Entanglement partners and monogamy in de Sitter universes

We consider measurement of the field operators $\hat{\varphi},\hat{\pi}$ at spatial points $x_{\text{A}}$ and $x_{\text{B}}$. The measurement process can be represented as the interaction between the field operators and dynamical variables of the measurement apparatus such as Unruh-DeWitt detectors [24]. In the present analysis, we do not specify details of the apparatus but just assume the interaction Hamiltonian between the field operators and the apparatus has the following form:

$$H_{\text{int}}=\sum_{j=\text{A,B}}\lambda_{j}(t)g_{j}(\hat{q}_{D},\hat{p}_{D})\otimes\int dx(w_{1j}(x)\hat{\varphi}(x)+w_{2j}(x)\hat{\pi}(x)),$$

(13)

where $g_{j}$ is a function of canonical variables of the measurement apparatus $(\hat{q}_{D},\hat{p}_{D})$, $w_{1j}(x),w_{2j}(x)$ are window functions defining a spatial local mode of the field at $x_{\text{A,B}}$, and $\lambda_{j}(t)$ is a switching function of the interaction. In the present analysis, we do not treat details of measurement protocols but only pay attention to the behavior of the local modes of the quantum field introduced by the window functions.

Let us introduce local operators at $x_{j}$ ($j=\text{A,B}$) using a window function $w_{j}(x)=w(x-x_{j})$ as

	$\displaystyle\hat{q}_{j}:=\int_{-\infty}^{\infty}dxw_{j}(x)\hat{\varphi}(x)=\int_{-\infty}^{\infty}dk\,\hat{\varphi}_{k}\,e^{ikx_{j}}w_{k},$		(14)
	$\displaystyle\hat{p}_{j}:=\int_{-\infty}^{\infty}dxw_{j}(x)\hat{\pi}(x)=\int_{-\infty}^{\infty}dk\,\hat{\pi}_{k}\,e^{ikx_{j}}w_{k},$		(15)

where the window function is assumed to be localized around $x_{j}$, and $w_{k}$ denotes the Fourier component of the window function:

$$w_{k}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dx\,w(x)e^{ikx},\quad w_{k}=w_{-k}^{*}.$$

(16)

We require that the window function is fixed so that the local operators $(\hat{q}_{j},\hat{p}_{j})$ define independent modes. In other words, they satisfy the canonical commutation relations given by

	$\displaystyle[\hat{q}_{i},\hat{p}_{j}]=i\int_{-\infty}^{\infty}dk\,e^{ik(x_{i}-x_{j})}|w_{k}|^{2}\equiv i\delta_{ij},$		(17)
	$\displaystyle[\hat{q}_{i},\hat{q}_{j}]=[\hat{p}_{i},\hat{p}_{j}]=0.$		(18)

Note that these commutators are independent of the state of the quantum field. Covariances for the local operators are

	$\displaystyle c_{1}(i,j)$	$\displaystyle:=\expectationvalue{\{\hat{q}_{i},\hat{q}_{j}\}}=4\int_{0}^{\infty}dk|w_{k}|^{2}|f_{k}|^{2}\cos k\Delta_{ij},$		(19)
	$\displaystyle c_{2}(i,j)$	$\displaystyle:=\expectationvalue{\{\hat{p}_{i},\hat{p}_{j}\}}=4\int_{0}^{\infty}dk|w_{k}|^{2}|g_{k}|^{2}\cos k\Delta_{ij},$		(20)
	$\displaystyle c_{3}(i,j)$	$\displaystyle:=\expectationvalue{\{\hat{q}_{i},\hat{p}_{j}\}}=2\int_{0}^{\infty}dk|w_{k}|^{2}\,i(f_{k}g_{k}^{*}-f_{k}^{*}g_{k})\cos(k\Delta_{ij}),$		(21)

where $\Delta_{ij}:=x_{i}-x_{j}$.

We adopt the de Sitter expansion of the scale factor $a_{\text{sc}}=-1/(H\eta),-\infty<\eta<0$, where $H$ is the Hubble constant. Mode functions corresponding to the Bunch-Davies vacuum state, which coincides with the Minkowski vacuum state in the short wavelength limit, are given by

$$f_{k}=\frac{1}{\sqrt{2|k|}}\left(1+\frac{1}{i|k|\eta}\right)e^{-i|k|\eta},\quad g_{k}=\sqrt{\frac{|k|}{2}}e^{-i|k|\eta}.$$

(35)

Covariances of the field operators are calculated as

	$\displaystyle M_{11}=\frac{1}{\pi}\int_{0}^{\infty}\frac{dk}{k}\left(1+\frac{1}{k^{2}\eta^{2}}\right)\cos k(x-y),$		(36)
	$\displaystyle M_{22}=\frac{1}{\pi}\int_{0}^{\infty}dk\,k\cos k(x-y),$		(37)
	$\displaystyle M_{12}=\frac{1}{\pi\eta}\int_{0}^{\infty}\frac{dk}{k}\cos k(x-y).$		(38)

We choose the UV and IR cutoff in the window functions as $k_{c}=\pi H/\delta,k_{0}=\pi H$. The IR cutoff represents the physical size of the whole universe $a_{\text{sc}}H^{-1}$ and the UV cutoff represents the physical size of the focusing spatial region $\delta\times a_{\text{sc}}H^{-1},0\leq\delta\leq 1$. Covariances (26), (27), (28) of the local modes are

	$\displaystyle c_{1}=\frac{1}{\pi H}\frac{\delta}{1-\delta}\int_{\delta}^{1}\frac{dz}{z}\left(1+\frac{a_{\text{sc}}^{2}\,\delta^{2}}{\pi^{2}z^{2}}\right)\cos\left(\frac{\pi z}{1+\delta}\right),$		(39)
	$\displaystyle c_{2}=\frac{\pi H}{\delta(1-\delta)}\int_{\delta}^{1}dzz\cos\left(\frac{\pi z}{1+\delta}\right),$		(40)
	$\displaystyle c_{3}=-\frac{1}{\pi H}\frac{a_{\text{sc}}H\delta}{1-\delta}\int_{\delta}^{1}\frac{dz}{z}\cos\left(\frac{\pi z}{1+\delta}\right).$		(41)

The left panel of Fig. 2 shows the evolution of negativity of the bipartite mode AB, with a fixed comoving size $\delta$. The initial nonzero negativity evolves to be zero at some specific value of the scale factor. The physical size of a local region is characterized by $\delta_{p}:=a_{\text{sc}}\delta$. The right panel of Fig. 2 shows the plot of negativity as a function of $(\delta_{p},a_{\text{sc}})$. For a fixed $\delta\in(0,1)$, this figure shows that entanglement (quantum correlation) between the two local modes A and B is lost after the physical size of the comoving region exceeds the Hubble horizon scale and the “classical” behavior of the quantum field emerges [2, 4, 3, 5, 6].

The disentanglement behavior in this figure can be intuitively understood as a result of the fact that “thermal” noise at the Gibbons-Hawking temperature $T_{H}=H/(2\pi)$ associated with the cosmological horizon destroys quantum correlations between A and B. The rest of this paper aims to provide a more quantitative understanding of the disentanglement phenomenon in terms of entanglement monogamy. The bipartite state AB is usually mixed because it is defined as a subsystem embedded in the total universe. As the monogamy relation proposed by Camalet [14, 15, 16, 17, 18] suggests, the amount of quantum correlation between A and B (i.e., internal correlation) is affected by the amount of quantum correlation between AB and its complement (i.e., external correlation). Therefore, we look for the partner modes that purify the bipartite mode AB and investigate the entanglement structure among them in the following sections.

The partners here are a special class of QIC. In [23, 22, 21], it has been shown that a linear map, denoted by $f_{\psi}$ for a pure Gaussian state $\ket{\psi}$, plays a key role in identifying a QIC. We here briefly review the results in [23, 22, 21], including the properties of $f_{\psi}$.

Let us first consider a system composed of $N$ harmonic oscillators, which is assumed to be in a pure Gaussian state $\ket{\psi}$. The canonical variables are defined by $\hat{\boldsymbol{r}}=(\hat{q}_{1},\hat{p}_{1},\dots,\hat{q}_{N},\hat{p}_{N})^{T}$. For simplicity, we assume that the first moments of the canonical variables vanish, i.e., $\langle\hat{\boldsymbol{r}}\rangle=0$, where $\langle\,\rangle$ denotes the expectation value in $\ket{\psi}$. The covariance matrix with respect to these canonical variables is defined by $\boldsymbol{m}:=\langle\{\hat{\boldsymbol{r}},\hat{\boldsymbol{r}}^{T}\}\rangle$.¹¹1$[\hat{\boldsymbol{r}},\hat{\boldsymbol{r}}^{T}]$ is the antisymmetric part of the operator $\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}^{T}$ and $\{\hat{\boldsymbol{r}},\hat{\boldsymbol{r}}^{T}\}$ is the symmetric part of the operator $\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}^{T}$. Because the total system is assumed to be in a pure Gaussian state, all the symplectic eigenvalues of $\boldsymbol{m}$ are one. That is, there exists a symplectic matrix $\boldsymbol{S}$ such that

$\displaystyle\boldsymbol{m}=\boldsymbol{S}\boldsymbol{S}^{T},$

(42)

where $\boldsymbol{S}$ satisfies $\boldsymbol{S}^{T}\boldsymbol{\Omega}_{N}\boldsymbol{S}=\boldsymbol{\Omega}_{N}$ and $[\hat{\boldsymbol{r}},\hat{\boldsymbol{r}}^{T}]=i\boldsymbol{\Omega}_{N}$ for

$$\boldsymbol{\Omega}_{N}:=\bigoplus_{i=1}^{N}\boldsymbol{J},\quad\boldsymbol{J}=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.$$

(43)

Note that the pure state condition (42) is equivalent to the following relation:

$$\boldsymbol{m}\boldsymbol{\Omega}_{N}\boldsymbol{m}=\boldsymbol{\Omega}_{N}.$$

(44)

If we introduce a new basis $\hat{\boldsymbol{r}}^{\prime}$ for the canonical variables by $\hat{\boldsymbol{r}}^{\prime}=\boldsymbol{S}^{-1}\hat{\boldsymbol{r}}$, we get

$$\boldsymbol{m}^{\prime}=\langle\{\hat{\boldsymbol{r}}^{\prime},\hat{\boldsymbol{r}}^{\prime T}\}\rangle=\boldsymbol{S}\boldsymbol{m}(\boldsymbol{S}^{T})^{-1}=\mathbb{I}_{2N},\quad[\hat{\boldsymbol{r}}^{\prime},\hat{\boldsymbol{r}}^{\prime T}]=i\boldsymbol{\Omega}_{N}.$$

(45)

The canonical variables defined by $\hat{\boldsymbol{r}}^{\prime}$ specify $N$ uncorrelated modes, each of which is in a pure state. Now we define a linear map $f_{\psi}$ by

$$f_{\psi}(\hat{q}_{i}^{\prime})=\hat{p}_{i}^{\prime},\quad f_{\psi}(\hat{p}_{i}^{\prime})=-\hat{q}_{i}^{\prime},$$

(46)

or equivalently,

$$f_{\psi}(\hat{\boldsymbol{r}}^{\prime})=\boldsymbol{\Omega}_{N}\hat{\boldsymbol{r}}^{\prime}.$$

(47)

This map has the following properties:

	$\displaystyle[\hat{\boldsymbol{r}}^{\prime},f_{\psi}(\hat{\boldsymbol{r}}^{\prime})^{T}]$	$\displaystyle=i\mathbb{I}_{2N},$		(48)
	$\displaystyle\langle\{\hat{\boldsymbol{r}}^{\prime},\hat{\boldsymbol{r}}^{\prime T}\}\rangle$	$\displaystyle=\mathbb{I}_{2N},\quad\langle\{\hat{\boldsymbol{r}}^{\prime},f_{\psi}(\hat{\boldsymbol{r}}^{\prime})^{T}\}\rangle=0,\quad\langle\{f_{\psi}(\hat{\boldsymbol{r}}^{\prime}),f_{\psi}(\hat{\boldsymbol{r}}^{\prime})^{T}\}\rangle=\mathbb{I}_{2N}.$		(49)

On the one hand, Eq. (48) means that $[\hat{r_{j}}^{\prime},f_{\psi}(\hat{r_{j}}^{\prime})]=i$; i.e., $(\hat{r_{j}}^{\prime},f_{\psi}(\hat{r_{j}}^{\prime}))$ defines a mode for any $j=1,\dots,2N$. On the other hand, Eq. (49) implies that the mode defined by $(\hat{r_{j}}^{\prime},f_{\psi}(\hat{r_{j}}^{\prime}))$ is in a pure state. Of course, these are straightforward consequences of the definition in Eq. (46).

Let us generalize this observation. Consider an operator $\hat{O}$ given by a linear combination of canonical variables. We first rescale this operator as $\hat{O}\to\hat{O}/\sqrt{2\langle\hat{O}^{2}\rangle}$ so that $\langle\hat{O}^{2}\rangle=1/2$. Expanding $\hat{O}$ in the basis $\hat{\boldsymbol{r}}^{\prime}$ as

$\displaystyle\hat{O}:=\sum_{i=1}^{N}w_{i}^{\prime}\hat{r}_{i}^{\prime}=\boldsymbol{w}^{\prime T}\hat{\boldsymbol{r}}^{\prime},$

(50)

where $\boldsymbol{w}^{\prime T}=(w_{1}^{\prime},w_{2}^{\prime},\dots,w_{2N}^{\prime})$, the normalization condition is equivalent to

$\displaystyle\boldsymbol{w}^{\prime T}\boldsymbol{w}^{\prime}=1.$

(51)

Introducing operators $(\hat{Q},\hat{P}):=(\hat{O},f_{\psi}(\hat{O}))$ for $\hat{O}$ with this normalization, from Eqs. (48), (49) and (51), we find

	$\displaystyle[\hat{Q},\hat{P}]$	$\displaystyle=i$		(52)
	$\displaystyle\langle\{\hat{Q},\hat{Q}\}\rangle$	$\displaystyle=1,\quad\langle\{\hat{Q},\hat{P}\}\rangle=\langle\{\hat{P},\hat{Q}\}\rangle=0,\quad\langle\{\hat{P},\hat{P}\}\rangle=1.$		(53)

Equation (52) means that $(\hat{Q},\hat{P})$ satisfies the canonical commutation relation and hence defines a mode. Further, Eq. (53) implies that the covariance matrix for this mode is equal to the $2\times 2$ identity matrix, implying that it is in a pure state.

In a series of studies [32, 23, 22, 21] on the carriers of information, the smallest subsystem in a pure state that carries the whole encoded information is termed a QIC. The encoded information can be fully retrieved by extracting a QIC from the system. Equations (52) and (53) imply that the mode defined by $(\hat{Q},\hat{P}):=(\hat{O},f_{\psi}(\hat{O}))$ is the QIC when the encoding operation is generated by $\hat{O}$. In a more general case where the encoding operation is generated by $\{\hat{O}_{i}\}_{i=1}^{n}$, where each of which is assumed to be a linear combination of canonical variables, it is proven [23] that the QIC is given by a subsystem composed of (at most) $n$ modes, which is algebraically defined by

$\displaystyle\{(\hat{O}_{i},f_{\psi}(\hat{O}_{i})\}_{i=1}^{n}.$

(54)

It is shown that operators $\{(\hat{Q}_{i},\hat{P}_{i})\}_{i}$ defined in Eq. (73) in [23] satisfy

	$\displaystyle[\hat{Q}_{j},\hat{P}_{k}]$	$\displaystyle=i\delta_{jk},\quad[\hat{Q}_{j},\hat{Q}_{k}]=[\hat{P}_{j},\hat{P}_{k}]=0$		(55)
	$\displaystyle\langle\{\hat{Q}_{j},\hat{Q}_{k}\}\rangle$	$\displaystyle=\delta_{jk},\quad\langle\{\hat{Q}_{j},\hat{P}_{k}\}\rangle=\langle\{\hat{P}_{j},\hat{Q}_{k}\}\rangle=0,\quad\langle\{\hat{P}_{j},\hat{P}_{k}\}\rangle=\delta_{jk},$		(56)

which generalizes Eqs. (55) and (56). It implies that the mode characterized by $\{(\hat{Q}_{i},\hat{P}_{i})\}_{i}$ is in a pure state when the total system is in $\ket{\psi}$, and therefore, they are the QIC as the encoded information is carried by them.

Although the subsystem composed of $n$ modes playing the role of QIC is uniquely determined, there are several ways to decompose it into $n$ independent modes. From Eq. (56), the Gaussian state $\ket{\psi}$ of the total system is decomposed into

$\displaystyle\ket{\psi}=\ket{\psi^{\prime}}_{12\cdots n}\otimes\ket{\psi^{\prime\prime}}_{\overline{12\cdots n}},\quad\ket{\psi^{\prime}}_{12\cdots n}:=\bigotimes_{j=1}^{n}\ket{0}_{j}$

(57)

where $\ket{0}_{j}$ denotes the “vacuum” state for the $j$th mode annihilated by $\hat{a}_{j}:=(\hat{Q}_{j}+i\hat{P}_{j})/\sqrt{2}$ and $\ket{\psi^{\prime\prime}}_{\overline{12\cdots n}}$ denotes a pure state for the complementary system. Since each of the $n$ modes is in a pure state in this decomposition, the analysis of the entanglement structure is not straightforward. In the following subsections, we introduce another decomposition that is more useful in analyzing the entanglement structure among partners.

For later convenience, we summarize here the properties of $f_{\psi}$. From the definition in Eq. (47), $f_{\psi}$ maps the original basis $\hat{\boldsymbol{r}}$ to

$$f_{\psi}(\hat{\boldsymbol{r}})=f_{\psi}(\boldsymbol{S}\hat{\boldsymbol{r}}^{\prime})=\boldsymbol{S}\boldsymbol{\Omega}_{N}\boldsymbol{S}^{-1}\hat{\boldsymbol{r}}=\boldsymbol{m}\boldsymbol{\Omega}_{N}\hat{\boldsymbol{r}},$$

(58)

where we have used $\boldsymbol{S}^{-1}=-\boldsymbol{\Omega}_{N}\boldsymbol{S}^{T}\boldsymbol{\Omega}_{N}$. From Eqs. (48) and (49), for any operators $\hat{O}$ and $\hat{O}^{\prime}$ given by linear combinations of canonical operators by (65) [22, 23], it can be directly checked that

	$\displaystyle\langle\hat{O}\rangle=0,\quad\langle f_{\psi}(\hat{O})\rangle=0,\quad f_{\psi}(f_{\psi}(\hat{O}))=-\hat{O}$		(59)
	$\displaystyle[\hat{O},f_{\psi}(\hat{O}^{\prime})]=i\langle\{\hat{O},\hat{O}^{\prime}\}\rangle,\quad\langle\{\hat{O},f_{\psi}(\hat{O}^{\prime})\}\rangle=i[\hat{O},\hat{O}^{\prime}],$		(60)
	$\displaystyle[f_{\psi}(\hat{O}),f_{\psi}(\hat{O}^{\prime})]=[\hat{O},\hat{O}^{\prime}],\quad\langle\{f_{\psi}(\hat{O}),f_{\psi}(\hat{O}^{\prime})\}\rangle=\langle\{\hat{O},\hat{O}^{\prime}\}\rangle$		(61)

hold.

So far, we have reviewed the properties of map $f_{\psi}$ in a harmonic oscillator system. The analyses can readily be extended to a scalar field by the following procedure. For a scalar field $\hat{\varphi}$ and its conjugate momentum $\hat{\pi}$ at a fixed time $t$, we denote

$\displaystyle\hat{\boldsymbol{R}}(x):=\begin{bmatrix}\hat{\varphi}(t,x)\\ \hat{\pi}(t,x)\end{bmatrix}.$

(62)

Here, for notational simplicity, we omit the time variable $t$ on the left-hand side. The equal-time commutation relations in Eq. (9) are written as

$$[\hat{\boldsymbol{R}}(x),\hat{\boldsymbol{R}}^{T}\!(y)]=i\begin{bmatrix}0&\delta(x-y)\\ -\delta(x-y)&0\end{bmatrix}=i\boldsymbol{J}\delta(x-y),$$

(63)

where $\boldsymbol{J}=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}$, while the covariance of the field operators in the state $\ket{\psi}$ are denoted by

$$\boldsymbol{M}(x,y):=\langle\{\hat{\boldsymbol{R}}(x),\hat{\boldsymbol{R}}^{T}\!(y)\}\rangle=\begin{bmatrix}M_{11}(x,y)&M_{12}(x,y)\\ M_{21}(x,y)&M_{22}(x,y)\end{bmatrix}=\begin{bmatrix}\langle\{\hat{\varphi}(x),\hat{\varphi}(y)\}\rangle&\langle\{\hat{\varphi}(x),\hat{\pi}(y)\}\rangle\\ \langle\{\hat{\pi}(x),\hat{\varphi}(y)\}\rangle&\langle\{\hat{\pi}(x),\hat{\pi}(y)\}\rangle\end{bmatrix}.$$

(64)

We introduce an operator

$$\hat{O}:=\int dx\left[w_{1}(x)\hat{\varphi}(x)+w_{2}(x)\hat{\pi}(x)\right]=\int dx\boldsymbol{W}^{T}\!(x)\hat{\boldsymbol{R}}(x),$$

(65)

where

$$\boldsymbol{W}(x)=\begin{bmatrix}w_{1}(x)\\ w_{2}(x)\end{bmatrix}$$

(66)

denotes weighting functions. In the analogy with Eq. (58), we define [23] a map $f_{\psi}$ by

	$\displaystyle f_{\psi}(\hat{O})$	$\displaystyle:=\int dxdydz\boldsymbol{W}^{T}(x)\boldsymbol{M}(x,y)\boldsymbol{J}\delta(y-z)\hat{\boldsymbol{R}}(z)$
		$\displaystyle=\int dx\boldsymbol{W}_{\!\!\!f_{\psi}(\hat{O})}^{T}(x)\hat{\boldsymbol{R}}(x),$		(67)

where $\boldsymbol{W}_{\!\!\!f_{\psi}(\hat{O})}$ is the window function defining $f_{\psi}(\hat{O})$, given by

$$\boldsymbol{W}_{\!\!\!f_{\psi}(\hat{O})}(x):=-\int dy\boldsymbol{J}\boldsymbol{M}(x,y)\boldsymbol{W}(y).$$

(68)

When the covariance of the field operator satisfies a purity condition

$\displaystyle\int dydz\boldsymbol{M}(x,y)\boldsymbol{J}\delta(y-z)\boldsymbol{M}(z,w)=\boldsymbol{J}\delta(x-w),$

(69)

which corresponds to Eq. (44), it is shown [22, 23] that the map $f_{\psi}$ satisfies all the properties in Eqs. (59), (60) and (61). Therefore, for operators $\{O_{i}\}_{i=1}^{n}$ given by linear combinations of field operators, the set of operators

$\displaystyle\{(\hat{O}_{i},f_{\psi}(\hat{O}_{i})\}_{i=1}^{n}$

(70)

defines (at most) $n$ modes in a pure state, provided that the field is in a pure Gaussian state $\ket{\psi}$. Note that Eq. (69) can be explicitly confirmed for the covariances given in Eqs. (36), (37) and (38).

Based on these results, we derive the partner formula for a single mode in Sec. III.2, which reproduces the formula in [19]. We further generalize it for the partner formula for two modes in Sec. III.3, which we shall use to analyze entanglement monogamy among local modes in a field. See Fig. 3 for the schematic picture of these setups.

As a practical application of the map $f_{\psi}$, we look for a partner mode that purifies a given single mode A [Fig. 3 (a)]. In particular, we apply the partner formula for a local mode $\hat{\boldsymbol{\xi}}_{\text{A}}=(\hat{q}_{\text{A}},\hat{p}_{\text{A}})^{T}$ at a spatial point ${\boldsymbol{x}}_{\text{A}}$ defined in the previous section. As we have seen in Sec. III.1, four operators

$$\hat{q}_{\text{A}},\hat{p}_{\text{A}},f_{\psi}(\hat{q}_{\text{A}}),f_{\psi}(\hat{p}_{\text{A}}),$$

(71)

define a two-mode system that is in a pure state, provided that the field is in a pure Gaussian state. To identify the partner mode of $\hat{\boldsymbol{\xi}}_{\text{A}}$, we here construct a mode generated by the operators in Eq. (71), which is orthonormal to the mode A. The covariance matrix for $\hat{\boldsymbol{\xi}}_{\text{A}}$ is

$$\boldsymbol{m}_{\text{A}}:=\expectationvalue{\{\hat{\boldsymbol{\xi}}_{\text{A}},\hat{\boldsymbol{\xi}}^{T}_{\text{A}}\}}=\begin{bmatrix}2\expectationvalue{\hat{q}_{\text{A}}^{2}}&\expectationvalue{\hat{q}_{\text{A}}\hat{p}_{\text{A}}+\hat{p}_{\text{A}}\hat{q}_{\text{A}}}\\ \expectationvalue{\hat{q}_{\text{A}}\hat{p}_{\text{A}}+\hat{p}_{\text{A}}\hat{q}_{\text{A}}}&2\expectationvalue{\hat{p}_{\text{A}}^{2}}\end{bmatrix}=\begin{bmatrix}c_{1}&c_{3}\\ c_{3}&c_{2}\end{bmatrix}.$$

(72)

Commutators and covariances between these operators are given by

$$[\hat{\boldsymbol{\xi}}_{\text{A}},\hat{\boldsymbol{\xi}}^{T}_{\text{A}}]=[f_{\psi}(\hat{\boldsymbol{\xi}}_{\text{A}}),f_{\psi}(\hat{\boldsymbol{\xi}}^{T}_{\text{A}})]=i\boldsymbol{J},\quad[\hat{\boldsymbol{\xi}}_{\text{A}},f_{\psi}(\hat{\boldsymbol{\xi}}^{T}_{\text{A}})]=i\boldsymbol{m}_{\text{A}},$$

(73)

and

$$\expectationvalue{\{\hat{\boldsymbol{\xi}}_{\text{A}},\hat{\boldsymbol{\xi}}^{T}_{\text{A}}\}}=\expectationvalue{\{f_{\psi}(\hat{\boldsymbol{\xi}}_{\text{A}}),f_{\psi}(\hat{\boldsymbol{\xi}}^{T}_{\text{A}})\}}=\boldsymbol{m}_{\text{A}},\quad\expectationvalue{\{\hat{\boldsymbol{\xi}}_{\text{A}},f_{\psi}(\hat{\boldsymbol{\xi}}^{T}_{\text{A}})\}}=-\boldsymbol{J}.$$

(74)

To extract a mode orthogonal to the original mode $\hat{\boldsymbol{\xi}}_{\text{A}}$ from $f_{\psi}(\hat{\boldsymbol{\xi}}_{\text{A}})$, we define operators

$$\hat{\boldsymbol{\zeta}}:=f_{\psi}(\hat{\boldsymbol{\xi}}_{\text{A}})-\boldsymbol{m}_{\text{A}}\boldsymbol{J}\hat{\boldsymbol{\xi}}_{\text{A}}.$$

(75)

They indeed commute with $\hat{\boldsymbol{\xi}}_{\text{A}}$ as

	$\displaystyle[\hat{\boldsymbol{\zeta}},\hat{\boldsymbol{\xi}}^{T}_{\text{A}}]$	$\displaystyle=[f_{\psi}(\hat{\boldsymbol{\xi}}_{\text{A}}),\hat{\boldsymbol{\xi}}^{T}_{\text{A}}]-[\boldsymbol{m}_{\text{A}}\boldsymbol{J}\hat{\boldsymbol{\xi}}_{\text{A}},\hat{\boldsymbol{\xi}}^{T}_{\text{A}}]$
		$\displaystyle=-\boldsymbol{m}_{\text{A}}-\boldsymbol{m}_{\text{A}}\boldsymbol{J}[\hat{\boldsymbol{\xi}}_{\text{A}},\hat{\boldsymbol{\xi}}^{T}_{\text{A}}]$
		$\displaystyle=0.$		(76)

Therefore, they define a mode orthogonal to $\hat{\boldsymbol{\xi}}_{\text{A}}$.