $\tfrac{\partial\rho}{\partial\nu_{i}}$. As the tangent vectors commute with $\rho$, we have

$\displaystyle c_{f}(\mathbf{L}_{\rho},\mathbf{R}_{\rho})\left(\frac{\partial\rho}{\partial\nu_{i}}\right)$

$\displaystyle=\frac{1}{f(1)}\left(\frac{2\hat{a}_{i}^{\dagger}\hat{a}_{i}}{\nu_{i}^{2}-1}-\frac{1}{\nu_{i}+1}\right).$

(20)

This leads to

	$\displaystyle g_{\rho}\left(\frac{\partial\rho}{\partial\nu_{i}},\frac{\partial\rho}{\partial\nu_{i}}\right)$	$\displaystyle=\frac{1}{f(1)}\frac{1}{\nu_{i}^{2}-1},$		(21)
	$\displaystyle g_{\rho}\left(\frac{\partial\rho}{\partial\nu_{j}},\frac{\partial\rho}{\partial\nu_{i}}\right)$	$\displaystyle=0\hskip 14.22636pt(i\neq j).$		(22)

Therefore, the metric is also diagonal in terms of infinitesimal changes of $\nu_{i}$ and we have diagonalized the metric. Note that Eq. (21) only depends on $f(1)$, not the whole shape of $f$. This is reminiscent of the up-to-scale uniqueness of monotone metrics on classical probabilities [29, 25].

To determine $g_{\rho}(i[\hat{H},\rho],i[\hat{H},\rho])$, first observe that unitary invariance from local phase shifts implies

$$g_{\rho}(i[\hat{x}_{i},\rho],i[\hat{x}_{i},\rho])=g_{\rho}(i[\hat{p}_{i},\rho],i[\hat{p}_{i},\rho])$$

(23)

and analogous equalities hold for phase conjugate Hamiltonians in Eq. (18). We may also trace out all but one or two systems, as a multi-mode thermal state is a tensor product of one-mode thermal states. This means that it suffices to compute the following four quantities:

$\displaystyle\begin{split}&g_{\rho}(i[\hat{x}_{1},\rho],i[\hat{x}_{1},\rho]),\\ &g_{\rho}\left(i\left[\frac{1}{2}\left(\hat{a}_{1}^{2}+\hat{a}_{1}^{\dagger 2}\right),\rho\right],i\left[\frac{1}{2}\left(\hat{a}_{1}^{2}+\hat{a}_{1}^{\dagger 2}\right),\rho\right]\right),\\ &g_{\rho}\left(i\left[\hat{a}_{1}\hat{a}_{2}+\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger},\rho\right],i\left[\hat{a}_{1}\hat{a}_{2}+\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger},\rho\right]\right),\\ &g_{\rho}\left(i\left[\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger},\rho\right],i\left[\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger},\rho\right]\right),\end{split}$

(24)

where $\rho$ is a one-mode thermal state with symplectic eigenvalue $\nu$ or a two-mode thermal state with symplectic eigenvalues $\nu_{1}$ and $\nu_{2}$.

As a computational tool, we use the Bargmann representation, which converts a density matrix into a function of complex variables. Essential properties of this representation are summarized in Appendix A. A one-mode thermal state with covariance matrix $\text{diag}(\nu,\nu)$ in number basis is given as

$$\rho=\frac{2}{\nu+1}\sum_{n=0}^{\infty}\left(\frac{\nu-1}{\nu+1}\right)^{n}\ket{n}\bra{n},$$

(25)

so the Bargmann representation is

$$K(z,w^{*})=\frac{2}{\nu+1}\text{Exp}\left(\frac{\nu-1}{\nu+1}zw^{*}\right).$$

(26)

Since thermal states are diagonal in number basis, the action of $c_{f}(\mathbf{L}_{\rho},\mathbf{R}_{\rho})$ is diagonal on $\ket{n}\bra{m}$:

$\displaystyle\begin{split}&c_{f}(\mathbf{L}_{\rho},\mathbf{R}_{\rho})(\ket{n}\bra{m})\\ &=\frac{\nu+1}{2}\left(\frac{\nu+1}{\nu-1}\right)^{m}f\left(\left(\frac{\nu-1}{\nu+1}\right)^{n-m}\right)^{-1}\ket{n}\bra{m}.\end{split}$

(27)

Therefore, we can obtain the Bargmann representation of both $i[\hat{x},\rho]$ and $c_{f}(\mathbf{L}_{\rho},\mathbf{R}_{\rho})(i[\hat{x},\rho])$. We have

	$\displaystyle\begin{split}&i[\hat{x},\rho]\\ &\hskip 5.69054pt\iff\frac{2\sqrt{2}i(z-w^{*})}{(\nu+1)^{2}}\text{Exp}\left(\frac{\nu-1}{\nu+1}zw^{*}\right),\end{split}$			(28)
	$\displaystyle\begin{split}&c_{f}(\mathbf{L}_{\rho},\mathbf{R}_{\rho})(i[\hat{x},\rho])\\ &\hskip 5.69054pt\iff\frac{\sqrt{2}i(z-w^{*})}{\nu-1}f\left(\frac{\nu+1}{\nu-1}\right)^{-1}\text{Exp}\left(zw^{*}\right).\end{split}$			(29)

The symmetry of $f$, $f(t)=tf(t^{-1})$, is used in obtaining Eq. (LABEL:Br:x). Now, computing $g_{\rho}(i[\hat{x},\rho],i[\hat{x},\rho])$ surmounts to evaluating a single integral:

$\displaystyle\begin{split}g_{\rho}(i[\hat{x},\rho],i[\rho,\hat{x}])&=\int\frac{d^{2}z\,d^{2}w}{\pi^{2}}\frac{4(z-w^{*})(z^{*}-w)}{(\nu+1)^{2}(\nu-1)}f\left(\frac{\nu+1}{\nu-1}\right)^{-1}\text{Exp}\left(\frac{\nu-1}{\nu+1}zw^{*}+z^{*}w-|z|^{2}-|w|^{2}\right)\\ &=\frac{2}{\nu-1}f\left(\frac{\nu+1}{\nu-1}\right)^{-1}.\end{split}$

(30)

The other values appearing in Eq. (24) are calculated in a similar procedure. The following general integral formula is useful:

$\displaystyle\begin{split}&\int\frac{d^{2}z\,d^{2}w}{\pi^{2}}z^{n}z^{*m}w^{k}w^{*\ell}\text{Exp}\left(\alpha zw^{*}+\beta z^{*}w-|z|^{2}-|w|^{2}\right)\\ &\hskip 14.22636pt=\delta_{n+k,m+\ell}\;\alpha^{\text{max}(m-n,0)}\beta^{\text{max}(n-m,0)}\frac{a!\;b!}{|n-m|!}\,_{2}F_{1}\left(a+1,b+1;\;|n-m|+1;\;\alpha\beta\right).\hskip 14.22636pt(\text{Re}(\alpha\beta)<1)\end{split}$

(31)

Here, $a=\text{max}(n,m)$, $b=\text{max}(k,\ell)$, and ${}_{2}F_{1}$ is the hypergeometric function. The results are

	$\displaystyle g_{\rho}\left(i\left[\frac{1}{2}\left(\hat{a}^{2}+\hat{a}^{\dagger 2}\right),\rho\right],i\left[\frac{1}{2}\left(\hat{a}^{2}+\hat{a}^{\dagger 2}\right),\rho\right]\right)$	$\displaystyle=\frac{4\nu^{2}}{(\nu-1)^{2}}f\left(\frac{(\nu+1)^{2}}{(\nu-1)^{2}}\right)^{-1},$		(32)
	$\displaystyle g_{\rho}\left(i\left[\hat{a}_{1}\hat{a}_{2}+\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger},\rho\right],i\left[\hat{a}_{1}\hat{a}_{2}+\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger},\rho\right]\right)$	$\displaystyle=\frac{2(\nu_{1}+\nu_{2})^{2}}{(\nu_{1}-1)(\nu_{2}-1)}f\left(\frac{\nu_{1}+1}{\nu_{1}-1}\frac{\nu_{2}+1}{\nu_{2}-1}\right)^{-1},$		(33)
	$\displaystyle g_{\rho}\left(i\left[\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger},\rho\right],i\left[\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger},\rho\right]\right)$	$\displaystyle=\frac{2(\nu_{1}-\nu_{2})^{2}}{(\nu_{1}+1)(\nu_{2}-1)}f\left(\frac{\nu_{1}-1}{\nu_{1}+1}\frac{\nu_{2}+1}{\nu_{2}-1}\right)^{-1}.$		(34)

In Eq. (32), $\rho$ is a one-mode thermal state with covariance matrix $\text{diag}(\nu,\nu)$, while in Eqs. (33, 34), $\rho$ is a two-mode thermal state with covariance matrix $\text{diag}(\nu_{1},\nu_{1},\nu_{2},\nu_{2})$. Eqs. (21, 30, 32–34) fully determine the monotone metric associated to an operator monotone function $f$ restricted to Gaussian states.

We have computed the metric in terms of infinitesimal changes in symplectic eigenvalues and infinitesimal unitaries. We now write the results in terms of infinitesimal changes in first-order moments and the covariance matrix. In terms of first-order moments and covariance matrix, a tangent vector is represented by a $2n$ column vector $d\overline{r}$ and a symmetric $2n\times 2n$ real matrix $dV$. Let $E_{i,j}$ be the $2n\times 2n$ matrix where the $(i,j)$ entry is 1 and all other entries are 0. Define the following matrices:

$\displaystyle\begin{split}N_{i}&:=E_{2i-1,2i-1}+E_{2i,2i},\\ S_{i}&:=E_{2i,2i-1}+E_{2i-1,2i},\\ T_{i}&:=E_{2i-1,2i-1}-E_{2i,2i},\\ S_{ij}&:=E_{2j,2i-1}+E_{2j-1,2i}+E_{2i,2j-1}+E_{2i-1,2j},\\ T_{ij}&:=E_{2j-1,2i-1}-E_{2j,2i}+E_{2i-1,2j-1}-E_{2i,2j},\\ A_{ij}&:=E_{2j,2i-1}-E_{2j-1,2i}-E_{2i,2j-1}+E_{2i-1,2j},\\ B_{ij}&:=E_{2j-1,2i-1}+E_{2j,2i}+E_{2i-1,2j-1}+E_{2i,2j}.\end{split}$

(35)

The indices run over $1\leq i\leq n$ or $1\leq i\neq j\leq n$.

Given a thermal state $\rho$ and its covariance matrix $V$, we have $\tfrac{\partial V}{\partial\nu_{i}}=N_{i}$, so $\tfrac{\partial\rho}{\partial\nu_{i}}$ corresponds to $N_{i}$. Under unitary transformations $\rho\mapsto e^{i\hat{H}t}\rho e^{-i\hat{H}t}$ where $\hat{H}$ is a homogeneous quadratic operator, the covariance matrix transforms as $V\mapsto e^{Xt}Ve^{X^{T}t}$ where $X$ is defined by $-i[\hat{H},\hat{r}_{i}]=\sum_{j=1}^{2n}X_{ij}\hat{r}_{j}$, for $1\leq i\leq 2n$. In terms of tangent vectors, $i[\hat{H},\rho]$ corresponds to $XV+VX^{T}$. Using that $V$ is diagonal, we have the following correspondences:

$\displaystyle\begin{split}i\left[\frac{1}{2}\left(\hat{a}_{i}^{2}+\hat{a}_{i}^{\dagger 2}\right),\rho\right]&\iff 2\nu_{i}S_{i},\\ i\left[\frac{i}{2}\left(\hat{a}_{i}^{2}-\hat{a}_{i}^{\dagger 2}\right),\rho\right]&\iff 2\nu_{i}T_{i},\\ i\left[\hat{a}_{i}\hat{a}_{j}+\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger},\rho\right]&\iff(\nu_{i}+\nu_{j})S_{ij},\\ i\left[i\left(\hat{a}_{i}\hat{a}_{j}-\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\right),\rho\right]&\iff(\nu_{i}+\nu_{j})T_{ij},\\ i\left[\hat{a}_{i}^{\dagger}\hat{a}_{j}+\hat{a}_{i}\hat{a}_{j}^{\dagger},\rho\right]&\iff(\nu_{i}-\nu_{j})A_{ij},\\ i\left[i\left(\hat{a}_{i}^{\dagger}\hat{a}_{j}-\hat{a}_{i}\hat{a}_{j}^{\dagger}\right),\rho\right]&\iff(\nu_{i}-\nu_{j})B_{ij}.\end{split}$

(36)

Finally, when $\hat{H}=\overline{p}\hat{x}-\overline{x}\hat{p}$, the first-order moments of $e^{i\hat{H}t}\rho e^{-i\hat{H}t}$ are $\overline{r}=(\overline{x}t,\overline{p}t)^{T}$. This gives the correspondences:

$\displaystyle i\left[\hat{x}_{i},\rho\right]\iff e_{2i},\;i\left[\hat{p}_{i},\rho\right]\iff-e_{2i-1},$

(37)

where $e_{i}$ is the $2n$ column vector with 1 at the $i$-th entry and all other entries 0.

The above correspondences show that the monotone metric restricted to Gaussian states is given as

$\displaystyle\begin{split}g_{\rho}((d\overline{r},dV),(d\overline{r},dV))&=\frac{1}{4}\sum_{i=1}^{n}\left[\frac{(\text{Tr }dV\,N_{i})^{2}}{f(1)(\nu_{i}^{2}-1)}+\frac{(\text{Tr }dV\,S_{i})^{2}+(\text{Tr }dV\,T_{i})^{2}}{(\nu_{i}-1)^{2}}f\left(\frac{(\nu_{i}+1)^{2}}{(\nu_{i}-1)^{2}}\right)^{-1}\right]\\ &+\frac{1}{8}\sum_{1\leq i<j\leq n}\left[\frac{(\text{Tr }dV\,S_{ij})^{2}+(\text{Tr }dV\,T_{ij})^{2}}{(\nu_{i}-1)(\nu_{j}-1)}f\left(\frac{\nu_{i}+1}{\nu_{i}-1}\frac{\nu_{j}+1}{\nu_{j}-1}\right)^{-1}\right.\\ &\hskip 56.9055pt+\left.\frac{(\text{Tr }dV\,A_{ij})^{2}+(\text{Tr }dV\,B_{ij})^{2}}{(\nu_{i}+1)(\nu_{j}-1)}f\left(\frac{\nu_{i}-1}{\nu_{i}+1}\frac{\nu_{j}+1}{\nu_{j}-1}\right)^{-1}\right]\\ &+\sum_{i=1}^{n}\frac{2}{\nu_{i}-1}f\left(\frac{\nu_{i}+1}{\nu_{i}-1}\right)^{-1}\left(d\overline{r}_{i}^{2}+d\overline{r}_{i+n}^{2}\right).\end{split}$

(38)

We have used that the matrices of Eq. (35) are orthogonal with respect to the inner product $\braket{X,Y}=\text{Tr}(X^{\dagger}Y)$. Each summand of Eq. (38) has a natural interpretation. The first corresponds to change in mean photon number of each mode, the second corresponds to single-mode squeezing of each mode, the third corresponds to two-mode squeezing of each pair of modes, the fourth corresponds to beam splitting of each pair of modes, and the last corresponds to displacement of each mode.

The results of Sec. III are directly applicable when one of the output states ($\rho_{0}$ or $\rho_{1}$) is a thermal state. At target absence, the signal mode is replaced with a thermal state, so only the idler mode needs to be in a thermal state. To directly apply the results of Sec. III, we first assume that the idler mode becomes a thermal state when tracing out the signal mode of the input state. This surmounts to assuming that the input state covariance matrix satisfies $a_{33}=a_{44}$, $a_{34}=0$.

The assumptions $a_{33}=a_{44}=1+2N_{I}$ and $a_{34}=0$ are not that restrictive, since in QI, we are able to perform any local unitary on the idler mode without affecting the decay constants. Therefore, Thm. IV.1 is actually applicable to any two-mode Gaussian state. The symplectic diagonalization of a single-mode state can be written analytically, so we can also write an analytic formula of the decay constants $\gamma_{\text{col}}$ and $\gamma_{\text{loc}}$ for arbitrary two-mode Gaussian states. However, the formula of symplectic diagonalization is lengthy, which leads to an incomprehensible equation for the decay constants. In the limit of large thermal background, $N_{B}\gg 1$, the formula for arbitrary two-mode Gaussian states becomes tractable as follows.