$\displaystyle\geq\left(\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))\right)^{-1}$
$\displaystyle\quad+\left(\Phi^{*}(\mathbf{X}^{\prime}\sqcup\mathbf{Y}^{\prime}:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))\right)^{-1}.$

###### Proof.

Let

$\displaystyle P_{0}$ $\displaystyle:L_{2}(\mathcal{A},\varphi)\to
L_{2}(B,\tau_{B})$ $\displaystyle P_{1}$
$\displaystyle:L_{2}(\mathcal{A},\varphi)\to\overline{\operatorname{alg}(C_{\ell},C_{r},\mathbf{X},\mathbf{Y})}^{\left\|\,\cdot\,\right\|_{\tau}}$
$\displaystyle P_{2}$
$\displaystyle:L_{2}(\mathcal{A},\varphi)\to\overline{\operatorname{alg}(D_{\ell},D_{r},\mathbf{X}^{\prime},\mathbf{Y}^{\prime})}^{\left\|\,\cdot\,\right\|_{\tau}}$

be the orthogonal projections onto their co-domains. Note that if

$Z\in\operatorname{alg}(C_{\ell},C_{r},\mathbf{X},\mathbf{Y})\quad\text{and}\quad\operatorname{alg}(D_{\ell},D_{r},\mathbf{X}^{\prime},\mathbf{Y}^{\prime}),$

then bi-freeness implies

$\widetilde{E}(ZZ^{\prime})=\widetilde{E}\left(Z\widetilde{E}(Z^{\prime})\right).$

Indeed, this is easily seen as if $Z$ and $Z^{\prime}$ are monomials, then any
cumulant of the monomial $ZZ^{\prime}$ corresponding to a bi-non-crossing
partition is non-zero if and only if it decomposes into a bi-non-crossing
partition on $Z$ union a bi-non-crossing partition on $Z^{\prime}$. Thus
$P_{1}P_{2}=P_{2}P_{1}=P_{0}$.

The remainder of the proof can then be read from [S1998]*Proposition 4.5,
[CS2020]*Proposition 5.8, or even [V1998-2]*Proposition 6.5. ∎

######  Proposition 7.4 (Bi-Free Cramer-Rao Inequality).

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, let
$\mathbf{X}\in A_{\ell}^{n}$ and $\mathbf{Y}\in A_{r}^{m}$ consist of self-
adjoint operators and let $(C_{\ell},C_{r})$ be a pair of $B$-algebras of $A$.
Then

$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))\tau\left(\sum^{n}_{i=1}X_{i}^{2}+\sum^{m}_{j=1}Y_{j}^{2}\right)\geq\left(\sum^{n}_{i=1}\tau_{B}(\eta_{\ell,i}(1))+\sum^{m}_{j=1}\tau_{B}(\eta_{r,j}(1))\right)^{2}.$

Moreover, equality holds if $(\mathbf{X},\mathbf{Y})$ are
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$-bi-semicircular
elements and

$\\{(C_{\ell},C_{r})\\}\cup\\{(\operatorname{alg}(B_{\ell},X_{i}),B_{r})\\}^{n}_{i=1}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},Y_{j}))\\}^{m}_{j=1}$

are bi-free with amalgamation over $B$ with respect to $E$. The converse holds
when $C_{\ell}=B_{\ell}$ and $C_{r}=B_{r}$.

###### Proof.

The result follow from the obvious modifications to [CS2020]*Proposition 5.10.
Also see [S1998]*Proposition 4.6 and [V1998-2]*Proposition 6.9. ∎

Similarly, limits behave as one expects based on [CS2020, V1998-2, S1998].

######  Proposition 7.5.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, let
$\mathbf{X}\in A_{\ell}^{n}$ and $\mathbf{Y}\in A_{r}^{m}$ consist of self-
adjoint operators and let $(C_{\ell},C_{r})$ be a pair of $B$-algebras of $A$.
Suppose further for each $k\in\mathbb{N}$ that $\mathbf{X}^{(k)}\in
A_{\ell}^{n}$ and $\mathbf{Y}^{(k)}\in A_{r}^{m}$ are tuples of self-adjoint
elements in $A$ such that

$\displaystyle\limsup_{k\to\infty}\left\|X^{(k)}_{i}\right\|<\infty,$
$\displaystyle\limsup_{k\to\infty}\left\|Y^{(k)}_{j}\right\|<\infty,$
$\displaystyle s\text{-}\lim_{k\to\infty}X^{(k)}_{i}=X_{i},\text{ and}$
$\displaystyle s\text{-}\lim_{k\to\infty}Y^{(k)}_{j}=Y_{j}$

for all $1\leq i\leq n$ and $1\leq j\leq m$ (where the strong limit is
computed as bounded linear maps acting on $L_{2}(A,\tau)$). Then

$\displaystyle\liminf_{k\to\infty}\Phi^{*}\left(\mathbf{X}^{(k)}\sqcup\mathbf{Y}^{(k)}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right)\geq\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))$

The proof of Proposition 7.5 becomes identical to [CS2020]*Proposition 5.12
once the following lemma is established. Also see [S1998]*Proposition 4.7 and
[V1998-2]*Proposition 6.10.

######  Lemma 7.6.

Under the assumptions of Proposition 7.5 along with the additional assumptions
that

$\xi_{k}=J_{\ell}\left(X_{1}^{(k)}:\left(\operatorname{alg}\left(C_{\ell},\widehat{\mathbf{X}}^{(k)}_{1}\right),\operatorname{alg}(C_{r},\mathbf{Y}^{(k)})\right),\eta_{\ell,1}\right)$

exist and are bounded in $L_{2}$-norm by some constant $K>0$, it follows that

$\xi=J_{\ell}\left(X_{1}:\left(\operatorname{alg}\left(C_{\ell},\widehat{\mathbf{X}}_{1}\right),\operatorname{alg}(C_{r},\mathbf{Y})\right),\eta_{\ell,1}\right)$

exists and is equal to

$w\text{-}\lim_{k\to\infty}P\left(\xi_{k}\right)$

where $P$ is the orthogonal projection of $L_{2}(A,\tau)$ onto
$\overline{\operatorname{alg}(C_{\ell},C_{r},\mathbf{X},\mathbf{Y})}^{\left\|\,\cdot\,\right\|_{\tau}}$.

If, in addition,

$\limsup_{k\to\infty}\left\|\xi_{k}\right\|_{2}\leq\left\|\xi\right\|_{2}$

then

$\lim_{k\to\infty}\left\|\xi_{k}-\xi\right\|_{2}=0.$

The same holds with $X_{1}$ replaced with $X_{i}$, and a similar result holds
for the right.

###### Proof.

The proof of this result follows from the same sequence of steps as
[CS2020]*Lemma 5.13 using the analytical operator-valued bi-free cumulants. ∎

######  Corollary 7.7.

Under the assumptions of Proposition 7.5, if in addition

$(B_{\ell}(\mathbf{X}),B_{r}(\mathbf{Y}))\qquad\text{and}\qquad\left(\operatorname{alg}\left(C_{\ell},\mathbf{X}^{(k)}\right),\operatorname{alg}\left(C_{r},\mathbf{Y}^{(k)}\right)\right)$

are bi-free for all $k$ and

$\lim_{k\to\infty}\left\|X^{(k)}_{i}\right\|=\lim_{k\to\infty}\left\|Y_{j}^{(k)}\right\|=0$

for all $1\leq i\leq n$ and $1\leq j\leq m$, then

$\displaystyle\lim_{k\to\infty}\Phi^{*}\left(\mathbf{X}+\mathbf{X}^{(k)}\sqcup\mathbf{Y}+\mathbf{Y}^{(k)}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right)=\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

Furthermore, if $C_{\ell}=B_{\ell}$, $C_{r}=B_{r}$, and

$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))<\infty,$

then

$J_{\ell}\left(X^{(k)}_{i}:\left(B_{\ell}\left(\widehat{(\mathbf{X}+\mathbf{X}^{(k)})}_{i}\right),B_{r}\left(\mathbf{Y}+\mathbf{Y}^{(k)}\right)\right),\eta_{\ell,i}\right)$

tends to

$J_{\ell}\left(X_{i}:\left(B_{\ell}\left(\hat{\mathbf{X}}_{i}\right),B_{r}\left(\mathbf{Y}\right)\right),\eta_{\ell,i}\right)$

in $\tau$-norm. A similar result holds for right bi-free conjugate variables.

###### Proof.

The proof is identical to [CS2020]*Corollary 5.14 and thus is omitted. ∎

######  Theorem 7.8.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, let
$\mathbf{X}\in A_{\ell}^{n}$ and $\mathbf{Y}\in A_{r}^{m}$ be tuples of self-
adjoint operators, let $(\\{S_{i}\\}^{n}_{i=1},\\{D_{j}\\}^{m}_{j=1})$ be a
collection of $(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ bi-
semicircular operators in $A$ and let $(C_{\ell},C_{r})$ be pairs of
$B$-algebras of $A$ such that

$\\{(\operatorname{alg}(C_{\ell},\mathbf{X}),\operatorname{alg}(C_{r},\mathbf{Y})\rangle)\\}\cup\\{(\operatorname{alg}(B_{\ell},S_{i}),B_{r})\\}^{n}_{i=1}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},D_{j}))\\}^{m}_{j=1}$

are bi-free. Then, the map

$h:[0,\infty)\ni
t\mapsto\Phi^{*}\left(\mathbf{X}+\sqrt{t}\mathbf{S}\sqcup\mathbf{Y}+\sqrt{t}\mathbf{D}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right)$

is decreasing, right continuous, and

$\frac{K_{2}^{2}}{K_{1}+K_{2}t}\leq h(t)\leq\frac{1}{t}K_{3}$

where

$\displaystyle K_{1}$
$\displaystyle=\tau\left(\sum^{n}_{i=1}X_{i}^{2}+\sum^{m}_{j=1}Y_{j}^{2}\right),$
$\displaystyle K_{2}$
$\displaystyle=\sum^{n}_{i=1}\tau_{B}(\eta_{\ell,i}(1_{B}))+\sum^{m}_{j=1}\tau_{B}(\eta_{r,j}(1_{B}))\text{
and}$ $\displaystyle K_{3}$
$\displaystyle=\sum^{n}_{i=1}\tau_{B}(\eta_{\ell,i}(1_{B}))^{2}+\sum^{m}_{j=1}\tau_{B}(\eta_{r,j}(1_{B}))^{2}.$

Moreover if $(\mathbf{X},\mathbf{Y})$ is the
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$-bi-semicircular
distribution and

$\\{(C_{\ell},C_{r})\\}\cup\\{(\operatorname{alg}(B_{\ell},X_{i}),B_{r})\\}^{n}_{i=1}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},Y_{j}))\\}^{m}_{j=1}$

are bi-free with amalgamation over $B$ with respect to $E$, then
$h(t)=\frac{K_{2}^{2}}{K_{1}^{2}+K_{2}t}$ for all $t$.

Finally, if $C_{\ell}=B_{\ell}$, $C_{r}=B_{r}$, and
$h(t)=\frac{K_{2}^{2}}{K_{1}+K_{2}t}$ for all $t$, then
$(\mathbf{X},\mathbf{Y})$ is the
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$-bi-semicircular
distribution.

###### Proof.

The proof becomes identical to [CS2020]*Theorem 5.15 using the above and the
fact that

$\tau\left(\left(X_{i}+\sqrt{t}S_{i}\right)^{2}\right)=\tau\left(X_{i}^{2}\right)+t\tau\left(S_{i}^{2}\right)=\tau\left(X_{i}^{2}\right)+t\tau_{B}\left(\eta_{\ell,i}(1_{B})\right),$

with an analogous computation on the right for use in the lower bound
computation, in conjunction with the Bi-Free Cramer-Rao Inequality
(Proposition 7.4). ∎

##  8\. Bi-Free Entropy with Respect to Completely Positive Maps

With the construction and properties of the bi-free Fisher information with
respect to completely positive maps complete, the construction and properties
of bi-free entropy with respect to completely positive maps follows easily by
extending results from [S1998] and [CS2020] with similar proofs.

######  Definition 8.1.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, let
$(C_{\ell},C_{r})$ be pairs of $B$-algebras of $A$, and let $\mathbf{X}\in
A_{\ell}^{n}$ and $\mathbf{Y}\in A_{r}^{m}$ be tuples of self-adjoint
operators. The _relative bi-free entropy of $(\mathbf{X},\mathbf{Y})$ with
respect to $(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ in the
presence of $(B_{\ell},B_{r})$_ is defined to be

$\displaystyle\chi^{*}$
$\displaystyle(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))$
$\displaystyle=\frac{K}{2}\ln(2\pi
e)+\frac{1}{2}\int^{\infty}_{0}\left(\frac{K}{1+t}-\Phi^{*}\left(\mathbf{X}+\sqrt{t}\mathbf{S}\sqcup\mathbf{Y}+\sqrt{t}\mathbf{D}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right)\right)\,dt,$

where

$K=\sum^{n}_{i=1}\tau_{B}(\eta_{\ell,i}(1_{B}))+\sum^{m}_{j=1}\tau_{B}(\eta_{r,j}(1_{B}))$

and $(\\{S_{i}\\}^{n}_{i=1},\\{D_{j}\\}^{m}_{j=1})$ is a collection of
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ bi-semicircular
operators such that

$\\{(\operatorname{alg}(C_{\ell},\mathbf{X}),\operatorname{alg}(C_{r},\mathbf{Y})\rangle)\\}\cup\\{(B_{\ell}(S_{i}),B_{r})\\}^{n}_{i=1}\cup\\{(B_{\ell},B_{r}(D_{j}))\\}^{m}_{j=1}$

are bi-free (note such semicircular operators can be included in $A$ by
Theorem 6.5).

In the case that $C_{\ell}=B_{\ell}$ and $C_{r}=B_{r}$, we use
$\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))$
to denote the bi-free entropy. If in addition $\eta_{\ell,i}=\eta_{r,j}=\eta$
for all $i$ and $j$, we use $\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:\eta)$ to
denote the bi-free entropy.

We note there is a slight change in the normalization used in Definition 8.1
over that used in [S1998]*Definition 8.1. Generally this makes no real
difference other than making some of the bounds in this section nice, such as
the following one.

######  Proposition 8.2.

In the context of Definition 8.1, if

$K_{1}=\tau\left(\sum^{n}_{i=1}X_{i}^{2}+\sum^{m}_{j=1}Y_{j}^{2}\right),$

then

$\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))\leq\frac{K}{2}\ln\left(\frac{2\pi
e}{K}K_{1}\right).$

Moreover, equality holds when $(\mathbf{X},\mathbf{Y})$ are
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$-bi-semicircular
operators such that
$\\{(C_{\ell},C_{r})\\}\cup\\{(\operatorname{alg}(B_{\ell},X_{i}),B_{r})\\}^{n}_{i=1}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},Y_{j}))\\}^{m}_{j=1}$
are bi-free and if $C_{\ell}=B_{\ell}$ and $C_{r}=B_{r}$, this is the only
setting where equality holds.

###### Proof.

The proof is identical to [CS2020]*Proposition 6.5 in conjunction with Theorem
7.8. ∎

######  Remark 8.3.

  1. (i)

In the case that $B=\mathbb{C}$ and $\eta$ is unital, Definition 8.1 produces
the non-microstate bi-free entropy from [CS2020]*Definition 6.1.

  2. (ii)

In the setting of Example 2.6, when $C_{r}=B_{r}$ and
$\eta_{\ell,i}=\eta_{r,j}=\eta$ for all $i$ and $j$, Definition 8.1 produces
the free entropy with respect to a completely positive map from
[S1998]*Definition 8.1 modulo an additive constant (which is 0 in the case
$\eta$ is unital).

Of course, due to the fact that the bi-free Fisher information from Section 7
behaves analogously to the Fisher informations considered in [CS2020, S1998],
results for the behaviour of entropy automatically generalize.

######  Proposition 8.4.

Using Remark 7.2, the following hold:

  1. (v)

In the context of Proposition 5.10 (i.e. reducing the $B$-$B$-non-commutative
probability space to a $D$-$D$-non-commutative probability space),

$\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))=\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\circ
F\\}^{n}_{i=1},\\{\eta_{r,j}\circ F\\}^{m}_{j=1})).$

  2. (vi)

For all $\lambda\in\mathbb{R}\setminus\\{0\\}$

$\displaystyle\chi^{*}$
$\displaystyle(\lambda\mathbf{X}\sqcup\lambda\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))$
$\displaystyle=\left(\sum^{n}_{i=1}\tau_{B}(\eta_{\ell,i}(1_{B}))+\sum^{m}_{j=1}\tau_{B}(\eta_{r,j}(1_{B}))\right)\ln|\lambda|+\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

  3. (vii)

In the context of Lemma 5.11 (i.e. $(D_{\ell},D_{r})$ a smaller pair of
$B$-algebras than $(C_{\ell},C_{r})$),

$\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(D_{\ell},D_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))\geq\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

  4. (viii)

In the context of Proposition 5.12 (i.e. adding in a bi-free pair of
$B$-algebras),

$\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(\operatorname{alg}(D_{\ell},C_{\ell}),\operatorname{alg}(D_{r},C_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))=\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

  5. (ix)

If in addition to the assumptions of Definition 8.1 we have
$\mathbf{X}^{\prime}\in A_{\ell}^{n^{\prime}}$, $\mathbf{Y}^{\prime}\in
A_{r}^{m^{\prime}}$,
$(\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1})$
is a collection of completely positive maps on $B$, and $(D_{\ell},D_{r})$ is
a pair of $B$-algebras, then

$\displaystyle\chi^{*}($
$\displaystyle\mathbf{X},\mathbf{X}^{\prime}\sqcup\mathbf{Y},\mathbf{Y}^{\prime}:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1}\cup\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}\cup\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$
$\displaystyle\leq\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))+\chi^{*}(\mathbf{X}^{\prime}\sqcup\mathbf{Y}^{\prime}:(D_{\ell},D_{r}),(\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$

  6. (x)

In the context of (ix) with the additional assumption that

$\left(\operatorname{alg}(C_{\ell},\mathbf{X}),\operatorname{alg}(C_{r},\mathbf{Y})\right)\qquad\text{and}\qquad\left(\operatorname{alg}(D_{\ell},\mathbf{X}^{\prime}),\operatorname{alg}(D_{r},\mathbf{Y}^{\prime})\right)$

are bi-free with amalgamation over $B$ with respect to $E$, Proposition 5.12
implies that

$\displaystyle\chi^{*}($
$\displaystyle\mathbf{X},\mathbf{X}^{\prime}\sqcup\mathbf{Y},\mathbf{Y}^{\prime}:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1}\cup\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}\cup\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$
$\displaystyle=\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))+\chi^{*}(\mathbf{X}^{\prime}\sqcup\mathbf{Y}^{\prime}:(D_{\ell},D_{r}),(\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$

Using Proposition 7.5, Theorem 7.8 and the same arguments as
[CS2020]*Proposition 6.7, the following holds.

######  Proposition 8.5.

Under the assumptions of Definition 8.1, if for each $k\in\mathbb{N}$ there
exists self-adjoint tuples $\mathbf{X}^{(k)}\in A^{n}_{\ell}$ and
$\mathbf{Y}^{(k)}\in A_{r}^{m}$ such that

$\displaystyle\limsup_{k\to\infty}\left\|X^{(k)}_{i}\right\|<\infty,$
$\displaystyle\limsup_{k\to\infty}\left\|Y^{(k)}_{j}\right\|<\infty,$
$\displaystyle s\text{-}\lim_{k\to\infty}X^{(k)}_{i}=X_{i},\text{ and}$
$\displaystyle s\text{-}\lim_{k\to\infty}Y^{(k)}_{j}=Y_{j}$

for all $1\leq i\leq n$ and $1\leq j\leq m$ (with the strong limit computed as
bounded linear maps acting on $L_{2}(A,\tau)$), then

$\displaystyle\limsup_{k\to\infty}\chi^{*}\left(\mathbf{X}^{(k)}\sqcup\mathbf{Y}^{(k)}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right)\leq\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

Using the previous proposition together with Theorem 7.8 and the same
arguments as [CS2020]*Proposition 6.8, the following holds.

######  Proposition 8.6.

Under the assumptions of Definition 8.1, suppose
$(\\{S_{i}\\}^{n}_{i=1},\\{D_{j}\\}^{m}_{j=1})$ is a collection of
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ bi-semicircular
operators such that

$(\operatorname{alg}(C_{\ell},\mathbf{X}),\operatorname{alg}(C_{r},\mathbf{Y})\rangle)\cup\\{(\operatorname{alg}(B_{\ell},S_{i}),B_{r})\\}^{n}_{i=1}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},D_{j}))\\}^{m}_{j=1}$

are bi-free. For $t\in[0,\infty)$, let

$g(t)=\chi^{*}\left(\mathbf{X}+\sqrt{t}\mathbf{S}\sqcup\mathbf{Y}+\sqrt{t}\mathbf{D}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right).$

Then $g:[0,\infty)\to\mathbb{R}\cup\\{-\infty\\}$ is a concave, continuous,
increasing function such that $g(t)\geq\frac{K}{2}\ln(2\pi et)$ where

$K=\sum^{n}_{i=1}\tau_{B}(\eta_{\ell,i}(1_{B}))+\sum^{m}_{j=1}\tau_{B}(\eta_{r,j}(1_{B}))$

and, when $g(t)\neq-\infty$,

$\lim_{\epsilon\to
0+}\frac{1}{\epsilon}(g(t+\epsilon)-g(t))=\frac{1}{2}\Phi^{*}\left(\mathbf{X}+\sqrt{t}\mathbf{S}\sqcup\mathbf{Y}+\sqrt{t}\mathbf{D}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right).$

Finally, using the Bi-Free Stam Inequality (Proposition 7.3) together with the
same proof as [CS2020]*Proposition 6.11 yields the following.

######  Proposition 8.7.

Under the assumptions of Definition 8.1, if

$\Phi^{*}\left(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})\right)<\infty,$

then

$\displaystyle\chi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(B_{\ell},B_{r}))\geq\frac{K}{2}\ln\left(\frac{2\pi
Ke}{\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))}\right)>-\infty,$

where

$K=\sum^{n}_{i=1}\tau_{B}(\eta_{\ell,i}(1_{B}))+\sum^{m}_{j=1}\tau_{B}(\eta_{r,j}(1_{B})).$

##  9\. Minimizing Bi-Free Fisher Information

In this section, we will prove Theorem 9.6 thereby describing the minimal
value of the bi-free Fisher information of non-self-adjoint pairs of operators
under certain distribution conditions. Throughout the section, we will be
working under the situation from Example 3.4 where $\mathcal{A}$ is a unital
C∗-algebra, $\varphi:\mathcal{A}\to\mathbb{C}$ is a state,
$B=M_{d}(\mathbb{C})$ (the $d\times d$ matrices with complex entries) and
$\tau_{B}=\operatorname{tr}_{d}$ (the normalized trace on
$M_{d}(\mathbb{C})$). Thus $A_{d}=\mathcal{A}\otimes M_{d}(\mathbb{C})\otimes
M_{d}(\mathbb{C})^{\text{op}}$, $E_{d}:A_{d}\to M_{d}(\mathbb{C})$ and
$\tau_{d}:A_{d}\to\mathbb{C}$ are defined such that

$E_{d}(Z\otimes b_{1}\otimes
b_{2})=\varphi(Z)b_{1}b_{2}\qquad\text{and}\qquad\tau_{d}(Z\otimes
b_{1}\otimes b_{2})=\varphi(Z)\operatorname{tr}_{d}(b_{1}b_{2}),$

for all $Z\in\mathcal{A}$ and $b_{1},b_{2}\in M_{d}(\mathbb{C})$. We recall
the following result that aids in computing moments in
$(A_{d},E_{d},\varepsilon)$ where $\\{E_{i,j}\\}^{n}_{i,j=1}\subseteq
M_{d}(\mathbb{C})$ are the canonical matrix units.

######  Lemma 9.1 ([S2016-1]*Lemma 3.7).

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space, let
$\chi\in\\{\ell,r\\}^{n}$, and let

$Z_{k}=\begin{cases}\sum^{d}_{i,j=1}z_{k;i,j}\otimes E_{i,j}\otimes
I_{d}&\text{if }\chi(k)=\ell\\\ \sum^{d}_{i,j=1}z_{k;i,j}\otimes I_{d}\otimes
E_{i,j}&\text{if }\chi(k)=r\end{cases}.$

Then

$E_{d}(Z_{1}\cdots Z_{n})=\sum^{d}_{\begin{subarray}{c}i_{1},\ldots,i_{n}=1\\\
j_{1},\ldots,j_{n}=1\end{subarray}}\varphi(z_{1;i_{1},j_{1}}\cdots
z_{n;i_{n},j_{n}})E_{\chi}((i_{1},\ldots,i_{n}),(j_{1},\ldots,j_{n}))$

where

$E_{\chi}((i_{1},\ldots,i_{n}),(j_{1},\ldots,j_{n})):=E_{i_{s_{\chi}(1)},j_{s_{\chi}(1)}}\cdots
E_{i_{s_{\chi}(n)},j_{s_{\chi}(n)}}\in M_{d}(\mathbb{C}).$

To discuss conjugate variables, we need to consider $L_{2}(A_{d},\tau_{d})$.
It is not difficult to verify that $L_{2}(A_{d},\tau_{d})$ can be identified
with the $d\times d$ matrices with entries in $L_{2}(\mathcal{A},\varphi)$
where

$(Z\otimes I_{d}\otimes I_{d})[\xi_{i,j}]=[Z\xi_{i,j}],$

$1_{\mathcal{A}}\otimes b\otimes I_{d}$ acts via left multiplication on
$[\xi_{i,j}]$ and $1_{\mathcal{A}}\otimes I_{d}\otimes b$ acts via right
multiplication on $[\xi_{i,j}]$ for all $[\xi_{i,j}]\in
M_{d}(L_{2}(\mathcal{A},\varphi))$ and $b\in M_{d}(\mathbb{C})$.

Next, we consider the subalgebra $D_{d}\subseteq M_{d}(\mathbb{C})$ of the
diagonal matrices. Clearly if $F_{d}:M_{d}(\mathbb{C})\to D_{d}$ is the
canonical conditional expectation onto the diagonal, then $(A_{d},F_{d}\circ
E_{d},\varepsilon|_{D_{d}\otimes D_{d}^{{\text{op}}}},\tau_{d})$ is also an
analytical $D_{d}$-$D_{d}$-non-commutative probability space (see Example
5.9).

To begin stating our main result, let $x,y\in\mathcal{A}$. For $d=2$, let

$X=x\otimes E_{1,2}\otimes I_{2}+x^{*}\otimes E_{2,1}\otimes
I_{2}\qquad\text{and}\qquad Y=y\otimes I_{2}\otimes E_{1,2}+y^{*}\otimes
I_{2}\otimes E_{2,1},$

which are then self-adjoint elements of $A_{2}$. The pair $(X,Y)$ is
intimately related to whether or not $(x,y)$ is bi-R-diagonal
([K2019]*Definition 2.12). Indeed, combining [K2019]*Proposition 2.21 and
[S2016-2]*Theorem 4.9, we obtain the following proposition.

######  Proposition 9.2.

As described above, the following conditions are equivalent:

  1. (i)

the pair $(x,y)$ is bi-R-diagonal,

  2. (ii)

the pair $(X,Y)$ is bi-R-cyclic,

  3. (iii)

the pair of algebras
$(\operatorname{alg}(D_{2},X),\operatorname{alg}(D_{2},Y))$ is bi-free from
$(M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r})$ with amalgamation over
$D_{2}$ with respect to $F_{2}\circ E_{2}$.

In addition, the joint moments of $(X,Y)$ with respect to $\tau_{d}$ are not
too difficult to describe. Indeed, for any $n\in\mathbb{N}$,
$Z_{1},Z_{2},\ldots,Z_{n}\in\\{X,Y\\}$, $\chi\in\\{\ell,r\\}^{n}$, and
$z_{1},\ldots,z_{n}\in\\{x,y\\}$ such that

$\chi(k)=\begin{cases}\ell&\text{if }Z_{k}=X\\\ r&\text{if
}Z_{k}=Y\end{cases}\qquad\text{and}\qquad z_{k}=\begin{cases}x&\text{if
}Z_{k}=X\\\ y&\text{if }Z_{k}=Y\end{cases},$

then

$E_{2}(Z_{1}\cdots
Z_{n})=\sum^{n}_{k=1}\sum_{p_{k}\in\\{1,*\\}}\varphi(z_{1}^{p_{1}}\cdots
z_{n}^{p_{n}})(E_{1,2})^{p_{s_{\chi}(1)}}(E_{1,2})^{p_{s_{\chi}(2)}}\cdots(E_{1,2})^{p_{s_{\chi}(n)}}.$

Thus, if $n$ is odd, we see that $\tau_{2}(Z_{1}\cdots Z_{n})=0$ and if $n$ is
even, we see that

$\tau_{2}(Z_{1}\cdots Z_{n})=\frac{1}{2}\left(\varphi(z_{1}^{p_{1}}\cdots
z_{n}^{p_{n}})+\varphi(z_{1}^{q_{1}}\cdots z_{n}^{q_{n}})\right),$

where

$p_{s_{\chi}(k)}=\begin{cases}1&\text{if }k\text{ is odd}\\\ *&\text{if
}k\text{ is even}\end{cases}\qquad\text{and}\qquad
q_{s_{\chi}(k)}=\begin{cases}*&\text{if }k\text{ is odd}\\\ 1&\text{if
}k\text{ is even}\end{cases}$

(that is, the $1$’s and $*$’s alternate in the $\chi$-ordering). Hence, the
joint moments of $(X,Y)$ with respect to $\tau_{d}$ depend only on specific
moments of $(x,y)$. We let $\Delta_{X,Y}$ denote the set of all pairs
$(x_{0},y_{0})$ in a C∗-non-commutative probability space
$(\mathcal{A}_{0},\varphi_{0})$ such that if we apply the above procedure to
$(x_{0},y_{0})$ resulting in $(X_{0},Y_{0})$, then $(X_{0},Y_{0})$ has the
same joint distribution as $(X,Y)$ (so $\Delta_{X,Y}=\Delta_{X_{0},Y_{0}}$).

One specific case worth mentioning is when $x$ and $y$ are normal operators
with $[\operatorname{alg}(x,x^{*}),\operatorname{alg}(y,y^{*})]=0$, thus
defining a probability measure $\mu$ on $\mathbb{C}^{2}$. Then, $X$ and $Y$
will be commuting self-adjoint operators and therefore their joint
distribution gives rise to a compactly supported probability measure $\mu_{0}$
on $\mathbb{R}^{2}$ with moments

$\tau_{2}(X^{n}Y^{m})=\begin{cases}0&\text{if }n+m\text{ is odd}\\\
\varphi((x^{*}x)^{i}(y^{*}y)^{j})&\text{if }n=2i\text{ and }m=2j\\\
\frac{1}{2}\varphi((x^{*}x)^{i}(xy^{*}+x^{*}y)(y^{*}y)^{j})&\text{if
}n=2i+1\text{ and }m=2j+1\end{cases}.$

One additional property is required in this section. In particular, as we are
attempting to generalize [NSS1999]*Theorem 1.1 which makes heavy use of
traciality, we need a condition that lets us bypass the issue that $\tau_{2}$
is not tracial on $A_{2}$.

######  Definition 9.3.

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space and let
$x,y\in\mathcal{A}$. We say that $(x,y)$ is _alternating adjoint flipping with
respect to $\varphi$_ if for any $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{2n}$,
and $z_{1},\ldots,z_{2n}\in\\{x,y\\}$ such that

$z_{k}=\begin{cases}x&\text{if }\chi(k)=\ell\\\ y&\text{if
}\chi(k)=r\end{cases},$

we have that

$\varphi(z_{1}^{p_{1}}\cdots z_{2n}^{p_{2n}})=\varphi(z_{1}^{q_{1}}\cdots
z_{2n}^{q_{2n}})$

where

$p_{s_{\chi}(k)}=\begin{cases}1&\text{if }k\text{ is odd}\\\ *&\text{if
}k\text{ is even}\end{cases}\qquad\text{and}\qquad
q_{s_{\chi}(k)}=\begin{cases}*&\text{if }k\text{ is odd}\\\ 1&\text{if
}k\text{ is even}\end{cases}.$

######  Remark 9.4.

If $(x,y)$ is alternating adjoint flipping, then the description of the joint
moments of $(X,Y)$ above reduce to a nicer expression. Furthermore we see that

$\varphi((x^{*}x)^{m})=\varphi((xx^{*})^{m})\qquad\text{and}\qquad\varphi((y^{*}y)^{m})=\varphi((yy^{*})^{m})$

for all $m\in\mathbb{N}$, so that $x^{*}x$ and $xx^{*}$ have the same
distribution and $y^{*}y$ and $yy^{*}$ have the same distribution, which would
be automatic if $\varphi$ was tracial when restricted to
$\operatorname{alg}(x,x^{*})$ and when restricted to
$\operatorname{alg}(y,y^{*})$ (a common assumption in bi-free probability).

Of course, bi-Haar unitary pairs are trivially seen to be alternating adjoint
flipping, since any joint moment with an equal number of adjoint and non-
adjoint terms is 1 and any joint moment with a differing number of adjoint and
non-adjoint terms is 0. Here is another example which is of use in this paper.

######  Example 9.5.

Let $\mathcal{H}$ be any Hilbert space of dimension at least $4$, let
$\mathcal{F}(\mathcal{H})$ denote the Fock space generated by $\mathcal{H}$,
let $\varphi_{0}$ be the vacuum vector state on
$\mathcal{B}(\mathcal{F}(\mathcal{H}))$ and let
$\\{e_{1},e_{2},e_{3},e_{4}\\}$ be an orthonormal set. For $i=1,2$ let
$s_{i}=l(e_{i})+l^{*}(e_{i})$ (i.e. left creation plus annihilation by
$e_{i}$) and for $j=1,2$ let $d_{j}=r(e_{j+2})+r^{*}(e_{j+2})$ (i.e. right
creation and annihilation by $e_{j+2}$). Thus
$(\\{s_{1},s_{2}\\},\\{d_{1},d_{2}\\})$ is a bi-free central limit
distribution with variance 1 and covariance 0.

Let

$c_{\ell}=\frac{1}{\sqrt{2}}(s_{1}+is_{2})\qquad\text{and}\qquad
c_{r}=\frac{1}{\sqrt{2}}(d_{1}+id_{2}).$

We call the pair $(c_{\ell},c_{r})$ a _bi-free circular pair_ (with mean 0,
variance 1, and covariance 0).

We claim that $(c_{\ell},c_{r})$ is an example of a bi-R-diagonal pair that is
alternating adjoint flipping with respect to $\varphi_{0}$. To see that
$(c_{\ell},c_{r})$ is bi-R-diagonal, we note that any bi-free cumulant for
$(\\{s_{1},s_{2}\\},\\{d_{1},d_{2}\\})$ of order 1, of order greater than 3,
or involving two different elements is 0. As

$\displaystyle\kappa_{1_{(\ell,\ell)}}(c_{\ell},c_{\ell})$
$\displaystyle=\frac{1}{2}\kappa_{1_{(\ell,\ell)}}(s_{1},s_{1})+(i)^{2}\frac{1}{2}\kappa_{1_{(\ell,\ell)}}(s_{2},s_{2})=0,$
$\displaystyle\kappa_{1_{(\ell,\ell)}}(c^{*}_{\ell},c^{*}_{\ell})$
$\displaystyle=\frac{1}{2}\kappa_{1_{(\ell,\ell)}}(s_{1},s_{1})+(-i)^{2}\frac{1}{2}\kappa_{1_{(\ell,\ell)}}(s_{2},s_{2})=0,$

and similar computations hold on the right, we have that $(c_{\ell},c_{r})$ is
bi-R-diagonal.

To see that $(c_{\ell},c_{r})$ is alternating adjoint flipping with respect to
$\varphi_{0}$, first note that $\varphi_{0}$ is tracial when restricted to
$\operatorname{alg}(s_{1},s_{2})$ as $s_{1}$ and $s_{2}$ are freely
independent with respect to $\varphi_{0}$. Hence for all $n\in\mathbb{N}$

$\varphi_{0}((c_{\ell}^{*}c_{\ell})^{n})=\varphi_{0}((c_{\ell}c_{\ell}^{*})^{n}).$

Moreover, as any monomial of odd length involving freely independent
semicircular variables is 0, we obtain that for all $n\in\mathbb{N}$ that

$\varphi_{0}(c_{\ell}(c_{\ell}^{*}c_{\ell})^{n})=0=\varphi_{0}(c_{\ell}^{*}(c_{\ell}c_{\ell}^{*})^{n}).$

Similarly, for all $n\in\mathbb{N}$ we have that

$\varphi_{0}((c_{r}^{*}c_{r})^{n})=\varphi_{0}((c_{r}c_{r}^{*})^{n})\qquad\text{and}\qquad\varphi_{0}(c_{r}(c_{r}^{*}c_{r})^{n})=0=\varphi_{0}(c_{r}^{*}(c_{r}c_{r}^{*})^{n}).$

To see the remaining moment conditions, first note that
$\\{c_{\ell},c_{\ell}^{*}\\}$ commutes with $\\{c_{r},c_{r}^{*}\\}$. Thus, as
the $\chi$-ordering is not changed by commutation of left and right operators,
it suffices to show that

$\displaystyle\varphi_{0}((c_{\ell}^{*}c_{\ell})^{n}(c_{r}c_{r}^{*})^{m})$
$\displaystyle=\varphi_{0}((c_{\ell}c_{\ell}^{*})^{n}(c_{r}^{*}c_{r})^{m}),$
$\displaystyle\varphi_{0}(c_{\ell}(c_{\ell}^{*}c_{\ell})^{n}(c_{r}c_{r}^{*})^{m})$
$\displaystyle=\varphi_{0}(c_{\ell}^{*}(c_{\ell}c_{\ell}^{*})^{n}(c_{r}^{*}c_{r})^{m}),$
$\displaystyle\varphi_{0}((c_{\ell}^{*}c_{\ell})^{n}c_{r}^{*}(c_{r}c_{r}^{*})^{m})$
$\displaystyle=\varphi_{0}(c_{\ell}^{*}(c_{\ell}c_{\ell}^{*})^{n}c_{r}(c_{r}^{*}c_{r})^{m})\text{
and}$
$\displaystyle\varphi_{0}(c_{\ell}(c_{\ell}^{*}c_{\ell})^{n}c_{r}^{*}(c_{r}c_{r}^{*})^{m})$
$\displaystyle=\varphi_{0}(c_{\ell}^{*}(c_{\ell}c_{\ell}^{*})^{n}c_{r}(c_{r}^{*}c_{r})^{m}),$

for all $n,m\in\mathbb{N}\cup\\{0\\}$. However, as
$\\{c_{\ell},c_{\ell}^{*}\\}$ is classically independent from
$\\{c_{r},c_{r}^{*}\\}$ since the joint bi-free cumulants vanish, each of the
8 above moment expressions simplifies to the $\varphi_{0}$-moment of the
$\\{c_{\ell},c_{\ell}^{*}\\}$ term times the $\varphi_{0}$-moment of the
$\\{c_{r},c_{r}^{*}\\}$. Thus, the desired moments are equal by the above
knowledge of the $\varphi_{0}$-moments of the $\\{c_{\ell},c_{\ell}^{*}\\}$
and the $\varphi_{0}$-moment of the $\\{c_{r},c_{r}^{*}\\}$.

With the above definitions, notation and constructions out of the way, our
main result is at hand.

######  Theorem 9.6.

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space and let
$x,y\in\mathcal{A}$ be such that $x^{*}x$ and $xx^{*}$ have the same
distribution with respect to $\varphi$ and $y^{*}y$ and $yy^{*}$ have the same
distribution with respect to $\varphi$. With $X$ and $Y$ as described above

$\min\left\\{\left.\Phi^{*}(\\{x_{0},x_{0}^{*}\\}\sqcup\\{y_{0},y_{0}^{*}\\}:(\mathbb{C},\mathbb{C}),\varphi)\,\right|(x_{0},y_{0})\in\Delta_{X,Y}\right\\}\geq
2\Phi^{*}(X\sqcup Y)$

and equality holds and is achieved for any pair $(x_{0},y_{0})$ that is
alternating adjoint flipping and bi-R-diagonal.

######  Remark 9.7.

Note that Theorem 9.6 is a generalization of [NSS1999]*Theorem 1.1 to the bi-
free setting. Prior to the acknowledgements of [NSS1999] it is mentioned that
the minimum in the free result can only be reached by an R-diagonal element
via the result from [V1999] that
$\Phi^{*}(x_{1},\ldots,x_{n}:B)=\Phi^{*}(x_{1},\ldots,x_{n})<\infty$ implies
$\\{x_{1},\ldots,x_{n}\\}$ is free from $B$. As there is no such known
analogous result in the bi-free case, we leave Theorem 9.6 as stated.

To begin the proof of Theorem 9.6, we note the following connecting the bi-
free Fisher informations of $(\\{x,x^{*}\\},\\{y,y^{*}\\})$ and $(X,Y)$ (and
thereby demonstrating the necessity of considering bi-free Fisher information
with respect to completely positive maps in this construction). We note that
the following is a generalization of [NSS1999]*Proposition 3.6 with a similar
but more complicated proof due to the $\chi$-ordering and additional variables
present.

######  Proposition 9.8.

Under the assumptions of Theorem 9.6, if $\eta:M_{2}(\mathbb{C})\to
M_{2}(\mathbb{C})$ is defined by

$\eta\left(\begin{bmatrix}a_{1,1}&a_{1,2}\\\
a_{2,1}&a_{2,2}\end{bmatrix}\right)=\begin{bmatrix}a_{2,2}&0\\\
0&a_{1,1}\end{bmatrix},$

then

$\Phi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\}:(\mathbb{C},\mathbb{C}),\varphi)=2\Phi^{*}(X\sqcup
Y:(M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r}),\eta).$

###### Proof.

First suppose the bi-free Fisher information
$\Phi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})$ is finite. Thus there exist

$\xi_{1},\xi_{2}\in\overline{\operatorname{alg}(x,x^{*},y,y^{*})}^{\left\|\,\cdot\,\right\|_{\varphi}}$

such that $\xi_{1}$ is the left bi-free conjugate variable for $x$ with
respect to $\varphi$ in the presence of $(x,\\{y,y^{*}\\})$ and $\xi_{2}$ is
the left bi-free conjugate variable for $x^{*}$ with respect to $\varphi$ in
the presence of $(x^{*},\\{y,y^{*}\\})$. Let

$\Xi=\begin{bmatrix}0&\xi_{2}\\\ \xi_{1}&0\end{bmatrix}\in
M_{2}(L_{2}(A_{2},\tau_{2})).$

We claim that
$\Xi=J_{\ell}\left(X:\left(M_{2}(\mathbb{C})_{\ell},\operatorname{alg}\left(M_{2}(\mathbb{C})_{r},Y\right)\right),\eta\right)$.
Since a similar result holds on the right and since

$\left\|\Xi\right\|_{\tau_{2}}=\frac{1}{2}\left(\left\|\xi_{1}\right\|_{\varphi}+\left\|\xi_{2}\right\|_{\varphi}\right),$

the result will follow in this case.

First we claim that
$\Xi\in\overline{\operatorname{alg}\left(X,Y,M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r}\right)}^{\tau_{2}}$.
Indeed, it is not difficult to verify that

$z\otimes E_{i_{1},j_{1}}\otimes
E_{i_{2},j_{2}}\in\operatorname{alg}\left(X,Y,M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r}\right)$

for all $z\in\\{x,x^{*},y,y^{*}\\}$ and $i_{1},i_{2},j_{1},j_{2}\in\\{1,2\\}$.
Thus, since
$\xi_{1},\xi_{2}\in\overline{\operatorname{alg}(x,x^{*},y,y^{*})}^{\left\|\,\cdot\,\right\|_{\varphi}}$,
the claim follows.

To complete the claim that $\Xi$ is the appropriate left bi-free conjugate
variable, we must show $\Xi$ satisfies the left bi-free conjugate variable
relations; that is, for all $n\in\mathbb{N}$, $b_{0},b_{1},\ldots,b_{n}\in
M_{2}(\mathbb{C})$, $\chi\in\\{\ell,r\\}^{n}$ with $\chi(n)=\ell$, and
$Z_{1},\ldots,Z_{n-1}\in A_{2}$ and $C_{1},\ldots C_{n-1}\in
1_{\mathcal{A}}\otimes M_{2}(\mathbb{C})\otimes M_{2}(\mathbb{C})^{\text{op}}$
where

$Z_{k}=\begin{cases}X&\text{if }\chi(k)=\ell\\\ Y&\text{if
}\chi(k)=r\end{cases}\qquad\text{and}\qquad
C_{k}=\begin{cases}L_{b_{k}}&\text{if }\chi(k)=\ell\\\ R_{b_{k}}&\text{if
}\chi(k)=r\end{cases},$

we have that

$\displaystyle\tau_{2}\left(L_{b_{0}}R_{b_{n}}Z_{1}C_{1}\cdots
Z_{n-1}C_{n-1}\Xi\right)=\sum_{\begin{subarray}{c}1\leq k<n\\\
\chi(k)=\ell\end{subarray}}\tau_{2}\left(L_{b_{0}}R_{b_{n}}\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}C_{p}\right)L_{\eta\left(E_{2}\left(C_{k}\prod_{p\in
V_{k}}Z_{p}C_{p}\right)\right)}\right),$ (1)

where $V_{k}=\\{k<m<n\,\mid\,\chi(m)=\ell\\}$. By linearity, it suffices to
consider $b_{k}=E_{i_{k},j_{k}}$ for all $k$ where $i_{k},j_{k}\in\\{1,2\\}$.
In that which follows, the proof is near identical to that of
[NSS1999]*Proposition 3.6 taking into account the $\chi$-order. For notational
purposes, for $k\in\\{1,2\\}$ let $\overline{k}=3-k$.

Let $q=s_{\chi}^{-1}(n)$ (i.e. $\Xi$ appears $q^{\mathrm{th}}$ in the
$\chi$-ordering). We begin by computing the left-hand side of (1). Using Lemma
9.1 (and recalling $\tau_{2}=\operatorname{tr}_{2}\circ E_{2}$), proceeding
via $\chi$-order using commutation, we obtain that

  * •

the only way the product produces a non-zero trace is if $i_{0}=j_{n}$,

  * •

the term $L_{b_{0}}XL_{b_{s_{\chi}(1)}}$ can be made to appear in the product
and is non-zero only if $j_{0}=\overline{i_{s_{\chi}(1)}}$,

  * •

the term $L_{b_{s_{\chi}(k-1)}}XL_{b_{s_{\chi}(k)}}$ can be made to occur for
all $2\leq k<q$ and is non-zero only if
$j_{s_{\chi}(k-1)}=\overline{i_{s_{\chi}(k)}}$,

  * •

the term $L_{b_{s_{\chi}(q-1)}}R_{b_{s_{\chi}(q+1)}}\Xi$ can be made to occur
and is non-zero only if $j_{s_{\chi}(q-1)}=\overline{i_{s_{\chi}(q+1)}}$,

  * •

the term $R_{b_{s_{\chi}(k+1)}}YR_{b_{s_{\chi}(k)}}$ can be made to occur for
all $q<k<n$ and is non-zero only if
$j_{s_{\chi}(k)}=\overline{i_{s_{\chi}(k+1)}}$ (recall the opposite
multiplication), and

  * •

the term $R_{b_{n}}YR_{b_{s_{\chi}(n)}}$ can be made to occur and is non-zero
only if $j_{s_{\chi}(n)}=\overline{i_{n}}$.

Note the discrepancy in notation around the $\Xi$ term due to the labelling of
the left and right $B$-operators (i.e. $b_{s_{\chi}(q)}=b_{n}$ is in the wrong
spot). Thus with

$(X)_{1,2}=x,\quad(X)_{2,1}=x^{*},\quad(Y)_{1,2}=y,\quad(Y)_{2,1}=y^{*},\quad(\Xi)_{1,2}=\xi_{2},\quad\text{and}\quad(\Xi)_{2,1}=\xi_{1},$

and

$\overline{Z_{k}}=\begin{cases}(X)_{j_{0},\overline{j_{0}}}&\text{if
}s_{\chi}(k)=1\\\
(X)_{j_{s_{\chi}(k-1)},\overline{j_{s_{\chi}(k-1)}}}&\text{if
}1<s_{\chi}(k)<q\\\
(\Xi)_{j_{s_{\chi}(q-1)},\overline{j_{s_{\chi}(q-1)}}}&\text{if
}s_{\chi}(k)=q\\\ (Y)_{j_{s_{\chi}(k)},\overline{j_{s_{\chi}(k)}}}&\text{if
}q<s_{\chi}(k)<n\\\ (Y)_{j_{s_{\chi}(n)},\overline{j_{s_{\chi}(n)}}}&\text{if
}s_{\chi}(k)=n\\\ \end{cases}$

we see that the left-hand side of (1) is

$\displaystyle\frac{1}{2}\delta_{j_{n},i_{0}}$
$\displaystyle\delta_{j_{0},\overline{i_{s_{\chi}(1)}}}\delta_{j_{{s_{\chi}}(1)},\overline{i_{s_{\chi}(2)}}}\cdots\delta_{j_{{s_{\chi}}(q-2)},\overline{i_{s_{\chi}(q-1)}}}\delta_{j_{{s_{\chi}}(1-2)},\overline{i_{s_{\chi}(q+1)}}}\delta_{j_{{s_{\chi}}(q+1)},\overline{i_{s_{\chi}(q+2)}}}\cdots\delta_{j_{{s_{\chi}}(n-1)},\overline{i_{s_{\chi}(n)}}}\delta_{j_{{s_{\chi}}(n)},\overline{i_{n}}}$
$\displaystyle\times\varphi\left(\overline{Z_{1}}\cdots\overline{Z_{n}}\right).$
(2)

where $\delta_{j,i}$ is the Kronecker delta. Moreover, using the conjugate
variable relations for $\xi_{1}$ and $\xi_{2}$, we see that

$\displaystyle\varphi\left(\overline{Z_{1}}\cdots\overline{Z_{n}}\right)=\sum_{\begin{subarray}{c}1\leq
k<n\\\
\chi(k)=\ell\end{subarray}}\delta_{j_{s_{\chi}(k-1)},\overline{j_{s_{\chi}(q-1)}}}\varphi\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}\overline{Z_{p}}\right)\varphi\left(\prod_{p\in
V_{k}}\overline{Z_{p}}\right)$ (3)

where the $\delta_{j_{s_{\chi}(k-1)},\overline{j_{s_{\chi}(q-1)}}}$ should be
$\delta_{j_{s_{\chi}(k-1)},\overline{j_{s_{\chi}(q-1)}}}$ when
$k=s_{\chi}^{-1}(1)$.

To complete the proof that equation (1) holds, we compute the right-hand side
of equation (1) and show the $k^{\mathrm{th}}$ term in the sum equals the
$k^{\mathrm{th}}$ term obtain in equation (2) using equation (3). Indeed, for
a fixed $1\leq k<n$ for which $\chi(k)=\ell$, we can compute

$M_{k}=E_{2}\left(C_{k}\prod_{p\in V_{k}}Z_{p}C_{p}\right),$

in a similar fashion to the above. Thus, to obtain a non-zero value, the
relations $j_{s_{\chi}(p-1)}=\overline{i_{s_{\chi}(p)}}$ for all $p\in V_{k}$
must hold. Moreover, one immediately obtains when $M_{k}\neq 0$ that

$M_{k}=\varphi\left(\prod_{p\in V_{k}}\overline{Z_{p}}\right)T_{k},$

for some $T_{k}\in M_{2}(\mathbb{C})$.

Next, notice $\eta\left(M_{k}\right)$ is equivalent to multiplying $M_{k}$ on
the left by $U=E_{1,2}+E_{2,1}$ (for right conjugate variables, one would
multiply on the right) and thus we consider $UM_{k}$ in place of
$\eta\left(M_{k}\right)$. At this point, notice by Lemma 9.1 that $UT_{k}$ can
be written as a product of $b_{p}$’s with $b_{s_{\chi}(q-1)}$ being the right-
most term. By commutation, $R_{b_{s_{\chi}(q+1)}}$ will act on the right of
$UT_{k}$ thereby multiplying by $b_{s_{\chi}(q+1)}$ on the right and forcing
$j_{s_{\chi}(q-1)}=\overline{i_{s_{\chi}(q+1)}}$ for a non-zero value to be
obtained. One then proceeds as above to show that a non-zero value is obtained
only if the above relations are satisfied and that the term that is produced
agrees with the $k^{\mathrm{th}}$ term of (3). Hence the proof is complete in
the case that $\Phi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})<\infty$.

To prove the result in the case that
$\Phi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})=\infty$, it suffices to show that
if $\Phi^{*}(X\sqcup Y:\eta)<\infty$ then
$\Phi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\}:(\mathbb{C},\mathbb{C}),\varphi)<\infty$.
Thus, suppose that $\Phi^{*}(X\sqcup Y:\eta)<\infty$. Hence
$\Xi=J_{\ell}\left(X:\left(M_{2}(\mathbb{C})_{\ell},\operatorname{alg}\left(M_{2}(\mathbb{C})_{r},Y\right)\right),\eta\right)$
exists and can be written as

$\Xi=\begin{bmatrix}\xi_{1,1}&\xi_{1,2}\\\ \xi_{2,1}&\xi_{2,2}\end{bmatrix}.$

We claim that

$\xi_{2,1}=J_{\ell}(x:(x^{*},\\{y,y^{*}\\}),\varphi)\qquad\text{and}\qquad\xi_{1,2}=J_{\ell}(x^{*}:(x,\\{y,y^{*}\\}),\varphi).$

As a similar result will hold on the right, we will obtain that
$\Phi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\}:(\mathbb{C},\mathbb{C}),\varphi)<\infty$
as desired.

As
$\Xi\in\overline{\operatorname{alg}\left(X,Y,M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r}\right)}^{\tau_{2}}$,
it is not difficult to see that
$\xi_{i,j}\in\overline{\operatorname{alg}(x,x^{*},y,y^{*})}^{\left\|\,\cdot\,\right\|_{\varphi}}$.
To see that $\xi_{2,1}$ and $\xi_{1,2}$ satisfy the appropriate left bi-free
conjugate variable relations, one need only use equation (1), choose
$b_{k}=E_{i_{k},j_{k}}$ satisfying the above required relations for a non-zero
value and expand both sides of equation (1) in an identical way to that above.
The resulting equations are exactly the left bi-free conjugate variable
relations required. ∎

Using the results from this paper, there is some immediate knowledge about the
bi-free Fisher information with respect to $\eta$ from Proposition 9.8. We
note that the following is a generalization of [NSS1999]*Proposition 3.7 with
a similar but more complicated proof due to the $\chi$-ordering and additional
variables present.

######  Proposition 9.9.

Under the assumptions and notation of Proposition 9.8,

$\Phi^{*}(X\sqcup Y:\eta)\geq\Phi^{*}(X\sqcup Y:\eta|_{D_{2}})$

and the inequality holds when
$(\operatorname{alg}((D_{2})_{\ell},X),\operatorname{alg}((D_{2})_{r},Y))$ is
bi-free from $(M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r})$ with
amalgamation over $D_{2}$ with respect to $F_{2}$. Moreover

$\Phi^{*}(X\sqcup Y:\eta|_{D_{2}})\geq\Phi^{*}(X\sqcup Y)$

and the inequality holds if $(x,y)$ is alternating adjoint flipping.

###### Proof.

Since $\eta=\eta\circ F$, we have that

$\Phi^{*}(X\sqcup Y:\eta)=\Phi^{*}(X\sqcup
Y:((M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r}),\eta\circ F)$

by Remark 7.2 part (v). Moreover

$\Phi^{*}(X\sqcup
Y:((M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r}),\eta\circ
F)\geq\Phi^{*}(X\sqcup Y:\eta|_{D_{2}})$

by Remark 7.2 part (vii). Furthermore, equality holds if
$(\operatorname{alg}((D_{2})_{\ell},X),\operatorname{alg}((D_{2})_{r},Y))$ is
bi-free from $(M_{2}(\mathbb{C})_{\ell},M_{2}(\mathbb{C})_{r})$ over $D_{2}$
with respect to $F_{2}$ by Remark 7.2 part (viii).

To see that $\Phi^{*}(X\sqcup Y:\eta|_{D_{2}})\geq\Phi^{*}(X\sqcup Y)$ we
assume that

$\Xi=J_{\ell}\left(X:\left((D_{2})_{\ell},\operatorname{alg}((D_{2})_{r},Y)\right),\eta|_{D_{2}}\right)\in\overline{\operatorname{alg}(X,Y,(D_{2})_{\ell},(D_{2})_{r})}^{\left\|\,\cdot\,\right\|_{\tau_{2}}}$

exists and show that $\Xi$ satisfies the left bi-free conjugate variable
relations for $X$ in the presence of $Y$. Thus if $P$ is the orthogonal
projection of $L_{2}(A_{2},\tau_{2})$ onto
$\overline{\operatorname{alg}(X,Y)}^{\left\|\,\cdot\,\right\|_{\tau_{2}}}$
then $P(\Xi)$ will also satisfy the left bi-free conjugate variable relations
for $X$ in the presence of $Y$. As a similar result will hold on the right,
the inequality $\Phi^{*}(X\sqcup Y:\eta|_{D_{2}})\geq\Phi^{*}(X\sqcup Y)$ will
be demonstrated.

By the defining property of $\Xi$, we know for all $n\in\mathbb{N}$,
$b_{0},b_{1},\ldots,b_{n}\in D_{2}$, $\chi\in\\{\ell,r\\}^{n}$ with
$\chi(n)=\ell$, and $Z_{1},\ldots,Z_{n-1}\in A_{2}$ and $C_{1},\ldots
C_{n-1}\in 1_{\mathcal{A}}\otimes M_{2}(\mathbb{C})\otimes
M_{2}(\mathbb{C})^{\text{op}}$ where

$Z_{k}=\begin{cases}X&\text{if }\chi(k)=\ell\\\ Y&\text{if
}\chi(k)=r\end{cases}\qquad\text{and}\qquad
C_{k}=\begin{cases}L_{b_{k}}&\text{if }\chi(k)=\ell\\\ R_{b_{k}}&\text{if
}\chi(k)=r\end{cases},$

that

$\displaystyle\tau_{2}\left(L_{b_{0}}R_{b_{n}}Z_{1}C_{1}\cdots
Z_{n-1}C_{n-1}\Xi\right)=\sum_{\begin{subarray}{c}1\leq k<n\\\
\chi(k)=\ell\end{subarray}}\tau_{2}\left(L_{b_{0}}R_{b_{n}}\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}C_{p}\right)L_{\eta\left((F\circ
E_{2})\left(C_{k}\prod_{p\in V_{k}}Z_{p}C_{p}\right)\right)}\right),$ (4)

where $V_{k}=\\{k<m<n\,\mid\,\chi(m)=\ell\\}$. We will use equation (4) where
$b_{k}=I_{2}$ for all $k$. To begin, notice that

$\eta\left((F\circ E_{2})\left(C_{k}\prod_{p\in
V_{k}}Z_{p}C_{p}\right)\right)=\tau_{2}\left(X^{|V_{k}|}\right)I_{2},$

as odd moments of $X$ are zero and as $x^{*}x$ and $xx^{*}$ have the same
distribution with respect to $\varphi$. Therefore

$\varphi((x^{*}x)^{m})=\varphi((xx^{*})^{m})=\tau_{2}(X^{2m}),$

for all $m\in\mathbb{N}$. Hence, equation (4) reduces to

$\tau_{2}\left(Z_{1}\cdots Z_{n-1}\Xi\right)=\sum_{\begin{subarray}{c}1\leq
k<n\\\
\chi(k)=\ell\end{subarray}}\tau_{2}\left((Z_{1},\ldots,Z_{n-1})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)\tau_{2}\left(X^{|V_{k}|}\right),$

which is exactly the desired formula.

To prove $\Phi^{*}(X\sqcup Y:\eta|_{D_{2}})\leq\Phi^{*}(X\sqcup Y)$ when
$(x,y)$ is alternating adjoint flipping thereby completing the proof, we
proceed in a similar (but more complicated) fashion. Suppose

$\Xi=J_{\ell}\left(X:\left(\mathbb{C},\operatorname{alg}(Y)\right)\right)\in\overline{\operatorname{alg}(X,Y)}^{\left\|\,\cdot\,\right\|_{\tau_{2}}}\subseteq\overline{\operatorname{alg}(X,Y,(D_{2})_{\ell},(D_{2})_{r})}^{\left\|\,\cdot\,\right\|_{\tau_{2}}}$

exists. We will demonstrate that $\Xi$ satisfies the left bi-free conjugate
variable relations for $X$ with respect to $\eta$ in the presence of
$((D_{2})_{\ell},\operatorname{alg}((D_{2})_{r},Y$)). As an analogous result
will hold on the right, this will complete the proof.

Write

$\Xi=\begin{bmatrix}\xi_{1,1}&\xi_{1,2}\\\ \xi_{2,1}&\xi_{2,2}\end{bmatrix}\in
M_{2}(L_{2}(\mathcal{A},\varphi))=L_{2}(A_{2},\tau_{2}).$

First we will demonstrate that $\xi_{1,1}=\xi_{2,2}=0$. To begin, let

$\displaystyle\mathcal{H}_{e}$
$\displaystyle=\overline{\mathrm{span}(Z_{1}\cdots
Z_{2n}\,\mid\,n\in\mathbb{N},Z_{k}\in\\{X,Y\\})}^{\left\|\,\cdot\,\right\|_{\tau_{2}}}\text{
and}$ $\displaystyle\mathcal{H}_{o}$
$\displaystyle=\overline{\mathrm{span}(Z_{1}\cdots
Z_{2n-1}\,\mid\,n\in\mathbb{N},Z_{k}\in\\{X,Y\\})}^{\left\|\,\cdot\,\right\|_{\tau_{2}}}.$

By the defining property of $\Xi$, we know that for all $n\in\mathbb{N}$,
$\chi\in\\{\ell,r\\}^{n}$ with $\chi(n)=\ell$, and $Z_{1},\ldots,Z_{n-1}\in
A_{2}$ where

$Z_{k}=\begin{cases}X&\text{if }\chi(k)=\ell\\\ Y&\text{if
}\chi(k)=r\end{cases}$

that

$\displaystyle\tau_{2}\left(Z_{1}\cdots
Z_{n-1}\Xi\right)=\sum_{\begin{subarray}{c}1\leq k<n\\\
\chi(k)=\ell\end{subarray}}\tau_{2}\left((Z_{1},\ldots,Z_{n-1})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)\tau_{2}\left(X^{|V_{k}|}\right),$
(5)

where $V_{k}=\\{k<m<n\,\mid\,\chi(m)=\ell\\}$. Note as $\tau_{2}$ evaluates
any odd product involving $X$ and $Y$ to 0 by Lemma 9.1, if $n-1$ is even,
then $\tau_{2}\left(Z_{1}\cdots Z_{n-1}\Xi\right)=0$. Therefore, since
$\Xi\in\mathcal{H}_{e}+\mathcal{H}_{o}$, we obtain that
$\Xi\in\mathcal{H}_{o}$.

Note for $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{2n-1}$,
$Z_{1},\ldots,Z_{2n-1}\in\\{X,Y\\}$ , and $z_{1},\ldots,z_{2n-1}\in\\{x,y\\}$
where

$Z_{k}=\begin{cases}X&\text{if }\chi(k)=\ell\\\ Y&\text{if
}\chi(k)=r\end{cases}\qquad\text{and}\qquad z_{k}=\begin{cases}x&\text{if
}\chi(k)=\ell\\\ y&\text{if }\chi(k)=r\end{cases}$

that in $M_{2}(L_{2}(\mathcal{A},\varphi))$ we have

$Z_{1}\cdots Z_{2n-1}=\begin{bmatrix}0&z_{1}^{p_{1}}z_{2}^{p_{2}}\cdots
z_{2n-1}^{p_{2n-1}}\\\ z_{1}^{q_{1}}z_{2}^{q_{2}}\cdots
z_{2n-1}^{q_{2n-1}}&0\end{bmatrix},$

where

$p_{s_{\chi}(k)}=\begin{cases}1&\text{if }k\text{ is odd}\\\ *&\text{if
}k\text{ is even}\end{cases}\qquad\text{and}\qquad
q_{s_{\chi}(k)}=\begin{cases}*&\text{if }k\text{ is odd}\\\ 1&\text{if
}k\text{ is even}\end{cases}.$

Therefore, as $\mathcal{H}_{o}$ is the
$\left\|\,\cdot\,\right\|_{\tau_{2}}$-limit of matrices of the above form and
as $\Xi\in\mathcal{H}_{o}$, we obtain that $\xi_{1,1}=\xi_{2,2}=0$ as desired.

Let

$\displaystyle H_{1}$ $\displaystyle=\mathrm{span}\left(z_{1}\cdots
z_{2n-1}\,\left|\,\begin{subarray}{c}n\in\mathbb{N},z_{k}\in\\{x,x^{*},y,y^{*}\\}\\\
\text{the powers of the }z_{k}\text{'s alternate between 1 and * in the
$\chi$-ordering}\\\ \text{and the first and last element in the
$\chi$-ordering have power 1 }\end{subarray}\right.\right)\quad\text{and}$
$\displaystyle H_{*}$ $\displaystyle=\mathrm{span}\left(z_{1}\cdots
z_{2n-1}\,\left|\,\begin{subarray}{c}n\in\mathbb{N},z_{k}\in\\{x,x^{*},y,y^{*}\\}\\\
\text{the powers of the }z_{k}\text{'s alternate between 1 and * in the
$\chi$-ordering}\\\ \text{and the first and last element in the
$\chi$-ordering have power *}\end{subarray}\right.\right).$

Using the above and the notation $\xi=\xi_{1,2}$ and $\xi^{*}=\xi_{2,1}$ (note
we do not claim that there is an involution operation on
$L_{2}(\mathcal{A},\varphi)$ as we do not know $\varphi$ is tracial), we see
that $\xi\in\overline{H_{1}}^{\left\|\,\cdot\,\right\|_{\varphi}}$,
$\xi^{*}\in\overline{H_{*}}^{\left\|\,\cdot\,\right\|_{\varphi}}$, and if we
have a $\left\|\,\cdot\,\right\|_{\varphi}$-limiting sequence using
$\\{x,x^{*},y,y^{*}\\}$ producing $\xi$ we can obtain an a
$\left\|\,\cdot\,\right\|_{\varphi}$-limiting sequence using
$\\{x,x^{*},y,y^{*}\\}$ producing $\xi^{*}$ by exchanging $x\leftrightarrow
x^{*}$ and $y\leftrightarrow y^{*}$. This, in conjunction with the alternating
adjoint flipping condition lets us show if $n\in\mathbb{N}$,
$\chi\in\\{\ell,r\\}^{2n}$, $Z_{1},\ldots,Z_{2n-1}\in\\{X,Y\\}$ and
$z_{1},\ldots,z_{2n-1}\in\\{x,y\\}$ where

$Z_{k}=\begin{cases}X&\text{if }\chi(k)=\ell\\\ Y&\text{if
}\chi(k)=r\end{cases}\qquad\text{and}\qquad z_{k}=\begin{cases}x&\text{if
}\chi(k)=\ell\\\ y&\text{if }\chi(k)=r\end{cases}$

that

$\displaystyle\varphi(z_{1}^{p_{1}}z_{2}^{p_{2}}\cdots
z_{2n-1}^{p_{2n-1}}\xi^{p_{2n}})=\varphi(z_{1}^{q_{1}}z_{2}^{q_{2}}\cdots
z_{2n-1}^{q_{2n-1}}\xi^{q_{2n}}),$ (6)

where

$p_{s_{\chi}(k)}=\begin{cases}1&\text{if }k\text{ is odd}\\\ *&\text{if
}k\text{ is even}\end{cases}\qquad\text{and}\qquad
q_{s_{\chi}(k)}=\begin{cases}*&\text{if }k\text{ is odd}\\\ 1&\text{if
}k\text{ is even}\end{cases}.$

Indeed, consider $z_{1}^{p_{1}}\cdots z_{2n-1}^{p_{2n-1}}\xi^{p_{2n}}$ with
$p_{2n}=1$ (the case $p_{2n}=*$ is analogous). As the terms preceding $\xi$ in
the $\chi$-ordering both must have $*$’s on them and as $\xi$ is a
$\left\|\,\cdot\,\right\|_{\varphi}$-limit of elements of $H_{1}$, we see that
$z_{1}^{p_{1}}\cdots z_{2n-1}^{p_{2n-1}}\xi^{p_{2n}}$ is a
$\left\|\,\cdot\,\right\|_{\varphi}$-limit of a linear combination of
monomials in $\\{x,x^{*},y,y^{*}\\}$ that alternate between $*$ and
non-$*$-terms in the $\chi$-ordering. As $x\leftrightarrow x^{*}$ and
$y\leftrightarrow y^{*}$ produces the same $\varphi$-moment by the alternating
adjoint flipping condition (as $z_{1}^{p_{1}}\cdots z_{2n-1}^{p_{2n-1}}$ and
every element of $H_{1}$ is of odd length) and produces a sequence that
converges to $z_{1}^{q_{1}}z_{2}^{q_{2}}\cdots
z_{2n-1}^{q_{2n-1}}\xi^{q_{2n}}$ with respect to
$\left\|\,\cdot\,\right\|_{\varphi}$, the claim is complete.

Returning to showing $\Xi$ satisfies the left bi-free conjugate variable
relations for $X$ with respect to $\eta$ in the presence of
$((D_{2})_{\ell},\operatorname{alg}((D_{2})_{r},Y$)), it suffices to
demonstrate that equation (4) holds for this $\Xi$. Furthermore, it suffices
to verify that equation (4) holds when $b_{k}=E_{i_{k},i_{k}}$ for all $k$. By
the same computations as done in the proof of Proposition 9.8 with
$j_{k}=i_{k}$ for all $k$, we see with $q=s_{\chi}^{-1}(n)$ that

$\displaystyle\tau_{2}\left(L_{b_{0}}R_{b_{n}}Z_{1}C_{1}\cdots
Z_{n-1}C_{n-1}\Xi\right)$
$\displaystyle=\begin{cases}\frac{1}{2}\varphi(z_{1}^{p_{1}}z_{2}^{p_{2}}\cdots
z_{n-1}^{p_{n-1}}\xi^{p_{n}})&\text{if }n\text{ is even and
}\left(i_{0},i_{s_{\chi}(1)},\ldots,i_{s_{\chi}(q-1)},i_{s_{\chi}(q+1)},\ldots,i_{s_{\chi}(n)}\right)=(1,2,\ldots,1,2)\\\
\frac{1}{2}\varphi(z_{1}^{q_{1}}z_{2}^{q_{2}}\cdots
z_{n-1}^{q_{n-1}}\xi^{q_{n}})&\text{if }n\text{ is even and
}\left(i_{0},i_{s_{\chi}(1)},\ldots,i_{s_{\chi}(q-1)},i_{s_{\chi}(q+1)},\ldots,i_{s_{\chi}(n)}\right)=(2,1,\ldots,2,1)\\\
0&\text{otherwise}\end{cases}$

and

$\displaystyle\sum_{\begin{subarray}{c}1\leq k<n\\\
\chi(k)=\ell\end{subarray}}\tau_{2}\left(L_{b_{0}}R_{b_{n}}\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}C_{p}\right)L_{\eta\left((F\circ
E_{2})\left(C_{k}\prod_{p\in V_{k}}Z_{p}C_{p}\right)\right)}\right)$
$\displaystyle=\begin{cases}\frac{1}{2}\sum_{\begin{subarray}{c}1\leq k<n\\\
\chi(k)=\ell\\\ |V_{k}|\text{
even}\end{subarray}}\varphi\left((z_{1}^{p_{1}},\ldots,z_{n-1}^{p_{n-1}})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)\varphi\left((z_{1}^{p_{1}},\ldots,z_{n-1}^{p_{n-1}})|_{V_{k}}\right)\\\
\qquad\qquad\text{if }n\text{ is even and
}\left(i_{0},i_{s_{\chi}(1)},\ldots,i_{s_{\chi}(q-1)},i_{s_{\chi}(q+1)},\ldots,i_{s_{\chi}(n)}\right)=(1,2,\ldots,1,2)\\\
\frac{1}{2}\sum_{\begin{subarray}{c}1\leq k<n\\\ \chi(k)=\ell\\\ |V_{k}|\text{
even}\end{subarray}}\varphi\left((z_{1}^{q_{1}},\ldots,z_{n-1}^{q_{n-1}})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)\varphi\left((z_{1}^{q_{1}},\ldots,z_{n-1}^{p_{qn-1}})|_{V_{k}}\right)\\\
\qquad\qquad\text{if }n\text{ is even and
}\left(i_{0},i_{s_{\chi}(1)},\ldots,i_{s_{\chi}(q-1)},i_{s_{\chi}(q+1)},\ldots,i_{s_{\chi}(n)}\right)=(2,1,\ldots,2,1)\\\
0,\qquad\quad\text{otherwise}\end{cases}$

where $V_{k}=\\{k<m<n\,\mid\,\chi(m)=\ell\\}$ (note only the terms where
$|V_{k}|$ is even survive from the $\eta\circ F\circ E_{2}$ expression due to
the form of $X$) and $z_{k}$, $p_{k}$, and $q_{k}$ are defined as usual in
this proof. Hence, it suffices to show when $n$ is even that

$\displaystyle\varphi(z_{1}^{p_{1}}z_{2}^{p_{2}}\cdots
z_{n-1}^{p_{n-1}}\xi^{p_{n}})$ $\displaystyle=\sum_{\begin{subarray}{c}1\leq
k<n\\\ \chi(k)=\ell\\\ |V_{k}|\text{
even}\end{subarray}}\varphi\left((z_{1}^{p_{1}},\ldots,z_{n-1}^{p_{n-1}})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)\varphi\left((z_{1}^{p_{1}},\ldots,z_{n-1}^{p_{n-1}})|_{V_{k}}\right),$
(7) $\displaystyle\varphi(z_{1}^{q_{1}}z_{2}^{q_{2}}\cdots
z_{n-1}^{q_{n-1}}\xi^{q_{n}})$ $\displaystyle=\sum_{\begin{subarray}{c}1\leq
k<n\\\ \chi(k)=\ell\\\ |V_{k}|\text{
even}\end{subarray}}\varphi\left((z_{1}^{q_{1}},\ldots,z_{n-1}^{q_{n-1}})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)\varphi\left((z_{1}^{q_{1}},\ldots,z_{n-1}^{p_{qn-1}})|_{V_{k}}\right).$
(8)

Note equations (7) and (8) are the same equation by the alternating adjoint
flipping condition and equation (6). Moreover, due to the defining property of
$\Xi$, we know with $n$ even that

$\displaystyle\tau_{2}(Z_{1}\cdots Z_{n-1}\Xi)$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq k<n\\\
\chi(k)=\ell\end{subarray}}\tau_{2}\left(\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}\right)L_{\eta\left((F\circ
E_{2})\left(\prod_{p\in V_{k}}Z_{p}\right)\right)}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq k<n\\\
\chi(k)=\ell\end{subarray}}\tau_{2}\left(\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}\right)\right)\tau_{2}\left(X^{|V_{k}|}\right).$

Due to the form of $X$ and the alternating adjoint flipping condition, we
immediately see that

$\tau_{2}\left(X^{|V_{k}|}\right)=\begin{cases}0&\text{if }|V_{k}|\text{ is
odd}\\\
\varphi\left((z_{1}^{p_{1}},\ldots,z_{n-1}^{p_{n-1}})|_{V_{k}}\right)&\text{if
}|V_{k}|\text{ is even}\end{cases}$

and, for $n$ even and $k$ such that $|V_{k}|$ is even, we have

$\displaystyle\tau_{2}\left(\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}\right)\right)$
$\displaystyle=\frac{1}{2}\left(\varphi\left((z_{1}^{p_{1}},\ldots,z_{n-1}^{p_{n-1}})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)+\varphi\left((z_{1}^{q_{1}},\ldots,z_{n-1}^{q_{n-1}})|_{V_{k}^{c}\setminus\\{k,n\\}}\right)\right)$
$\displaystyle=\varphi\left((z_{1}^{p_{1}},\ldots,z_{n-1}^{p_{n-1}})|_{V_{k}^{c}\setminus\\{k,n\\}}\right),$

thereby completing the proof. ∎

###### Proof of Theorem 9.6.

The proof follows immeditaley by combining Propositions 9.2, 9.8, and 9.9. ∎

##  10\. Maximizing Bi-Free Entropy

In this section, we will prove Theorem 10.3 obtaining an upper bound for the
bi-free entropy of a pair of operators and their adjoints based on the entropy
of a pair of matrices and demonstrate when equality is obtained. In
particular, this generalizes an essential result from [NSS1999]*Section 5.

To begin, we must establish a formula for the bi-free entropy of non-self-
adjoint operators.

######  Definition 10.1.

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space and let
$\\{X_{i},X_{i}^{*}\\}^{n}_{i=1}\cup\\{X^{\prime}_{i}\\}^{n^{\prime}}_{i=1}\cup\\{Y_{j},Y_{j}^{*}\\}^{m}_{j=1}\cup\\{Y^{\prime}_{j}\\}^{m^{\prime}}_{j=1}\subseteq\mathcal{A}$
where $X^{\prime}_{i}$ and $Y^{\prime}_{j}$ are self-adjoint for all $i$ and
$j$. The _bi-free entropy of
$(\\{\mathbf{X},\mathbf{X}^{*},\mathbf{X}^{\prime}\\},\\{\mathbf{Y},\mathbf{Y}^{*},\mathbf{Y}^{\prime}\\})$_
is defined to be

$\displaystyle\chi^{*}(\mathbf{X},\mathbf{X}^{*},\mathbf{X}^{\prime}\sqcup\mathbf{Y},\mathbf{Y}^{*},\mathbf{Y}^{\prime})=\frac{2n+2m+n^{\prime}+m^{\prime}}{2}\ln(2\pi
e)+\frac{1}{2}\int^{\infty}_{0}\left(\frac{2n+2m+n^{\prime}+m^{\prime}}{1+t}-g(t)\right)\,dt,$

where

$g(t)=\Phi^{*}\left(\mathbf{X}+\sqrt{t}\mathbf{C}_{\ell},\mathbf{X}^{*}+\sqrt{t}\mathbf{C}^{*}_{\ell},\mathbf{X}^{\prime}+\sqrt{t}\mathbf{S}\sqcup\mathbf{Y}+\sqrt{t}\mathbf{C}_{r},\mathbf{Y}^{*}+\sqrt{t}\mathbf{C}^{*}_{r},\mathbf{Y}^{\prime}+\sqrt{t}\mathbf{D}\right),$

where $\mathbf{S}$ and $\mathbf{D}$ consist of semicircular variables of mean
0, variance 1, covariance 0 and $\mathbf{C}_{\ell}$ and $\mathbf{C}_{r}$
consist of circular variables of mean 0, variance 1 and covariance 0 such that

$(\\{\mathbf{X},\mathbf{X}^{*},\mathbf{X}^{\prime}\\},\\{\mathbf{Y},\mathbf{Y}^{*},\mathbf{Y}^{\prime}\\})\cup\\{(S_{i},1)\\}^{n^{\prime}}_{i=1}\cup\\{(1,D_{j})\\}^{m^{\prime}}_{j=1}\cup\\{(\\{C_{\ell,i},C_{\ell,i}^{*}\\},1)\\}^{n}_{i=1}\cup\\{(1,\\{C_{r,j},C_{r,j}^{*}\\})\\}^{m}_{j=1}$

are bi-free.

######  Remark 10.2.

Given any C∗-non-commutative probability space $(\mathcal{A},\varphi)$, it is
always possible to find a larger C∗-non-commutative probability space that
contains the necessary bi-free elements from Definition 10.1. Indeed, one need
only consider the scalar reduced free product of the appropriate spaces and
use Definition 2.7 to obtain bi-freeness. The fact that the state is positive
follows as it will be a vector state.

In Definition 10.1, the bi-free Fisher information for tuples without self-
adjoint elements is that in this paper with $\eta=\varphi$ and $B=\mathbb{C}$.

In the simplest case, one may ask why we do not simply define

$\chi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})=\chi^{*}\left(\left\\{\Re(x),\Im(x)\right\\}\sqcup\left\\{\Re(y),\Im(y)\right\\}\right)$

to trivially reduce to the self-adjoint case in a similar fashion to Remark
7.2 part (iv) and why the integrand in Definition 10.1 is well-defined. Both
of these questions are answered via Remark 7.2 part (iv) as

$\displaystyle\Phi^{*}\left(\left\\{x+\sqrt{t}c_{\ell},x^{*}+\sqrt{t}c_{\ell}^{*}\right\\}\sqcup\left\\{y+\sqrt{t}c_{r},y^{*}+\sqrt{t}c_{r}^{*}\right\\}\right)$
$\displaystyle=\frac{1}{2}\Phi^{*}\left(\left\\{\Re(x)+\sqrt{t}\Re(c_{\ell}),\Im(x)+\sqrt{t}\Im(c_{\ell})\right\\}\sqcup\left\\{\Re(y)+\sqrt{t}\Re(c_{r}),\Im(y)+\sqrt{t}\Im(c_{r})\right\\}\right)$
$\displaystyle=\frac{1}{2}\Phi^{*}\left(\left\\{\Re(x)+\frac{\sqrt{t}}{\sqrt{2}}s_{1},\Im(x)+\frac{\sqrt{t}}{\sqrt{2}}s_{2}\right\\}\sqcup\left\\{\Re(y)+\frac{\sqrt{t}}{\sqrt{2}}d_{1},\Im(y)+\frac{\sqrt{t}}{\sqrt{2}}d_{2}\right\\}\right)$
$\displaystyle=\Phi^{*}\left(\left\\{\sqrt{2}\Re(x)+\sqrt{t}s_{1},\sqrt{2}\Im(x)+\sqrt{t}s_{2}\right\\}\sqcup\left\\{\sqrt{2}\Re(y)+\sqrt{t}d_{1},\sqrt{2}\Im(y)+\sqrt{t}d_{2}\right\\}\right)$

where $s_{1},s_{2},d_{1}$, and $d_{2}$ are as in Example 9.5. Hence the
integrand in Definition 10.1 is well-defined with

$\displaystyle\chi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})$
$\displaystyle=\chi^{*}\left(\left\\{\sqrt{2}\Re(x),\sqrt{2}\Im(x)\right\\}\sqcup\left\\{\sqrt{2}\Re(y),\sqrt{2}\Im(y)\right\\}\right)$
$\displaystyle=\chi^{*}\left(\left\\{\Re(x),\Im(x)\right\\}\sqcup\left\\{\Re(y),\Im(y)\right\\}\right)+4\ln(\sqrt{2}).$

We normalize Definition 10.1 so that the following holds and generalizes
[NSS1999]*Theorem 1.4 in the case $d=1$.

######  Theorem 10.3.

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space and let
$x,y\in\mathcal{A}$ be such that $x^{*}x$ and $xx^{*}$ have the same
distribution with respect to $\varphi$ and $y^{*}y$ and $yy^{*}$ have the same
distribution with respect to $\varphi$. With $X$ and $Y$ as in Section 9,

$\chi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})\leq 2\chi^{*}(X\sqcup Y)$

and equality holds whenever the pair $(x,y)$ is bi-R-diagonal and alternating
adjoint flipping.

To prove Theorem 10.3, we need two technical lemmata. For the first, note the
following does not immediately follow from Remark 2.8 as being bi-free over
$M_{2}(\mathbb{C})$ with respect to $E_{2}$ does not imply being bi-free with
respect to $\tau_{2}$.

######  Lemma 10.4.

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space, let
$x,y\in\mathcal{A}$ be such that $x^{*}x$ and $xx^{*}$ have the same
distribution with respect to $\varphi$ and $y^{*}y$ and $yy^{*}$ have the same
distribution with respect to $\varphi$, and let $(c_{\ell},c_{r})$ be a bi-
free circular pair in $\mathcal{A}$ with mean 0, variance 1 and covariance 0
such that

$(\\{x,x^{*}\\},\\{y,y^{*}\\})\cup\\{(\\{c_{\ell},c^{*}_{\ell}\\},1)\\}\cup\\{(1,\\{c_{r},c_{r}^{*}\\})\\}$

are bi-free with respect to $\varphi$. Using the notation of Section 10, if

$S_{\ell}=c_{\ell}\otimes E_{1,2}\otimes I_{2}+c_{\ell}^{*}\otimes
E_{2,1}\otimes I_{2}\in A_{2}\quad\text{and}\quad S_{r}=c_{r}\otimes
I_{2}\otimes E_{1,2}+c_{r}^{*}\otimes I_{2}\otimes E_{2,1}\in A_{2},$

then $S_{\ell}$ and $S_{r}$ have semicircular distributions with respect to
$\tau_{2}$ of mean 0 and variance 1 and

$\\{(X,Y)\\}\cup\\{(S_{\ell},1_{A_{2}})\\}\cup\\{(1_{A_{2}},S_{r})\\}$

are bi-free with respect to $\tau_{2}$.

###### Proof.

As $\\{(\\{c_{\ell},c^{*}_{\ell}\\},1)\\}\cup\\{(1,\\{c_{r},c_{r}^{*}\\})\\}$
are bi-free with respect to $\varphi$ by Example 9.5, Remark 2.8 implies that
$(S_{\ell},1)$ and $(1,S_{r})$ are bi-free with respect to $E_{2}$. Moreover,
as $c_{\ell}$ and $c_{r}$ commute, $s_{\ell}$ and $s_{r}$ commute. Hence we
see for all $n,m\in\mathbb{N}$ that

$\displaystyle\tau_{2}(s_{\ell}^{n}s_{r}^{m})=\operatorname{tr}_{2}(E_{2}(s_{\ell}^{n}s_{r}^{m}))=\operatorname{tr}_{2}(E_{2}(s_{\ell}^{n})E_{2}(s_{r}^{m}))=\begin{cases}0&\text{if
}n\text{ or }m\text{ is odd}\\\
\varphi\left((c_{\ell}^{*}c_{\ell})^{\frac{n}{2}}\right)\varphi\left((c_{r}^{*}c_{r})^{\frac{m}{2}}\right)&\text{if
$n$ and $m$ are even}\end{cases},$

by Example 9.5 and the alternating adjoint flipping condition. Therefore, as
$c_{\ell}^{*}c_{\ell}$ and $c_{r}^{*}c_{r}$ are known to have the same
distributions as the square of a semicircular element of mean 0 and variance 1
(see [VDN1992]*Section 5.1), we obtain that $(s_{\ell},s_{r})$ is the bi-free
central limit distribution with mean 0, variance 1 and covariance 0 with
respect to $\tau_{2}$. Hence
$\\{(S_{\ell},1_{A_{2}})\\}\cup\\{(1_{A_{2}},S_{r})\\}$ are bi-free with
respect to $\tau_{2}$.

To complete the proof, it suffices to show that
$\\{(X,Y)\\}\cup\\{(S_{\ell},S_{r})\\}$ are bi-free with respect to
$\tau_{2}$. Therefore, by [CNS2015-2], it suffices to show for all
$n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$, non-constant
$\gamma\in\\{1,2\\}^{n}$, and $Z_{k}\in\\{X,Y,S_{\ell},S_{r}\\}$ where

$Z_{k}=\begin{cases}X&\text{if }\chi(k)=\ell\text{ and }\gamma(k)=1\\\
Y&\text{if }\chi(k)=r\text{ and }\gamma(k)=1\\\ S_{\ell}&\text{if
}\chi(k)=\ell\text{ and }\gamma(k)=2\\\ S_{r}&\text{if }\chi(k)=r\text{ and
}\gamma(k)=2\end{cases}$

that

$\displaystyle\tau_{2}(Z_{1}\cdots
Z_{n})=\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\
\pi\leq\gamma\end{subarray}}\kappa^{\tau_{2}}_{\pi}(Z_{1},\ldots,Z_{n})$ (9)

where $\gamma$ is representing the partition
$\\{\\{k\,\mid\,\gamma(k)=1\\},\\{k\mid\,\gamma(k)=2\\}\\}$. Note if
$z_{1},\ldots,z_{n}\in\\{x,y,c_{\ell},c_{r}\\}$ are such that

$z_{k}=\begin{cases}x&\text{if }Z_{k}=X\\\ y&\text{if }Z_{k}=Y\\\
c_{\ell}&\text{if }Z_{k}=S_{\ell}\\\ c_{r}&\text{if }Z_{k}=S_{r}\end{cases}$

then by Lemma 9.1 and the fact that
$(\\{x,x^{*}\\},\\{y,y^{*}\\})\cup\\{(c_{\ell},1)\\}\cup\\{(1,c_{r})\\}$ are
bi-free with respect to $\varphi$, we have that

$\displaystyle\tau_{2}(Z_{1}\cdots Z_{n})$
$\displaystyle=\begin{cases}0&\text{if }n\text{ is odd}\\\
\frac{1}{2}\left(\varphi\left(z_{1}^{p_{1}}\cdots
z_{n}^{p_{n}}\right)+\varphi\left(z_{1}^{q_{1}}\cdots
z_{n}^{q_{n}}\right)\right)&\text{if }n\text{ is even}\end{cases}$
$\displaystyle=\begin{cases}0&\text{if }n\text{ is odd}\\\
\frac{1}{2}\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\
\pi\leq\gamma\end{subarray}}\kappa^{\varphi}_{\pi}\left(z_{1}^{p_{1}},\ldots,z_{n}^{p_{n}}\right)+\kappa^{\varphi}_{\pi}\left(z_{1}^{q_{1}},\ldots,z_{n}^{q_{n}}\right)&\text{if
}n\text{ is even}\end{cases},$

where

$p_{s_{\chi}(k)}=\begin{cases}1&\text{if }k\text{ is odd}\\\ *&\text{if
}k\text{ is even}\end{cases}\qquad\text{and}\qquad
q_{s_{\chi}(k)}=\begin{cases}*&\text{if }k\text{ is odd}\\\ 1&\text{if
}k\text{ is even}\end{cases}.$

To show this agrees with the right-hand side of equation (9), we divide the
discussion into several cases. To this end, let

$I_{X,Y}=\\{k\,\mid\,\gamma(k)=1\\}\qquad\text{and}\qquad
I_{S}=\\{k\,\mid\,\gamma(k)=2\\}.$

First suppose $n$ is odd. If $|I_{S}|$ is odd, then the right-hand side of
equation (9) is zero as there must be a cumulant involving an odd number of
$S_{\ell}$ and $S_{r}$ and $\\{(S_{\ell},S_{r})\\}$ is a bi-free central limit
distribution with 0 mean. Otherwise, $|I_{X,Y}|$ is odd. In this case, we may
rearrange the sum on the right-hand side of equation (9) to add over all
$\pi\in\operatorname{BNC}(\chi)$ with $\pi\leq\gamma$ that form the same
partition when restricted to $I_{S}$. Since summing over such partitions
yields a product of moment terms in the $X$’s and $Y$’s where the sum of the
lengths of the moments is $|I_{X,Y}|$ and since all odd moment terms involving
only $X$’s and $Y$’s is zero by Lemma 9.1, this portion of the sum yields
zero. Hence, equation (9) holds when $n$ is odd.

In the case $n$ is even, note if $|I_{S}|$ is odd, then the right-hand side of
equation (9) is still zero. However,

$\frac{1}{2}\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\
\pi\leq\gamma\end{subarray}}\kappa^{\varphi}_{\pi}\left(z_{1}^{p_{1}},\ldots,z_{n}^{p_{n}}\right)+\kappa^{\varphi}_{\pi}\left(z_{1}^{q_{1}},\ldots,z_{n}^{q_{n}}\right)=0$

as there must be a cumulant involving an odd number of
$(\\{c_{\ell},c_{\ell}^{*}\\},\\{c_{r},c_{r}^{*}\\})$ and $(c_{\ell},c_{r})$
is a bi-free circular pair. Thus, we may assume that $n$, $|I_{S}|$, and
$|I_{X,Y}|$ are even.

Under these assumptions, we claim that

$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\
\pi\leq\gamma\end{subarray}}\kappa^{\varphi}_{\pi}\left(z_{1}^{p_{1}},\ldots,z_{n}^{p_{n}}\right)+\kappa^{\varphi}_{\pi}\left(z_{1}^{q_{1}},\ldots,z_{n}^{q_{n}}\right)=\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\
\pi\leq\gamma\end{subarray}}\kappa^{\tau_{2}}_{\pi}(Z_{1},\ldots,Z_{n}).$

To see this, again we need only consider $\pi\in\operatorname{BNC}(\chi)$ that
form pair partitions when restricted to $I_{S}$ and no block of $\pi$ contains
both an element of $\\{k\,\mid\,\chi(k)=\ell\\}$ and of
$\\{k\,\mid\,\chi(k)=r\\}$, since $\\{(S_{\ell},1)\\}\cup\\{(1,S_{r})\\}$ are
bi-free with respect to $\tau_{2}$ and
$\\{(\\{c_{\ell},c^{*}_{\ell}\\},1)\\}\cup\\{(1,\\{c_{r},c_{r}^{*}\\})\\}$ are
bi-free with respect to $\varphi$. For such a partition $\pi$, if we let
$\widehat{\pi}$ be the largest partition on $I_{X,Y}$ such that
$\widehat{\pi}\cup\pi|_{I_{S}}$ is an element of $\operatorname{BNC}(\chi)$,
then by adding over all $\sigma\in\operatorname{BNC}(\chi)$ with
$\sigma\leq\gamma$ and $\sigma|_{I_{S}}=\pi|_{I_{S}}$, it suffices to show
that

$\displaystyle\frac{1}{2}$
$\displaystyle\left(\varphi_{\widehat{\pi}}\left((z_{1}^{p_{1}},\ldots,z_{n}^{p_{n}})|_{I_{X,Y}}\right)\kappa^{\varphi}_{\pi|_{I_{S}}}\left(\left(z_{1}^{p_{1}},\ldots,z_{n}^{p_{n}}\right)|_{I_{S}}\right)+\varphi_{\widehat{\pi}}\left((z_{1}^{q_{1}},\ldots,z_{n}^{q_{n}})|_{I_{X,Y}}\right)\kappa^{\varphi}_{\pi|_{I_{S}}}\left(\left(z_{1}^{q_{1}},\ldots,z_{n}^{q_{n}}\right)|_{I_{S}}\right)\right)$
$\displaystyle=(\tau_{2})_{\widehat{\pi}}\left(\left(Z_{1},\ldots,Z_{n}\right)|_{I_{X,Y}}\right).$
(10)

Note that

$\kappa^{\varphi}_{\pi|_{I_{S}}}\left(\left(z_{1}^{p_{1}},\ldots,z_{n}^{p_{n}}\right)|_{I_{S}}\right)=0\quad\text{
or
}\quad\kappa^{\varphi}_{\pi|_{I_{S}}}\left(\left(z_{1}^{q_{1}},\ldots,z_{n}^{q_{n}}\right)|_{I_{S}}\right)=0$

if and only if $\pi$ has a block with two $*$-terms or two non-$*$-terms, as
$\pi|_{I_{S}}$ is a pair partition and $(c_{\ell},c_{r})$ is a bi-circular
pair. In this case we would have that $\widehat{\pi}$ has a block of odd
length and thus the right-hand side of equation (10) is also zero, as any odd
$\tau_{2}$-moment involving $X$ and $Y$ is zero. Otherwise, both
$\varphi$-cumulants are 1 and this forces every block of $\widehat{\pi}$ to be
of even length and alternate between $1$ and $*$ in the $\chi$-ordering. Since

$\displaystyle\varphi((x^{*}x)^{m})=\varphi((xx^{*})^{m})$
$\displaystyle=\tau_{2}(X^{2m})\quad\text{and}\quad\varphi((y^{*}y)^{m})=\varphi((yy^{*})^{m})=\tau_{2}(Y^{2m})$

and since (by the assumption that $\pi$ does not contain a block containing
elements of $\\{k\,\mid\,\chi(k)=\ell\\}$ and of $\\{k\,\mid\,\chi(k)=r\\}$)
there is a single block of $\widehat{\pi}$ containing elements of
$\\{k\,\mid\,\chi(k)=\ell\\}$ and $\\{k\,\mid\,\chi(k)=r\\}$, adding the two
$\varphi$-terms together produces exactly the $\tau_{2}$ term in equation
(10). ∎

######  Lemma 10.5.

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space, let
$x,y\in\mathcal{A}$, and let $(c_{\ell},c_{r})$ be a bi-free circular pair in
$\mathcal{A}$ with mean 0, variance 1 and covariance 0 such that

$(\\{x,x^{*}\\},\\{y,y^{*}\\})\cup\\{(\\{c_{\ell},c^{*}_{\ell}\\},1)\\}\cup\\{(1,\\{c_{r},c_{r}^{*}\\})\\}$

are bi-free with respect to $\varphi$. Then:

  1. (i)

If $(x,y)$ is bi-R-diagonal, then
$\left(x+\sqrt{t}c_{\ell},y+\sqrt{t}c_{r}\right)$ is bi-R-diagonal for all
$t\in(0,\infty)$.

  2. (ii)

If $(x,y)$ is alternating adjoint flipping, then
$\left(x+\sqrt{t}c_{\ell},y+\sqrt{t}c_{r}\right)$ is alternating adjoint
flipping for all $t\in(0,\infty)$.

  3. (iii)

If $x^{*}x$ and $xx^{*}$ (respectively $y^{*}y$ and $yy^{*}$) have the same
distribution with respect to $\varphi$, then
$(x+\sqrt{t}c_{\ell})^{*}(x+\sqrt{t}c_{\ell})$ and
$(x+\sqrt{t}c_{\ell})(x+\sqrt{t}c_{\ell})^{*}$ (respectively
$(y+\sqrt{t}c_{r})^{*}(y+\sqrt{t}c_{r})$ and
$(y+\sqrt{t}c_{r})(y+\sqrt{t}c_{r})^{*})$ have the same distribution with
respect to $\varphi$.

###### Proof.

As $(c_{\ell},c_{r})$ is bi-R-diagonal by Example 9.5 and as sums and scalar
multiples of bi-R-diagonal pairs are bi-R-diagonal by [K2019]*Proposition 3.1,
(i) follows.

To see that (ii) holds, first we claim for all $n\in\mathbb{N}$,
$\chi\in\\{\ell,r\\}^{2n}$, and
$z_{1},\ldots,z_{n}\in\\{x,y,c_{\ell},c_{r}\\}$ such that

$z_{k}\in\begin{cases}\\{x,c_{\ell}\\}&\text{if }\chi(k)=\ell\\\
\\{y,c_{r}\\}&\text{if }\chi(k)=r\end{cases},$

we have that

$\varphi(z_{1}^{p_{1}}\cdots z_{2n}^{p_{2n}})=\varphi(z_{1}^{q_{1}}\cdots
z_{2n}^{q_{2n}}),$

where

$p_{s_{\chi}(k)}=\begin{cases}1&\text{if }k\text{ is odd}\\\ *&\text{if
}k\text{ is even}\end{cases}\qquad\text{and}\qquad
q_{s_{\chi}(k)}=\begin{cases}*&\text{if }k\text{ is odd}\\\ 1&\text{if
}k\text{ is even}\end{cases}.$

Recall that

$\varphi(z_{1}^{p_{1}}\cdots
z_{2n}^{p_{2n}})=\sum_{\pi\in\operatorname{BNC}(\chi)}\kappa_{\pi}(z_{1}^{p_{1}},\ldots,z_{2n}^{p_{2n}})$

and the bi-free cumulant is zero if any block of $\pi$ contains both an
element of $\\{x,x^{*},y,y^{*}\\}$ and an element of
$\\{c_{\ell},c^{*}_{\ell},c_{r},c_{r}^{*}\\}$ As the only cumulants involving
$c_{\ell},c^{*}_{\ell},c_{r},c^{*}_{r}$ with non-zero values are

$\kappa_{1_{(\ell,\ell)}}(c_{\ell},c^{*}_{\ell})=1=\kappa_{1_{(\ell,\ell)}}(c^{*}_{\ell},c_{\ell})\qquad\text{and}\qquad\kappa_{1_{(r,r)}}(c_{r},c^{*}_{r})=1=\kappa_{1_{(r,r)}}(c^{*}_{r},c_{r}),$

for any fixed $\pi\in\operatorname{BNC}(\chi)$ for which the blocks containing
$\\{c_{\ell},c^{*}_{\ell},c_{r},c_{r}^{*}\\}$ do not cause the bi-free
cumulant to be zero, we may add over all elements of
$\operatorname{BNC}(\chi)$ with the same blocks as $\pi$ for those indices
corresponding to elements of $\\{c_{\ell},c^{*}_{\ell},c_{r},c_{r}^{*}\\}$ to
obtain a product of moments involving $\\{x,x^{*},y,y^{*}\\}$, each of which
is of even length and alternates between $1$ and $*$ in the $\chi$-ordering.
We may then use the alternating adjoint flipping condition on $(x,y)$ to
exchange the powers and reverse this cumulant reduction process to obtain
$\varphi(z_{1}^{q_{1}}\cdots z_{2n}^{q_{2n}})$, thereby completing the claim.
Thus (ii) then follows by linearity.

To see that (iii) holds, we desire to show that

$\varphi\left(\left((x+\sqrt{t}c_{\ell})^{*}(x+\sqrt{t}c_{\ell})\right)^{n}\right)=\varphi\left(\left((x+\sqrt{t}c_{\ell})(x+\sqrt{t}c_{\ell})^{*}\right)^{n}\right)$

for all $n\in\mathbb{N}$. To see how the left-hand side can be changed into
the right-hand side, arguments similar to the proof of Lemma 10.4 are used.
First, we expand out the product and expand the moment using linearity. Then,
for each moment term, we expand via the free cumulants and use the fact that
mixed free cumulants vanish. Cumulants involving an odd number of $c_{\ell}$
and $c_{\ell}^{*}$ vanish and thus we can consider only pair partitions when
restricted to entries involving $c_{\ell}$ and $c_{\ell}^{*}$. Any cumulant
involving just $c_{\ell}$ or just $c_{\ell}^{*}$ vanishes and can be ignored.
By adding over all partitions with the same blocks on $c_{\ell}$ and
$c_{\ell}^{*}$ that do not vanish yields a product of moment terms of the form
$\varphi((x^{*}x)^{m})$ and $\varphi((xx^{*})^{m})$. For any such terms,
viewing the $(2n)^{\mathrm{th}}$ term as the first term doesn’t change the
value, as the distributions of $x^{*}x$ and $xx^{*}$ are the same, thereby
effectively moving the $x$ or $c_{\ell}$ term at the end to the beginning. One
then reverses the above process and obtains the right-hand side as desired. ∎

###### Proof of Theorem 10.3..

As per Remark 10.2, we may assume without loss of generality that there exists
a bi-free circular pair $(c_{\ell},c_{r})$ (with mean 0, variance 1 and
covariance 0) in $\mathcal{A}$ such that

$\\{(\\{x,x^{*}\\},\\{y,y^{*}\\})\\}\cup\\{(\\{c_{\ell},c_{\ell}^{*}\\},1)\\}\cup\\{(1,\\{c_{r},c_{r}^{*}\\})\\}$

are bi-free. Therefore, as $\\{(X,Y)\\}\cup\\{(S_{\ell},S_{r})\\}$ are bi-free
with respect to $\tau_{2}$ by Lemma 10.4, we obtain that

$\chi^{*}(X\sqcup Y)=\ln(2\pi
e)+\frac{1}{2}\int^{\infty}_{0}\left(\frac{2}{1+t}-\Phi^{*}\left(X+\sqrt{t}S_{\ell}\sqcup
Y+\sqrt{t}S_{r}\right)\right)\,dt.$

However, as

$\displaystyle X+\sqrt{t}S_{\ell}$ $\displaystyle=(x+\sqrt{t}c_{\ell})\otimes
E_{1,2}\otimes I_{2}+(x+\sqrt{t}c_{\ell})^{*}\otimes E_{2,1}\otimes I_{2},$
$\displaystyle Y+\sqrt{t}S_{r}$ $\displaystyle=(y+\sqrt{t}c_{r})\otimes
I_{2}\otimes E_{1,2}+(y+\sqrt{t}c_{r})^{*}\otimes I_{2}\otimes E_{2,1}$

and as Lemma 10.5 part (iii) shows that the assumptions of Theorem 9.6 as
satisfied, we obtain that

$\displaystyle\Phi^{*}\left(\\{x+\sqrt{t}c_{\ell},(x+\sqrt{t}c_{\ell})^{*}\\}\sqcup\\{y+\sqrt{t}c_{r},(y+\sqrt{t}c_{r})^{*}\\}\right)\geq
2\Phi^{*}\left(X+\sqrt{t}S_{\ell}\sqcup Y+\sqrt{t}S_{r}\right)$ (11)

for all $t\in(0,\infty)$. Hence the inequality

$\chi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})\leq 2\chi^{*}(X\sqcup Y)$

follows by comparing the above bi-free entropy formula with that from
Definition 10.1.

In the case that $(x,y)$ is bi-R-diagonal and alternating adjoint flipping,
Lemma 10.5 implies $(x+\sqrt{t}c_{\ell},y+\sqrt{t}c_{r})$ is bi-R-diagonal and
alternating adjoint flipping for all $t\in(0,\infty)$, thus equality holds in
equation (11) by Theorem 9.6. ∎

##  11\. Other Results

In this section, we will examine other results from [NSS1999] that generalize
to the bi-free setting. As these results are less connected to bi-free entropy
with respect to a completely positive map and have proofs that can be adapted
from [NSS1999] using the same modifications from Sections 9 and 10 to deal
with the $\chi$-ordering, we simply state these results.

######  Theorem 11.1 (Generalization of [NSS1999]*Theorem 1.2).

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space, let
$d\in\mathbb{N}$, and let $(A_{d},E_{d},\varepsilon,\tau_{d})$ be as in
Section 9. Then

  1. (i)

For all
$\\{x_{i,j}\\}^{d}_{i,j=1},\\{y_{i,j}\\}^{d}_{i,j=1}\subseteq\mathcal{A}$, if

$X=\sum^{d}_{i,j=1}x_{i,j}\otimes E_{i,j}\otimes I_{d}\qquad\text{and}\qquad
Y=\sum^{d}_{i,j=1}y_{i,j}\otimes I_{d}\otimes E_{i,j},$

then

$\Phi^{*}\left(\\{x_{i,j},x^{*}_{i,j}\\}^{d}_{i,j=1}\sqcup\\{y_{i,j},y_{i,j}^{*}\\}^{d}_{i,j=1}\right)\geq
d^{3}\Phi^{*}(X,X^{*}\sqcup Y,Y^{*}).$

Moreover, equality holds if $(\\{X,X^{*}\\},\\{Y,Y^{*}\\})$ is bi-free from
$(M_{d}(\mathbb{C})_{\ell},M_{d}(\mathbb{C})_{r})$ with respect to $\tau_{d}$.

  2. (ii)

If in (i) $X$ and $Y$ are self-adjoint, then

$\Phi^{*}\left(\\{x_{i,j}\\}^{d}_{i,j=1}\sqcup\\{y_{i,j}\\}^{d}_{i,j=1}\right)\geq
d^{3}\Phi^{*}(X\sqcup Y)$

with equality holding if $(\\{X\\},\\{Y\\})$ is bi-free from
$(M_{d}(\mathbb{C})_{\ell},M_{d}(\mathbb{C})_{r})$ with respect to $\tau_{d}$.

######  Theorem 11.2 (Generalization of [NSS1999]*Theorem 1.5).

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space, let
$d\in\mathbb{N}$, and let $(A_{d},E_{d},\varepsilon,\tau_{d})$ be as in
Section 9. Then

  1. (i)

For all
$\\{x_{i,j}\\}^{d}_{i,j=1},\\{y_{i,j}\\}^{d}_{i,j=1}\subseteq\mathcal{A}$, if

$X=\sum^{d}_{i,j=1}x_{i,j}\otimes E_{i,j}\otimes I_{d}\qquad\text{and}\qquad
Y=\sum^{d}_{i,j=1}y_{i,j}\otimes I_{d}\otimes E_{i,j},$

then

$\chi^{*}\left(\\{x_{i,j},x^{*}_{i,j}\\}^{d}_{i,j=1}\sqcup\\{y_{i,j},y_{i,j}^{*}\\}^{d}_{i,j=1}\right)\geq
d^{2}\left(\chi^{*}(X,X^{*}\sqcup Y,Y^{*})-2\ln(d)\right).$

Moreover, equality holds if $(\\{X,X^{*}\\},\\{Y,Y^{*}\\})$ is bi-free from
$(M_{d}(\mathbb{C})_{\ell},M_{d}(\mathbb{C})_{r})$ with respect to $\tau_{d}$.

  2. (ii)

If in (i) $X$ and $Y$ are self-adjoint, then

$\chi^{*}\left(\\{x_{i,j}\\}^{d}_{i,j=1}\sqcup\\{y_{i,j}\\}^{d}_{i,j=1}\right)\leq
d^{2}\left(\chi^{*}(X\sqcup Y)-\ln(d)\right)$

with equality holding if $(\\{X\\},\\{Y\\})$ is bi-free from
$(M_{d}(\mathbb{C})_{\ell},M_{d}(\mathbb{C})_{r})$ with respect to $\tau_{d}$.

To prove Theorem 11.2, we note it is essential to prove the following.

######  Lemma 11.3 (Generalization of [NSS1999]*Proposition 5.3).

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space, let
$D_{\ell},D_{r}$ be $*$-subalgebras of $A$, let $d\in\mathbb{N}$, and let
$(A_{d},E_{d},\varepsilon,\tau_{d})$ be as in Section 9.

  1. (i)

If $\\{c_{\ell,i,j}\\}^{d}_{i,j=1}$ and $\\{c_{r,i,j}\\}^{d}_{i,j=1}$ are
circular elements of $\mathcal{A}$ with mean 0 and variance 1 such that

$\\{(D_{\ell},D_{r})\\}\cup\\{(\\{c_{\ell,i,j},c^{*}_{\ell,i,j}\\},1)\\}^{d}_{i,j=1}\cup\\{(1,\\{c_{r,i,j},c^{*}_{r,i,j})\\}^{d}_{i,j=1}$

are bi-free, then

$C_{\ell}=\sum^{d}_{i,j=1}c_{\ell,i,j}\otimes E_{i,j}\otimes
I_{d}\quad\text{and}\quad C_{r}=\sum^{d}_{i,j=1}c_{r,i,j}\otimes I_{d}\otimes
E_{i,j}$

are circular elements of mean 0 and variance $d$ such that
$(\\{C_{\ell},C_{\ell}^{*}\\},1)$, $(1,\\{C_{r},C^{*}_{r}\\})$, and
$(D_{\ell}\otimes M_{d}(\mathbb{C})\otimes I_{D},D_{r}\otimes I_{d}\otimes
M_{d}(\mathbb{C}))$ are bi-free with respect to $\tau_{d}$.

  2. (ii)

If $\\{s_{\ell,i,j}\\}^{d}_{i,j=1}$ and $\\{s_{r,i,j}\\}^{d}_{i,j=1}$ are
elements of $\mathcal{A}$ with mean 0 and variance 1 such that $s_{\ell,i,i}$
and $s_{r,i,i}$ are semicircular elements for all $i$, $s_{\ell,i,j}$ and
$s_{r,i,j}$ are circular elements for all $i,j$,
$s_{\ell,i,j}^{*}=s_{\ell,j,i}$ and $s_{r,i,j}^{*}=s_{r,j,i}$ for all $i,j$,
and

$\\{(D_{\ell},D_{r})\\}\cup\\{(s_{\ell,i,i},1)\\}^{d}_{i=1}\cup\\{(1,s_{r,i,i})\\}^{d}_{i=1}\cup\\{(\\{s_{\ell,i,j},s^{*}_{\ell,i,j}\\},1)\\}_{1\leq
i<j\leq d}\cup\\{(1,\\{s_{r,i,j},ss^{*}_{r,i,j})\\}_{1\leq i<j\leq d}$

are bi-free, then

$S_{\ell}=\sum^{d}_{i,j=1}s_{\ell,i,j}\otimes E_{i,j}\otimes
I_{d}\quad\text{and}\quad S_{r}=\sum^{d}_{i,j=1}s_{r,i,j}\otimes I_{d}\otimes
E_{i,j}$

are semicircular elements of mean 0 and variance $d$ such that $(S_{\ell},1)$,
$(1,S_{r})$, and $(D_{\ell}\otimes M_{d}(\mathbb{C})\otimes I_{D},D_{r}\otimes
I_{d}\otimes M_{d}(\mathbb{C}))$ are bi-free with respect to $\tau_{d}$.

Note the proof of Lemma 11.3 is obtained by first generalizing [NSS1999]*Lemma
5.4, which clearly holds due to the bi-free cumulant characterization of the
conjugate variables. The proof then proceeds via constructing the appropriate
conjugate variables for either $(C_{\ell},C_{r})$ or $(S_{\ell},S_{r})$ using
the techniques from Section 9.

## References

  * [1]
CharlesworthI.An alternating moment condition for bi-freenessAdv.
Math.3462019546–568@article{C2019, author = {Charlesworth, I.}, title = {An
alternating moment condition for bi-freeness}, journal = {Adv. Math.}, volume
= {346}, year = {2019}, pages = {546-568}}

  * [3]
CharlesworthI.NelsonB.SkoufranisP.Combinatorics of bi-free probability with
amalgamationComm. Math. Phys.33822015801–847@article{CNS2015-1, author =
{Charlesworth, I.}, author = {Nelson, B.}, author = {Skoufranis, P.}, title =
{Combinatorics of Bi-Free Probability with Amalgamation}, journal = {Comm.
Math. Phys.}, volume = {338}, number = {2}, year = {2015}, pages = {801-847}}

  * [5]
CharlesworthI.NelsonB.SkoufranisP.On two-faced families of non-commutative
random variablesCanad. J. Math.67620151290–1325@article{CNS2015-2, author =
{Charlesworth, I.}, author = {Nelson, B.}, author = {Skoufranis, P.}, title =
{On two-faced families of non-commutative random variables}, journal = {Canad.
J. Math.}, volume = {67}, number = {6}, year = {2015}, pages = {1290-1325}}

  * [7]
CharlesworthI.SkoufranisP.Analogues of entropy in bi-free probability theory:
microstatesarXiv:1902:03874201938@article{CS2019, author = {Charlesworth, I.},
author = {Skoufranis, P.}, title = {Analogues of Entropy in Bi-Free
Probability Theory: Microstates}, eprint = {arXiv:1902:03874}, year = {2019},
pages = {38}}

  * [9]
CharlesworthI.SkoufranisP.Analogues of entropy in bi-free probability theory:
non-microstatesAdv. Math.3752020107367@article{CS2020, author = {Charlesworth,
I.}, author = {Skoufranis, P.}, title = {Analogues of Entropy in Bi-Free
Probability Theory: Non-Microstates}, journal = {Adv. Math.}, volume = {375},
year = {2020}, pages = {107367}}

  * [11]
GuY.SkoufranisP.Conditionally bi-free independence with amalgamationInt. Math.
Res. Not.20182320187359–7419@article{GS2018, author = {Gu, Y.}, author =
{Skoufranis, P.}, title = {Conditionally Bi-Free Independence with
Amalgamation}, journal = {Int. Math. Res. Not.}, volume = {2018}, number =
{23}, year = {2018}, pages = {7359–7419}}

  * [13]
KatsimpasG.On bi-r-diagonal pairs of
operatorsarXiv:1902.01041201942@article{K2019, author = {Katsimpas, G.}, title
= {On Bi-R-Diagonal Pairs of Operators}, eprint = {arXiv:1902.01041}, year =
{2019}, pages = {42}}

  * [15]
NicaA.ShlyakhtenkoD.SpeicherR.Some minimization problems for the free analogue
of the fisher informationAdv. Math.14121999282–321@article{NSS1999, author =
{Nica, A.}, author = {Shlyakhtenko, D.}, author = {Speicher, R.}, title =
{Some minimization problems for the free analogue of the Fisher information},
journal = {Adv. Math.}, volume = {141}, number = {2}, year = {1999}, pages =
{282-321}}

  * [17]
NicaA.SpeicherR.Lectures on the combinatorics of free probabilityCambridge
University Press132006Cambridge@book{NS2006, author = {Nica, A.}, author =
{Speicher, R.}, title = {Lectures on the combinatorics of free probability},
publisher = {Cambridge University Press}, volume = {13}, year = {2006}, place
= {Cambridge}}

  * [19]
ShlyakhtenkoD.Free entropy with respect to a completely positive mapAm. J.
Math.122145–812000@article{S2000, author = {Shlyakhtenko, D.}, title = {Free
entropy with respect to a completely positive map}, journal = {Am. J. Math.},
volume = {122}, number = {1}, pages = {45-81}, year = {2000}}

  * [21]
SkoufranisP.Independences and partial $R$-transforms in bi-free
probabilityAnn. Inst. Henri Poincaré Probab.
Stat.5231437–14732016@article{S2016-1, author = {Skoufranis, P.}, title =
{Independences and Partial $R$-Transforms in Bi-Free Probability}, journal =
{Ann. Inst. Henri Poincar\'{e} Probab. Stat.}, volume = {52}, number = {3},
pages = {1437-1473}, year = {2016}}

  * [23]
SkoufranisP.On operator-valued bi-free distributionsAdv.
Math.203638–7152016@article{S2016-2, author = {Skoufranis, P.}, title = {On
Operator-Valued Bi-Free Distributions}, journal = {Adv. Math.}, volume =
{203}, pages = {638-715}, year = {2016}}

  * [25]
SpeicherR.Combinatorial theory of the free product with amalgamation and
operator-valued free probability theoryAmerican Mathematical
Soc.6271998@book{S1998, author = {Speicher, R.}, title = {Combinatorial theory
of the free product with amalgamation and operator-valued free probability
theory}, publisher = {American Mathematical Soc.}, volume = {627}, year =
{1998}}

  * [27]
VoiculescuD. V.The analogues of entropy and of fisher’s information measure in
free probability theory, v: noncommutative hilbert transformsInvent.
Math.13211998189–227@article{V1998-2, author = {Voiculescu, D. V.}, title =
{The analogues of entropy and of Fisher's information measure in free
probability theory, V: Noncommutative Hilbert transforms}, journal = {Invent.
Math.}, volume = {132}, number = {1}, year = {1998}, pages = {189-227}}

  * [29]
VoiculescuD. V.The analogues of entropy and of fisher’s information measure in
free probability theory, vi: liberation and mutual free informationAdv.
Math.14621999101–166@article{V1999, author = {Voiculescu, D. V.}, title = {The
analogues of entropy and of Fisher's information measure in free probability
theory, VI: Liberation and Mutual Free Information}, journal = {Adv. Math.},
volume = {146}, number = {2}, year = {1999}, pages = {101-166}}

  * [31]
VoiculescuD. V.Free probability for pairs of faces iComm. Math.
Phys.2014332955–980@article{V2014, author = {Voiculescu, D. V.}, title = {Free
probability for pairs of faces I}, journal = {Comm. Math. Phys.}, year =
{2014}, volume = {332}, pages = {955-980}}

  * [33]
VoiculescuD. V.DykemaK. J.NicaA.Free random variablesCRM Monograph
SeriesAmerican Mathematical Society1Providence, RI1992@book{VDN1992, author =
{Voiculescu, D. V.}, author = {Dykema, K. J.}, author = {Nica, A.}, title =
{Free Random Variables}, series = {CRM Monograph Series}, publisher =
{American Mathematical Society}, volume = {1}, place = {Providence, RI}, year
= {1992}}

  * [35]