# Minimum number-phase uncertainty states via weighted Bergman spaces

Yi C. Huang  School of Mathematical Sciences, Nanjing Normal University,
Nanjing 210023, People’s Republic of China<EMAIL_ADDRESS>https://orcid.org/0000-0002-1297-7674

###### Abstract.

The number-phase uncertainty result of Luo via the Hardy space on unit disc
(Phys Lett A, 2000) is extended in this paper to the scale of weighted Bergman
spaces. The minimum uncertainty states are thereby explicitly identified.

###### Key words and phrases:

Phase operator; uncertainty relation; Bergman spaces

###### 2020 Mathematics Subject Classification:

81Q10, 30H20

##  1\. Introduction

We are interested in the number-phase observable on the quantum mechanical
Hilbert space ${\mathcal{H}}$ of harmonic oscillator with one degree of
freedom. The (complex) inner product for ${\mathcal{H}}$ is denoted by
$\langle\cdot,\cdot\rangle$. As usual, we shall adopt Dirac’s bra-ket
notations $|\phi\rangle$ and $\langle\psi|$ to represent the vector $\phi$ in
${\mathcal{H}}$ and the linear functional

$\langle\psi|:\phi\mapsto\langle\psi|\phi\rangle$

that acts on ${\mathcal{H}}$. Let $N$ be the number operator with normalised
eigenstates denoted by $|n\rangle$, $n=0,1,2,\cdots$. Then $N$ can be
expressed formally as

$N=\sum_{n=0}^{\infty}n|n\rangle\langle n|.$

Let $\Phi$ be the exponential phase operator proposed by Dirac [4] (see also
Susskind and Glogower [14], Lévy-Leblond [9] and Newton [12] and the
references therein)

$\Phi=\sum_{n=0}^{\infty}|n\rangle\langle n+1|.$

Note that $\Phi$ annihilates $|0\rangle$, and sends $|n+1\rangle$ to
$|n\rangle$ for $n=0,1,2,\cdots$. Apparently,

$[\Phi,N]=\Phi.$ (1)

The formal adjoint of $\Phi$ is given by

$\Phi^{*}=\sum_{n=0}^{\infty}|n+1\rangle\langle n|.$

Accordingly, we have the operator identity $\langle 0|\Phi^{*}=0$ on
${\mathcal{H}}$.

As a concrete representation via analytic functions, we take ${\mathcal{H}}$
as the weighted Bergman space ${\mathcal{H}}_{\lambda}$ (see e.g. Hedenmalm,
Korenblum and Zhu [7]) defined by

${\mathcal{H}}_{\lambda}:=\bigg{\\{}f:{\mathbb{D}}\rightarrow{\mathbb{C}},\,\text{holomorphic},\,\langle
f,f\rangle=\frac{\lambda-1}{\pi}\int\kern-8.0pt\int_{{\mathbb{D}}}f(z)\overline{f(z)}(1-z\bar{z})^{\lambda-2}dzd\bar{z}<\infty\bigg{\\}}.$

Here $\lambda>1$ is a weight parameter. An orthonormal basis of
${\mathcal{H}}_{\lambda}$ is

$\bigg{\\{}e_{n}(z):=\sqrt{\frac{\Gamma(n+\lambda)}{n!\Gamma(\lambda)}}z^{n}:n\geq
0\bigg{\\}}.$

For a harmonic oscillator model based on ${\mathcal{H}}_{\lambda}$, see for
example Luo [10]. The degenerate case $\lambda=1$ corresponds to the Hardy
space in complex analysis

${\mathcal{H}}_{1}:=\bigg{\\{}f:{\mathbb{D}}\rightarrow{\mathbb{C}},\,\text{holomorphic},\,\langle
f,f\rangle=\lim_{r\rightarrow
1}\int_{0}^{2\pi}f(re^{i\theta})\overline{f(re^{i\theta})}\frac{d\theta}{2\pi}<\infty\bigg{\\}},$

with orthonormal (Taylor) basis $\\{z^{n}:n\geq 0\\}$. For further function
theoretic studies on ${\mathcal{H}}_{1}$ and
$\\{{\mathcal{H}}_{\lambda}\\}_{\lambda>1}$, see Garnett’s monograph [6] and
the aforementioned [7].

######  Example 1.1.

In above analytic representation, we have

$Nf(z)=z\frac{\partial}{\partial z}f(z),$

and we denote by $\Phi_{\lambda}$ and $\Phi_{\lambda}^{*}$ the corresponding
exponential phase operators. For $\lambda=1$ we encounter with the
backward/forward shifts (see Nikol’skiĭ’s treatise [13])

$\Phi_{1}f(z)=\frac{f(z)-f(0)}{z},$ $\Phi_{1}^{*}f(z)=zf(z).$

Hence, $\Phi_{1}N=N$ and the Leibniz rule

$N\Phi_{1}=N-\Phi_{1}$

leads to the commutation relation (1).

Let $f\in{\mathcal{H}}$ be a state with unit norm. For any operator $A$ (not
necessarily Hermitian) on ${\mathcal{H}}$, the expectation of $A$ in the state
$f$ is defined as

$\langle A\rangle=\langle A\rangle_{f}:=\langle f,Af\rangle.$

The variance of $A$ in $f$ is then defined as

$(\Delta A)^{2}=(\Delta_{f}A)^{2}:=\bigg{\langle}(A-\langle
A\rangle)(A-\langle A\rangle)^{*}\bigg{\rangle}.$

The minimum uncertainty states are the coherent states that minimise the
uncertainty relation under investigation. In this paper, we are interested in
the number-phase uncertainty relation and we aim to minimise the quantity
$(\Delta_{f}N)^{2}(\Delta_{f}\Phi_{\lambda})^{2}$.

######  Theorem 1.2.

Let $\lambda>1$. The minimum uncertainty states for the number-phase pair
$(N,\Phi)$ in ${\mathcal{H}}_{\lambda}({\mathbb{D}})$ can be parametrised as

$\displaystyle(w,k)\in{\mathbb{C}}\times{\mathbb{Z}}_{+}\mapsto
f_{w,k}(z)=cz^{k}\sum_{n=0}^{\infty}\frac{w^{n}}{n!}e_{n}(z).$ (2)

Here ${\mathbb{Z}}_{+}=\\{0,1,2,\cdots\\}$ and the normalisation constant
$c\in{\mathbb{C}}$ is determined by

$I_{k,\lambda}(|w|^{2})=(\bar{c}c)^{-1},$ (3)

where

$I_{k,\lambda}(t):=\sum_{n=0}^{\infty}\frac{t^{n}}{(n!)^{2}}\frac{(n+k)!}{n!}\frac{\Gamma(n+\lambda)}{\Gamma(n+k+\lambda)}.$

In particular,

$\langle
N\rangle_{f_{w,k}}-w\overline{\langle\Phi_{\lambda}\rangle_{f_{w,k}}}=k.$ (4)

Related results can be found in Carruthers and Nieto [3], Lerner, Huang and
Walters [8] and Luo [11]. For a nice survey exploring the number-phase
statistics via analytic functions, see Vourdas [15] (and also the papers [2,
16]).

##  2\. Proof of Theorem 1.2

Derivation of (2).—Using Cauchy-Schwarz and noting that $N$ is Hermitian,

$\displaystyle(\Delta N)^{2}(\Delta\Phi_{\lambda})^{2}$
$\displaystyle=\bigg{\langle}(N-\langle N\rangle)^{*}f,(N-\langle
N\rangle)^{*}f\bigg{\rangle}$
$\displaystyle\qquad\qquad\times\bigg{\langle}(\Phi_{\lambda}-\langle\Phi_{\lambda}\rangle)^{*}f,(\Phi_{\lambda}-\langle\Phi_{\lambda}\rangle)^{*}f\bigg{\rangle}$
$\displaystyle\geq\bigg{|}\bigg{\langle}(N-\langle
N\rangle)f,\left(\Phi^{*}_{\lambda}-\overline{\langle\Phi_{\lambda}\rangle}\right)f\bigg{\rangle}\bigg{|}^{2}.$

In using Cauchy-Schwarz, the equality holds iff there exists
$w\in{\mathbb{C}}$ such that

$(N-\langle
N\rangle)f=w\left(\Phi_{\lambda}^{*}-\overline{\langle\Phi_{\lambda}\rangle}\right)f,$

or by introducing $k=\langle
N\rangle-w\overline{\langle\Phi_{\lambda}\rangle}$,

$Nf=w\Phi_{\lambda}^{*}f+kf.$ (5)

We can solve the operational part of (5),

$Ng=w\Phi_{\lambda}^{*}g,$

by the (normalised) eigenstate expansion, and

$g(z)=g(0)\sum_{n=0}^{\infty}\frac{w^{n}}{n!}e_{n}(z).$

Thus, we solve (5) with

$f(z)=cz^{k}\sum_{n=0}^{\infty}\frac{w^{n}}{n!}e_{n}(z),$

where $c$ is a normalisation constant, and for $f\in{\mathcal{H}}_{\lambda}$
it is necessary that $k\in{\mathbb{Z}}_{+}$.

Derivation of (3).—Recall that $f$ has unit norm. Using
$\|e_{n+k}\|_{{\mathcal{H}}_{\lambda}}=1$ for all $n\geq 0$,

$\langle
f,f\rangle=\bar{c}c\sum_{n=0}^{\infty}\frac{(w\overline{w})^{n}}{(n!)^{2}}\frac{(n+k)!}{n!}\frac{\Gamma(n+\lambda)}{\Gamma(n+k+\lambda)}=1.$

This gives (3). For convenience, let

$G(n,k):=\frac{(n+k)!}{n!}\frac{\Gamma(n+\lambda)}{\Gamma(n+k+\lambda)}.$

Note that for $\lambda=1$, $G(n,k)\equiv 1$.

Derivation of (4).—We compute

$\displaystyle\langle N\rangle_{f_{w,k}}$
$\displaystyle=\bar{c}c\sum_{n=0}^{\infty}\frac{(w\overline{w})^{n}}{(n!)^{2}}G(n,k)(n+k)$
$\displaystyle=k+\bar{c}c\sum_{n=0}^{\infty}\frac{(w\overline{w})^{n+1}}{(n!)(n+1)!}G(n+1,k).$

Since
$w\overline{\langle\Phi_{\lambda}\rangle}=\overline{\langle\overline{w}\Phi_{\lambda}\rangle}$
and

$\displaystyle\overline{w}\Phi_{\lambda}f(z)$
$\displaystyle=c(\overline{w}w)\sum_{n=0}^{\infty}\frac{w^{n}}{(n+1)!}\sqrt{G(n+1,k)}e_{n+k}(z),$

we compute

$w\overline{\langle\Phi_{\lambda}\rangle}=\bar{c}c\sum_{n=0}^{\infty}\frac{(w\overline{w})^{n+1}}{n!(n+1)!}G(n+1,k).$

Thus, the consistency equation (4) is verified.

##  3\. Conclusion

We extended S. Luo’s number-phase uncertainty result [11] to the scale of
weighted Bergman spaces. His Hardy space result is the $\lambda\rightarrow 1$
limit of Theorem 1.2:

$f_{w,k}(z)\rightarrow
cz^{k}\sum_{n=0}^{\infty}\frac{w^{n}}{n!}z^{n}=cz^{k}e^{wz}.$

which are shifted Barut-Girardello states, see e.g. Brif [1]. The weighted
Bergman spaces are useful in harmonic analysis, functional inequalities and
quantum mechanical studies, see e.g. Luo [10, Sect. 4-5] and Frank [5]. It
would be (mathematically) interesting to consider the representation via other
classes of analytic functions.

## Declarations

Ethical Approval. Not applicable.

Declaration of competing interest. The author declares that he has no
competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

Availability of data and materials. Data sharing not applicable to this
article as no data sets were generated or analysed during the current study.

Funding. Research supported by National NSF grant of China (no. 11801274).

Acknowledgement. The author would like to thank Professors Min LI (HUST), Yuan
SUN (NJNU) and Haiyan XU (GDUT) for helpful communications.

## References

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#  $F$-manifold color algebras

Ming Ding, Zhiqi Chen and Jifu Li  Department of Mathematics and information
science, Guangzhou University, Guangzhou, P.R. China
<EMAIL_ADDRESS>School of Mathematical Sciences and LPMC,
Nankai University, Tianjin, P.R. China<EMAIL_ADDRESS>School of
Science, Tianjin University of technology, Tianjin, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce the notion of $F$-manifold color algebras and
study their properties which extend some results for $F$-manifold algebras.

###### Key words and phrases:

$F$-manifold color algebra, coherence $F$-manifold color algebra,
pre-$F$-manifold color algebra

##  1\. Introduction

The notion of Frobenius manifolds was invented by Dubrovin [10] in order to
give a geometrical expression of the Witten-Dijkgraaf-Verlinde-Verlinde
equations. In 1999, Hertling and Manin [12] introduced the concept of
$F$-manifolds as a relaxation of the conditions of Frobenius manifolds.
Inspired by the investigation of algebraic structures of $F$-manifolds, the
notion of an $F$-manifold algebra is given by Dotsenko [8] in 2019 to relate
the operad $F$-manifold algebras to the operad pre-Lie algebras. An
$F$-manifold algebra is defined as a triple $(A,\cdot,[,])$ satisfying the
following Hertling-Manin relation,

$P_{x\cdot y}(z,w)=x\cdot P_{y}(z,w)+y\cdot P_{x}(z,w),\ \ \ \forall
x,y,z,w\in A,$

where $(A,\cdot)$ is a commutative associative algebra, $(A,[,])$ is a Lie
algebra and $P_{x}(y,z)=[x,y\cdot z]-[x,y]\cdot z-y\cdot[x,z]$. A pre-Lie
algebra is a vector space $A$ with a bilinear multiplication $\cdot$
satisfying

$(x\cdot y)\cdot z-x\cdot(y\cdot z)=(y\cdot x)\cdot z-y\cdot(x\cdot z),\ \ \
\forall x,y,z\in A.$

Pre-Lie algebras have attracted a great deal of attentions in many areas of
mathematics and physics (see [1, 2, 4, 5, 9, 16] and so on).

Recently, Liu, Sheng and Bai [13] introduced the concept of pre-$F$-manifold
algebras which gives rise to $F$-manifold algebras. They also studied
representations of $F$-manifold algebras, and constructed many other examples
of $F$-manifold algebras.

In this paper, we introduce the notions of an $F$-manifold color algebra and a
pre-$F$-manifold color algebra which can be viewed as natural generalizations
of an $F$-manifold algebra and a pre-$F$-manifold algebra, and then extend
some properties of an $F$-manifold algebra in [13] to the color case.

The paper is organized as follows. In Section 2, we summarize some basic
concepts of Lie color algebras, pre-Lie color algebras, representations of
$\varepsilon$-commutative associative algebras and Lie color algebras. In
Section 3, we introduce the concept of an $F$-manifold color algebra and study
its representations. We also introduce the notion of a coherence $F$-manifold
color algebra and obtain that an $F$-manifold color algebra with a
nondegenerate symmetric bilinear form is a coherence $F$-manifold color
algebra. In Section 4, we introduce the concept of a pre-$F$-manifold color
algebra and show that a pre-$F$-manifold color algebra can naturally give rise
to an $F$-manifold color algebra.

Throughout this paper, we assume that $\mathbb{K}$ is an algebraically closed
field of characteristic zero and all the vector spaces are finite dimensional
over $\mathbb{K}$.

##  2\. Lie color algebras and relative algebraic structures

The concept of a Lie color algebra was introduced by Scheunert [17] which has
been widely studied. In this section, we collect some definitions and the
representations of Lie color algebras and some Lie color admissible algebras
such as pre-Lie color algebras and $\varepsilon$-commutative associative
algebras. One can refer to [3, 6, 7, 11, 14, 15, 17, 18, 19, 20] for more
details.

######  Definition 2.1.

Let $G$ be an abelian group. A map $\varepsilon:G\times
G\rightarrow\mathbb{K}\backslash\\{0\\}$ is called a skew-symmetric
bicharacter on $G$ if the following identities hold,

  1. (i)

$\varepsilon(a,b)\varepsilon(b,a)=1$,

  2. (ii)

$\varepsilon(a,b+c)=\varepsilon(a,b)\varepsilon(a,c)$,

  3. (iii)

$\varepsilon(a+b,c)=\varepsilon(a,c)\varepsilon(b,c)$,

for all $a,b,c\in G$.

By the definition, it is obvious that
$\varepsilon(a,0)=\varepsilon(0,a)=1,\varepsilon(a,a)=\pm 1\;\mbox{for
all}\;a\in G$.

######  Definition 2.2.

Let $G$ be an abelian group with bicharacter $\varepsilon$ as above. The
$G$-graded vector space

$A=\bigoplus_{g\in G}A_{g}$

is called a pre-Lie color algebra, if $A$ has a bilinear multiplication
operation $\cdot$ satisfying

  1. (1)

$A_{a}\cdot A_{b}\subseteq A_{a+b},$

  2. (2)

$(x\cdot y)\cdot z-x\cdot(y\cdot z)=\varepsilon(a,b)((y\cdot x)\cdot
z-y\cdot(x\cdot z)),$

for all $x\in A_{a},y\in A_{b},z\in A_{c}$, and $a,b,c\in G$.

######  Definition 2.3.

Let $G$ be an abelian group with bicharacter $\varepsilon$ as above. The
$G$-graded vector space

$A=\bigoplus_{g\in G}A_{g}$

is called a Lie color algebra, if $A$ has a bilinear multiplication
$[,]:A\times A\rightarrow A$ satisfying

  1. (1)

$[A_{a},A_{b}]\subseteq A_{a+b},$

  2. (2)

$[x,y]=-\varepsilon(a,b)[y,x],$

  3. (3)

$\varepsilon(c,a)[x,[y,z]]+\varepsilon(b,c)[z,[x,y]]+\varepsilon(a,b)[y,[z,x]]=0,$

for all $x\in A_{a},y\in A_{b},z\in A_{c}$, and $a,b,c\in G$.

######  Remark 2.4.

It is well known that a pre-Lie algebra $(A,\cdot)$ with a commutator
$[x,y]=x\cdot y-y\cdot x$ becomes a Lie algebra. Similarly, one has a pre-Lie
color algebra’s version, that is a pre-Lie color algebra
$(A,\cdot,\varepsilon)$ with a commutator $[x,y]=x\cdot
y-\varepsilon(x,y)y\cdot x$ becomes a Lie color algebra.

An element $x$ of the $G$-graded vector space $A$ is called homogeneous of
degree $a$ if $x\in A_{a}$. In the rest of this paper, for homogeneous
elements $x\in A_{a}$, $y\in A_{b}$, $z\in A_{c}$, we will write
$\varepsilon(x,y)$ instead of $\varepsilon(a,b)$, $\varepsilon(x+y,z)$ instead
of $\varepsilon(a+b,c)$, and so on. Furthermore, when the skew-symmetric
bicharacter $\varepsilon(x,y)$ appears, it means that $x$ and $y$ are both
homogeneous elements.

By an $\varepsilon$-commutative associative algebra $(A,\cdot,\varepsilon)$,
we mean that $(A,\cdot)$ is a $G$-graded associative algebra with the
following $\varepsilon$-commutativity

$x\cdot y=\varepsilon(x,y)y\cdot x$

for all homogeneous elements $x,y\in A$.

Let $V$ be a $G$-graded vector space. A representation $(V,\mu)$ of an
$\varepsilon$-commutative associative algebra $(A,\cdot,\varepsilon)$ is a
linear map $\mu:A\longrightarrow\mathfrak{gl}(V)$ satisfying

$\mu(x)v\in V_{a+b},\ \ \ \ \mu(x\cdot y)=\mu(x)\circ\mu(y)$

for all $v\in V_{a},x\in A_{b},y\in A_{c}.$ A representation $(V,\rho)$ of a
Lie color algebra $(A,[,],\varepsilon)$ is a linear map
$\rho:A\longrightarrow\mathfrak{gl}(V)$ satisfying

$\rho(x)v\in V_{a+b},\ \ \ \
\rho([x,y])=\rho(x)\circ\rho(y)-\varepsilon(x,y)\rho(y)\circ\rho(x)$

for all $v\in V_{a},x\in A_{b},y\in A_{c}.$

We consider the dual space $V^{*}$ of the $G$-graded vector space $V$. Then
$V^{*}=\bigoplus_{a\in G}V^{*}_{a}$ is also a $G$-graded space, where

$V^{*}_{a}=\\{\alpha\in V^{*}|\alpha(x)=0,b\neq-a,\forall x\in V_{b},b\in
G\\}.$

Define a linear map $\mu^{*}:A\longrightarrow\mathfrak{gl}(V^{*})$ by

$\mu^{*}(x)\alpha\in V^{*}_{a+c},\ \ \ \
\langle\mu^{*}(x)\alpha,v\rangle=-\varepsilon(x,\alpha)\langle\alpha,\mu(x)v\rangle$

for all $x\in A_{a},v\in V_{b},\alpha\in V^{*}_{c}.$

It is easy to see that

(1) If $(V,\mu)$ is a representation of an $\varepsilon$-commutative
associative algebra $(A,\cdot,\varepsilon)$, then $(V^{*},-\mu^{*})$ is a
representation of $(A,\cdot,\varepsilon)$;

(2) If $(V,\mu)$ is a representation of a Lie color algebra
$(A,[,],\varepsilon)$, then $(V^{*},\mu^{*})$ is a representation of
$(A,[,],\varepsilon)$.

##  3\. $F$-manifold color algebras and representations

In this section, we will introduce the notion of $F$-manifold color algebras,
and generalize some results in [13] to the color case.

######  Definition 3.1.

An $F$-manifold color algebra is a quadruple $(A,\cdot,[,],\varepsilon)$,
where $(A,\cdot,\varepsilon)$ is an $\varepsilon$-commutative associative
algebra and $(A,[,],\varepsilon)$ is a Lie color algebra, such that for all
homogeneous elements $x,y,z,w\in A$, the color Hertling-Manin relation holds:

(1) $P_{x\cdot y}(z,w)=x\cdot P_{y}(z,w)+\varepsilon(x,y)y\cdot P_{x}(z,w),$

where $P_{x}(y,z)$ is defined by

(2) $P_{x}(y,z)=[x,y\cdot z]-[x,y]\cdot z-\varepsilon(x,y)y\cdot[x,z].$

######  Remark 3.2.

When $G=\\{0\\}$ and $\varepsilon(0,0)=1$, then an $F$-manifold color algebra
is exactly an $F$-manifold algebra.

######  Definition 3.3.

Let $(A,\cdot,[,],\varepsilon)$ be an $F$-manifold color algebra. A
representation of $A$ is a triple $(V,\rho,\mu)$, such that $(V,\rho)$ is a
representation of the Lie color algebra $(A,[,],\varepsilon)$ and $(V,\mu)$ is
a representation of the $\varepsilon$-commutative associative algebra
$(A,\cdot,\varepsilon)$ satisfying

$\displaystyle R_{\rho,\mu}(x_{1}\cdot
x_{2},x_{3})=\mu(x_{1})R_{\rho,\mu}(x_{2},x_{3})+\varepsilon(x_{1},x_{2})\mu(x_{2})R_{\rho,\mu}(x_{1},x_{3}),$
$\displaystyle\mu(P_{x_{1}}(x_{2},x_{3}))=\varepsilon(x_{1},x_{2}+x_{3})S_{\rho,\mu}(x_{2},x_{3})\mu(x_{1})-\mu(x_{1})S_{\rho,\mu}(x_{2},x_{3}),$

where $R_{\rho,\mu},S_{\rho,\mu}:A\otimes A\rightarrow\mathfrak{gl}(V)$ are
defined by

(3) $\displaystyle R_{\rho,\mu}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\rho(x_{1})\mu(x_{2})-\varepsilon(x_{1},x_{2})\mu(x_{2})\rho(x_{1})-\mu([x_{1},x_{2}]),$
(4) $\displaystyle S_{\rho,\mu}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\mu(x_{1})\rho(x_{2})+\varepsilon(x_{1},x_{2})\mu(x_{2})\rho(x_{1})-\rho(x_{1}\cdot
x_{2})$

for all homogeneous elements $x_{1},x_{2},x_{3}\in A$.

In the rest of the paper, we sometimes denote $R_{\rho,\mu},S_{\rho,\mu}$ by
$R,S$ respectvely if there is no risk of confusion.

######  Example 3.4.

Let $(A,\cdot,[,],\varepsilon)$ be an $F$-manifold color algebra. Then
$(A,\mathrm{ad},\mathcal{L})$ is a representation of $A$, where
$\mathrm{ad}:A\longrightarrow\mathfrak{gl}(A)$ defined by
$\mathrm{ad}_{x}y=[x,y]$, and multiplication operator
$\mathcal{L}:A\longrightarrow\mathfrak{gl}(A)$ are defined by
$\mathcal{L}_{x}y=x\cdot y$ for all homogeneous elements $x,y\in A$.

###### Proof.

It is known that $(V,\mathrm{ad})$ is a representation of the Lie color
algebra $(A,[,],\varepsilon)$ and $(V,\mathcal{L})$ is a representation of the
$\varepsilon$-commutative associative algebra $(A,\cdot,\varepsilon)$.

Note that, for all homogeneous elements $x,y,z,w\in A$, we have

$\displaystyle R(x,y)z$ $\displaystyle=$
$\displaystyle(ad_{x}\mathcal{L}_{y}-\varepsilon(x,y)\mathcal{L}_{y}ad_{x}-\mathcal{L}_{[x,y]})\cdot
z$ $\displaystyle=$ $\displaystyle[x,y\cdot
z]-\varepsilon(x,y)y\cdot[x,y]-[x,y]\cdot z$ $\displaystyle=$ $\displaystyle
P_{x}(y,z).$

Thus

$\displaystyle P_{x\cdot y}(z,w)=x\cdot P_{y}(z,w)+\varepsilon(x,y)y\cdot
P_{x}(z,w)$

implies the equation

$\displaystyle R(x\cdot
y,z)w=\mathcal{L}_{x}R(y,z)w+\varepsilon(x,y)\mathcal{L}_{y}R(x,z)w.$

On the other hand

$\displaystyle
S(y,z)w=(\mathcal{L}_{y}ad_{z}+\varepsilon(y,z)\mathcal{L}_{z}ad_{y}-ad_{y\cdot
z})w$ $\displaystyle=$ $\displaystyle
y\cdot[z,w]+\varepsilon(y,z)z\cdot[y,w]-[y\cdot z,w]$ $\displaystyle=$
$\displaystyle-\varepsilon(z,w)y\cdot[w,z]-\varepsilon(y,w)\varepsilon(z,w)[w,y]\cdot
z+\varepsilon(y+z,w)[w,y\cdot z]$ $\displaystyle=$
$\displaystyle\varepsilon(y+z,w)([w,y\cdot z]-[w,y]\cdot
z-\varepsilon(w,y)y\cdot[w,z])$ $\displaystyle=$
$\displaystyle\varepsilon(y+z,w)P_{w}(y,z).$

Thus

$\displaystyle\varepsilon(x,y+z)S(y,z)\mathcal{L}_{x}w-\mathcal{L}_{x}S(y,z)w$
$\displaystyle=$ $\displaystyle\varepsilon(x,y+z)S(y,z)(x\cdot w)-x\cdot
S(y,z)w$ $\displaystyle=$
$\displaystyle\varepsilon(x,y+z)\varepsilon(y+z,x+w)P_{x\cdot
w}(y,z)-\varepsilon(y+z,w)x\cdot P_{w}(y,z)$ $\displaystyle=$
$\displaystyle\varepsilon(y+z,w)\\{P_{x\cdot w}(y,z)-x\cdot P_{w}(y,z)\\}$
$\displaystyle=$ $\displaystyle\varepsilon(y+z,w)\varepsilon(x,w)w\cdot
P_{x}(y,z)$ $\displaystyle=$ $\displaystyle\varepsilon(x+y+z,w)w\cdot
P_{x}(y,z)$ $\displaystyle=$ $\displaystyle P_{x}(y,z)w.$

Hence, $(A,\mathrm{ad},\mathcal{L})$ is a representation of $A$. ∎

Let $(A,\cdot,[,],\varepsilon)$ be an $F$-manifold color algebra and
$(V,\rho,\mu)$ a representation of $A$. Note that $A\oplus V$ is a G-graded
vector space. In the following, if we write $x+v\in A\oplus V$ as a
homogeneous element, it means that $x,v$ are of the same degree as $x+v$. For
any homogeneous elements $x_{1}+v_{1},x_{2}+v_{2}\in A\oplus V$, define

$[x_{1}+v_{1},x_{2}+v_{2}]_{\rho}=[x_{1},x_{2}]+\rho(x_{1})v_{2}-\varepsilon(x_{1},x_{2})\rho(x_{2})v_{1}.$

It is well known that $(A\oplus V,[,]_{\rho},\varepsilon)$ is a Lie color
algebra. Moreover, if one defines

$(x_{1}+v_{1})\cdot_{\mu}(x_{2}+v_{2})=x_{1}\cdot
x_{2}+\mu(x_{1})v_{2}+\varepsilon(x_{1},x_{2})\mu(x_{2})v_{1},$

then $(A\oplus V,\cdot_{\mu},\varepsilon)$ is an $\varepsilon$-commutative
associative algebra. In fact, we have

######  Proposition 3.5.

With the above notations, let $(A,\cdot,[,],\varepsilon)$ be an $F$-manifold
color algebra and $(V,\rho,\mu)$ a representation of $A$. Then $(A\oplus
V,\cdot_{\mu},[,]_{\rho},\varepsilon)$ is an $F$-manifold color algebra.

###### Proof.

We only need to prove that the color Hertling-Manin relation holds.

For any homogeneous elements $x_{1}+v_{1},x_{2}+v_{2},x_{3}+v_{3}\in A\oplus
V$, we have

$\displaystyle P_{x_{1}+v_{1}}(x_{2}+v_{2},x_{3}+v_{3})$ $\displaystyle=$
$\displaystyle[x_{1}+v_{1},(x_{2}+v_{2})\cdot_{\mu}(x_{3}+v_{3})]_{\rho}-[x_{1}+v_{1},x_{2}+v_{2}]_{\rho}\cdot_{\mu}(x_{3}+v_{3})$
$\displaystyle-\varepsilon(x_{1},x_{2})(x_{2}+v_{2})\cdot_{\mu}[x_{1}+v_{1},x_{3}+v_{3}]_{\rho}$
$\displaystyle=$ $\displaystyle[x_{1},x_{2}\cdot
x_{3}]+\rho(x_{1})\\{\mu(x_{2})v_{3}+\varepsilon(x_{2},x_{3})\mu(x_{3})v_{2}\\}-\varepsilon(x_{1},x_{2}+x_{3})\rho(x_{2}\cdot
x_{3})v_{1}-I-II.$

where

$\displaystyle I$ $\displaystyle=$
$\displaystyle\\{[x_{1},x_{2}]+\rho(x_{1})v_{3}-\varepsilon(x_{1},x_{2})\rho(x_{2})v_{1}\\}\cdot_{\mu}(x_{3}+v_{3})$
$\displaystyle=$ $\displaystyle[x_{1},x_{2}]\cdot
x_{3}+\mu([x_{1},x_{2}])v_{3}+\varepsilon(x_{1}+x_{2},x_{3})\mu(x_{3})\\{\rho(x_{1})v_{2}-\varepsilon(x_{1},x_{2})\rho(x_{2})v_{1}\\},$

and

$\displaystyle II$ $\displaystyle=$
$\displaystyle\varepsilon(x_{1},x_{2})(x_{2}+v_{2})\cdot_{\mu}\\{[x_{1},x_{3}]+\rho(x_{1})v_{3}-\varepsilon(x_{1},x_{3})\rho(x_{3})v_{1}\\}$
$\displaystyle=$
$\displaystyle\varepsilon(x_{1},x_{2})\\{x_{2}\cdot[x_{1},x_{3}]+\mu(x_{2})(\rho(x_{1})v_{3}-\varepsilon(x_{1},x_{3})\rho(x_{3})v_{1})$
$\displaystyle+\varepsilon(x_{2},x_{1}+x_{3})\mu([x_{1},x_{3}])v_{2}\\}.$

Thus

$\displaystyle P_{x_{1}+v_{1}}(x_{2}+v_{2},x_{3}+v_{3})$ $\displaystyle=$
$\displaystyle
P_{x_{1}}(x_{2},x_{3})+\\{\rho(x_{1})\mu(x_{2})-\mu([x_{1},x_{2}])-\varepsilon(x_{1},x_{2})\mu(x_{2})\rho(x_{1})\\}v_{3}$
$\displaystyle+\\{\varepsilon(x_{2},x_{3})\rho(x_{1})\mu(x_{3})-\varepsilon(x_{1}+x_{2},x_{3})\mu(x_{3})\rho(x_{1})$
$\displaystyle-\varepsilon(x_{1},x_{2})\varepsilon(x_{2},x_{1}+x_{3})\mu([x_{1},x_{3}])\\}v_{2}+\\{-\varepsilon(x_{1},x_{2}+x_{3})\rho(x_{2}\cdot
x_{3})$
$\displaystyle+\varepsilon(x_{1}+x_{2},x_{3})\varepsilon(x_{1},x_{2})\mu(x_{3})\rho(x_{2})+\varepsilon(x_{1},x_{2})\varepsilon(x_{1},x_{3})\mu(x_{2})\rho(x_{3})\\}v_{1}$
$\displaystyle=$ $\displaystyle
P_{x_{1}}(x_{2},x_{3})+R(x_{1},x_{2})v_{3}+\varepsilon(x_{2},x_{3})R(x_{1},x_{3})v_{2}+\varepsilon(x_{1},x_{2}+x_{3})S(x_{2},x_{3})v_{1}.$

Hence, for any homogeneous element $x_{4}+v_{4}\in A\oplus V$, we have

$\displaystyle
P_{(x_{1}+v_{1})\cdot_{\mu}(x_{2}+v_{2})}(x_{3}+v_{3},x_{4}+v_{4})$
$\displaystyle=$ $\displaystyle P_{x_{1}\cdot
x_{2}+\mu(x_{1})v_{2}+\varepsilon(x_{1},x_{2})\mu(x_{2})v_{1}}(x_{3}+v_{3},x_{4}+v_{4})$
$\displaystyle=$ $\displaystyle P_{x_{1}\cdot x_{2}}(x_{3},x_{4})+R(x_{1}\cdot
x_{2},x_{3})v_{4}+\varepsilon(x_{3},x_{4})R(x_{1}\cdot x_{2},x_{4})v_{3}$
$\displaystyle+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})S(x_{3},x_{4})(\mu(x_{1})v_{2}+\varepsilon(x_{1},x_{2})\mu(x_{2})v_{1}).$

On the other hand

$\displaystyle(x_{1}+v_{1})\cdot_{\mu}P_{x_{2}+v_{2}}(x_{3}+v_{3},x_{4}+v_{4})$
$\displaystyle=$
$\displaystyle(x_{1}+v_{1})\cdot_{\mu}\\{P_{x_{2}}(x_{3},x_{4})+R(x_{2},x_{3})v_{4}+\varepsilon(x_{3},x_{4})R(x_{2},x_{4})v_{3}+\varepsilon(x_{2},x_{3}+x_{4})S(x_{3},x_{4})v_{2}\\}$
$\displaystyle=$ $\displaystyle x_{1}\cdot
P_{x_{2}}(x_{3},x_{4})+\mu(x_{1})\\{R(x_{2},x_{3})v_{4}+\varepsilon(x_{3},x_{4})R(x_{2},x_{4})v_{3}+\varepsilon(x_{2},x_{3}+x_{4})S(x_{3},x_{4})v_{2}\\}$
$\displaystyle+\varepsilon(x_{1},x_{2}+x_{3}+x_{4})\mu(P_{x_{2}}(x_{3},x_{4}))v_{1},$

and

$\displaystyle\varepsilon(x_{1},x_{2})(x_{2}+v_{2})\cdot_{\mu}P_{x_{1}+v_{1}}(x_{3}+v_{3},x_{4}+v_{4})$
$\displaystyle=$ $\displaystyle\varepsilon(x_{1},x_{2})\\{x_{2}\cdot
P_{x_{1}}(x_{3},x_{4})+\mu(x_{2})\\{R(x_{1},x_{3})v_{4}+\varepsilon(x_{3},x_{4})R(x_{1},x_{4})v_{3}$
$\displaystyle+\varepsilon(x_{1},x_{3}+x_{4})S(x_{3},x_{4})v_{1}\\}+\varepsilon(x_{2},x_{1}+x_{3}+x_{4})\mu(P_{x_{1}}(x_{3},x_{4}))v_{2}\\}.$

Thus

$\displaystyle(x_{1}+v_{1})\cdot_{\mu}P_{x_{2}+v_{2}}(x_{3}+v_{3},x_{4}+v_{4})+\varepsilon(x_{1},x_{2})(x_{2}+v_{2})\cdot_{\mu}P_{x_{1}+v_{1}}(x_{3}+v_{3},x_{4}+v_{4})$
$\displaystyle=$ $\displaystyle x_{1}\cdot
P_{x_{2}}(x_{3},x_{4})+\varepsilon(x_{1},x_{2})x_{2}\cdot
P_{x_{1}}(x_{3},x_{4})$
$\displaystyle+\\{\mu(x_{1})R(x_{2},x_{3})+\varepsilon(x_{1},x_{2})\mu(x_{2})(R(x_{1},x_{3}))\\}v_{4}$
$\displaystyle+\\{\varepsilon(x_{3},x_{4})\mu(x_{1})R(x_{2},x_{4})+\varepsilon(x_{1},x_{2})\varepsilon(x_{3},x_{4})\mu(x_{2})R(x_{1},x_{4})\\}v_{3}$
$\displaystyle+\\{\varepsilon(x_{2},x_{3}+x_{4})\mu(x_{1})S(x_{3},x_{4})+\varepsilon(x_{1},x_{2})\varepsilon(x_{2},x_{1}+x_{3}+x_{4})\mu(P_{x_{1}}(x_{3},x_{4}))\\}v_{2}$
$\displaystyle+\varepsilon(x_{1},x_{2}+x_{3}+x_{4})\\{\mu(x_{2})S(x_{3},x_{4})+\mu(P_{x_{2}}(x_{3},x_{4}))\\}v_{1}$
$\displaystyle=$ $\displaystyle
P_{(x_{1}+v_{1})\cdot_{\mu}(x_{2}+v_{2})}(x_{3}+v_{3},x_{4}+v_{4}).$

By the definition of an $F$-manifold color algebra, the conclusion follows
immediately. ∎

Let $(V,\rho,\mu)$ be a representation of an $F$-manifold algebra, then the
triple $(V^{*},\rho^{*},-\mu^{*})$ is not necessarily a representation of this
$F$-manifold algebra [13]. In the following, we prove the version of an
$F$-manifold color algebra.

######  Proposition 3.6.

Let $(A,\cdot,[,],\varepsilon)$ be an $F$-manifold color algebra, $(V,\rho)$ a
representation of the Lie color algebra $(A,[,],\varepsilon)$, and $(V,\mu)$ a
representation of the $\varepsilon$-commutative associative algebra
$(A,\cdot,\varepsilon)$, such that, for any homogeneous elements $x,y,z\in A$,

$\displaystyle R_{\rho,\mu}(x\cdot
y,z)=\varepsilon(x,y+z)R_{\rho,\mu}(y,z)\mu(x)+\varepsilon(y,z)R_{\rho,\mu}(x,z)\mu(y),$
$\displaystyle\mu(P_{x}(y,z))=-\varepsilon(x,y+z)T_{\rho,\mu}(y,z)\mu(x)+\mu(x)T_{\rho,\mu}(y,z),$

where $R_{\rho,\mu}$ is given by (3) and $T_{\rho,\mu}:A\otimes
A\rightarrow\mathfrak{gl}(V)$ is defined by

$\displaystyle T_{\rho,\mu}(x,y)$ $\displaystyle=$
$\displaystyle-\varepsilon(x,y)\rho(y)\mu(x)-\rho(x)\mu(y)+\rho(x\cdot y),$

then $(V^{*},\rho^{*},-\mu^{*})$ is a representation of $A$.

###### Proof.

Assume that $x,y,z\in A,v\in V,\alpha\in V^{*}$ are all homogeneous elements.
First, we claim the following two identities:

$\displaystyle\langle R_{\rho^{*},-\mu^{*}}(x,y)(\alpha),v\rangle$
$\displaystyle=$
$\displaystyle\langle\alpha,\varepsilon(x+y,\alpha)R_{\rho,\mu}(x,y)v\rangle;$
$\displaystyle\langle S_{\rho^{*},-\mu^{*}}(x,y)(\alpha),v\rangle$
$\displaystyle=$
$\displaystyle\langle\alpha,\varepsilon(x+y,\alpha)T_{\rho,\mu}(x,y)v\rangle.$

The claims follow from some direct calculations, respectively:

$\displaystyle\langle R_{\rho^{*},-\mu^{*}}(x,y)(\alpha),v\rangle$
$\displaystyle=$
$\displaystyle\langle(-\rho^{*}(x)\mu^{*}(y)+\varepsilon(x,y)\mu^{*}(y)\rho^{*}(x)+\mu^{*}([x,y])\alpha,v\rangle$
$\displaystyle=$
$\displaystyle\varepsilon(x,y+\alpha)\langle\mu^{*}(y)\alpha,\rho(x)v\rangle-\varepsilon(x,y)\varepsilon(y,x+\alpha)\langle(\rho^{*}(x)\alpha,\mu(y)v\rangle$
$\displaystyle-\varepsilon(x+y,\alpha)\langle\alpha,\mu([x,y])v\rangle$
$\displaystyle=$
$\displaystyle-\varepsilon(x,y)\varepsilon(x+y,\alpha)\langle\alpha,\mu(y)\rho(x)v\rangle+\varepsilon(y,\alpha)\varepsilon(x,\alpha)\langle\alpha,\rho(x)\mu(y)v\rangle$
$\displaystyle-\varepsilon(x+y,\alpha)\langle\alpha,\mu([x,y])v\rangle$
$\displaystyle=$
$\displaystyle\langle\alpha,\varepsilon(x+y,\alpha)\\{-\varepsilon(x,y)\mu(y)\rho(x)+\rho(x)\mu(y)-\mu([x,y])\\}v\rangle$
$\displaystyle=$
$\displaystyle\langle\alpha,\varepsilon(x+y,\alpha)R_{\rho,\mu}(x,y)v\rangle,$

and

$\displaystyle\langle S_{\rho^{*},-\mu^{*}}(x,y)(\alpha),v\rangle$
$\displaystyle=$
$\displaystyle\langle\\{-\mu^{*}(x)\rho^{*}(y)-\varepsilon(x,y)\mu^{*}(y)\rho^{*}(x)-\rho^{*}(x\cdot
y)\\}\alpha,v\rangle$ $\displaystyle=$
$\displaystyle-\varepsilon(x,y+\alpha)\varepsilon(y,\alpha)\langle\alpha,\rho(y)\mu(x)v\rangle-\varepsilon(y,\alpha)\varepsilon(x,\alpha)\langle\alpha,\rho(x)\mu(y)v\rangle$
$\displaystyle+\varepsilon(x+y,\alpha)\langle\alpha,\rho(x\cdot y)v\rangle$
$\displaystyle=$
$\displaystyle\langle\alpha,\varepsilon(x+y,\alpha)\\{-\varepsilon(x,y)\rho(y)\mu(x)-\rho(x)\mu(y)+\rho(x\cdot
y)\\}v\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha,\varepsilon(x+y,\alpha)T_{\rho,\mu}(x,y)v\rangle.$

With the above identities, we have

$\displaystyle\langle\\{R_{\rho^{*},-\mu^{*}}(x\cdot
y,z)+\mu^{*}(x)R_{\rho^{*},-\mu^{*}}(y,z)+\varepsilon(x,y)\mu^{*}(y)R_{\rho^{*},-\mu^{*}}(x,z)\\}\alpha,v\rangle$
$\displaystyle=$
$\displaystyle\langle\alpha,\varepsilon(x+y+z,\alpha)R_{\rho,\mu}(x\cdot
y,z)v\rangle-\varepsilon(x,y+z+\alpha)\varepsilon(y+z,\alpha)\langle\alpha,R_{\rho,\mu}(y,z)\mu(x)v\rangle$
$\displaystyle-\varepsilon(x+z,\alpha)\varepsilon(y,z+\alpha)\langle\alpha,R_{\rho,\mu}(x,z)\mu(y)v\rangle$
$\displaystyle=$
$\displaystyle\varepsilon(x+y+z,\alpha)\langle\alpha,\\{R_{\rho,\mu}(x\cdot
y,z)-\varepsilon(x,y+z)R_{\rho,\mu}(y,z)\mu(x)-\varepsilon(y,z)R_{\rho,\mu}(x,z)\mu(y)\\}v\rangle$
$\displaystyle=$ $\displaystyle 0.$

And

$\displaystyle\langle\\{-\mu^{*}(P_{x}(y,z))+\varepsilon(x,y+z)S_{\rho^{*},-\mu^{*}}(y,z)\mu^{*}(x)-\mu^{*}(x)S_{\rho^{*},-\mu^{*}}(y,z)\\}\alpha,v\rangle$
$\displaystyle=$
$\displaystyle\varepsilon(x+y+z,\alpha)\langle\alpha,\mu(P_{x}(y,z))v\rangle+\varepsilon(x,y+z)\varepsilon(y+z,x+\alpha)\langle\mu^{*}(x)\alpha,T_{\rho,\mu}(y,z)v\rangle$
$\displaystyle+\varepsilon(x,y+z+\alpha)\langle
S_{\rho^{*},-\mu^{*}}(y,z)\alpha,\mu(x)v\rangle$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,\alpha)\langle\alpha,\mu(P_{x}(y,z))v\rangle-\varepsilon(y+z,\alpha)\varepsilon(x,\alpha)\langle\alpha,\mu(x)T_{\rho,\mu}(y,z)v\rangle$
$\displaystyle+\varepsilon(x,y+z+\alpha)\varepsilon(y+z,\alpha)\langle\alpha,T_{\rho,\mu}(y,z)\mu(x)v\rangle$
$\displaystyle=$
$\displaystyle\varepsilon(x+y+z,\alpha)\langle\alpha,\\{\mu(P_{x}(y,z))-\mu(x)T_{\rho,\mu}(y,z)+\varepsilon(x,y+z)T_{\rho,\mu}(y,z)\mu(x)\\}v\rangle$
$\displaystyle=$ $\displaystyle 0.$

Thus, the conclusion follows immediately from the definition of
representations and the hypothesis. ∎

######  Definition 3.7.

A coherence $F$-manifold color algebra is an $F$-manifold color algebra such
that, for all homogeneous elements $x,y,z,w\in A$, the following hold

$\displaystyle P_{x\cdot y}(z,w)$ $\displaystyle=$
$\displaystyle\varepsilon(x,y+z)P_{y}(z,x\cdot
w)+\varepsilon(y,z)P_{x}(z,y\cdot w),$ $\displaystyle P_{x}(y,z)w$
$\displaystyle=$ $\displaystyle-\varepsilon(x,y+z)T(y,z)(x\cdot
w)+xT(y,z)(w),$

where

$T(y,z)(w)=-\varepsilon(y,z)[z,y\cdot w]-[y,z\cdot w]+[y\cdot z,w].$

######  Proposition 3.8.

Let $(A,\cdot,[,],\varepsilon)$ be an $F$-manifold color algebra, and
$\mathfrak{B}$ a nondegenerate symmetric bilinear form on $A$ satisfying

$\mathfrak{B}(x\cdot y,z)=\mathfrak{B}(x,y\cdot
z),\;\;\mathfrak{B}([x,y],z)=\mathfrak{B}(x,[y,z]),$

for all homogeneous elements $x,y,z\in A$, then $(A,\cdot,[,],\varepsilon)$ is
a coherence $F$-manifold color algebra.

###### Proof.

First, we prove that

$\mathfrak{B}(P_{x}(y,z),w)=\varepsilon(x+y,z)\mathfrak{B}(z,P_{x}(y,w))$

for all homogeneous elements $x,y,z,w\in A$.

In fact, we have

$\displaystyle\mathfrak{B}(P_{x}(y,z),w)$ $\displaystyle=$
$\displaystyle\mathfrak{B}([x,y\cdot z]-[x,y]\cdot
z-\varepsilon(x,y)y\cdot[x,z],w)$ $\displaystyle=$
$\displaystyle-\varepsilon(x,y+z)\mathfrak{B}([y\cdot
z,x],w)-\varepsilon(x+y,z)\mathfrak{B}(z,[x,y]\cdot
w)-\varepsilon(x,y)\varepsilon(y,x+z)\mathfrak{B}([x,z],y\cdot w)$
$\displaystyle=$ $\displaystyle-\varepsilon(x,y+z)\mathfrak{B}(y\cdot
z,[x,w])-\varepsilon(x+y,z)\mathfrak{B}(z,[x,y]\cdot
w)+\varepsilon(y,z)\varepsilon(x,z)\mathfrak{B}(z,[x,y\cdot w])$
$\displaystyle=$
$\displaystyle-\varepsilon(x,y+z)\varepsilon(y,z)\mathfrak{B}(z,y\cdot[x,w])-\varepsilon(x+y,z)\mathfrak{B}(z,[x,y]\cdot
w)+\varepsilon(x+y,z)\mathfrak{B}(z,[x,y\cdot w])$ $\displaystyle=$
$\displaystyle\varepsilon(x+y,z)\mathfrak{B}(z,-\varepsilon(x,y)y\cdot[x,w]-[x,y]\cdot
w+[x,y\cdot w])$ $\displaystyle=$
$\displaystyle\varepsilon(x+y,z)\mathfrak{B}(z,P_{x}(y,w)).$

By the above relation, for all homogeneous elements $x,y,z,w_{1},w_{2}\in A$,
we have

$\displaystyle\mathfrak{B}(P_{x\cdot
y}(z,w_{1})-\varepsilon(x,y+z)P_{y}(z,x\cdot
w_{1})-\varepsilon(y,z)P_{x}(z,y\cdot w_{1}),w_{2})$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x\cdot
y}(z,w_{2}))-\varepsilon(x,y+z)\varepsilon(y+z,x+w_{1})\mathfrak{B}(x\cdot
w_{1},P_{y}(z,w_{2}))$ $\displaystyle-\varepsilon(y,z)\varepsilon(x\cdot
z,y\cdot w_{1})\mathfrak{B}(y\cdot w_{1},P_{x}(z,w_{2}))$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x\cdot
y}(z,w_{2}))-\varepsilon(x,y+z)\varepsilon(y\cdot z,x\cdot
w_{1})\varepsilon(x,w_{1})\mathfrak{B}(w_{1},x\cdot P_{y}(z,w_{2}))$
$\displaystyle-\varepsilon(y,z)\varepsilon(x\cdot z,y\cdot
w_{1})\varepsilon(y,w_{1})\mathfrak{B}(w_{1},y\cdot P_{x}(z,w_{2}))$
$\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x\cdot
y}(z,w_{2}))-\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},x\cdot
P_{y}(z,w_{2}))$
$\displaystyle-\varepsilon(y,z)\varepsilon(x+z,y)\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},y\cdot
P_{x}(z,w_{2}))$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x\cdot
y}(z,w_{2}))-\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},x\cdot
P_{y}(z,w_{2}))$
$\displaystyle-\varepsilon(x,y)\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},y\cdot
P_{x}(z,w_{2}))$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x\cdot
y}(z,w_{2})-x\cdot P_{y}(z,w_{2})-\varepsilon(x,y)y\cdot P_{x}(z,w_{2}))$
$\displaystyle=$ $\displaystyle 0.$

We claim the following identity:

$\mathfrak{B}(T(y,z)(x\cdot
w_{1}),w_{2})=\varepsilon(y+z,w_{1}+w_{2})\mathfrak{B}(w_{1},P_{w_{2}}(y,z)).$

In fact, we have

$\displaystyle\mathfrak{B}(T(y,z)(x\cdot w_{1}),w_{2})$ $\displaystyle=$
$\displaystyle\mathfrak{B}(-\varepsilon(y,z)[z,y\cdot w_{1}]-[y,z\cdot
w_{1}]+[y\cdot z,w_{1}],w_{2})$ $\displaystyle=$
$\displaystyle\varepsilon(y,z)\varepsilon(z,y+w_{1})\mathfrak{B}(y\cdot
w_{1},[z,w_{2}])+\varepsilon(y,z+w_{1})\mathfrak{B}(z\cdot
w_{1},[y,w_{2}])-\varepsilon(y+z,w_{1})\mathfrak{B}(w_{1},[yz,w_{2}])$
$\displaystyle=$
$\displaystyle\varepsilon(z,w_{1})\varepsilon(y,w_{1})\mathfrak{B}(w_{1},y\cdot[z,w_{2}])+\varepsilon(y,z+w_{1})\varepsilon(z,w_{1})\mathfrak{B}(w_{1},z\cdot[y,w_{2}])$
$\displaystyle-\varepsilon(y+z,w_{1})\mathfrak{B}(w_{1},[y\cdot z,w_{2}])$
$\displaystyle=$
$\displaystyle\varepsilon(y+z,w_{1})\mathfrak{B}(w_{1},y\cdot[z,w_{2}])+\varepsilon(y+z,w_{1})\varepsilon(y,z)\mathfrak{B}(w_{1},z\cdot[y,w_{2}])-\varepsilon(y+z,w_{1})\mathfrak{B}(w_{1},[y\cdot
z,w_{2}])$ $\displaystyle=$
$\displaystyle\varepsilon(y+z,w_{1})\mathfrak{B}(w_{1},y\cdot[z,w_{2}]+\varepsilon(y,z)z\cdot[y,w_{2}]-[y\cdot
z,w_{2}])$ $\displaystyle=$
$\displaystyle\varepsilon(y+z,w_{1})\mathfrak{B}(w_{1},\varepsilon(y+z,w_{2})P_{w_{2}}(y,z))$
$\displaystyle=$
$\displaystyle\varepsilon(y+z,w_{1}+w_{2})\mathfrak{B}(w_{1},P_{w_{2}}(y,z)).$

With the above identity, we have

$\displaystyle\mathfrak{B}(P_{x}(y,z)\cdot
w_{1}+\varepsilon(x,y+z)T(y,z)(x\cdot w_{1})-x\cdot T(y,z)(w_{1}),w_{2})$
$\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x}(y,z)w_{2})+\varepsilon(x,y+z)\varepsilon(y+z,x+w_{1}+w_{2})\mathfrak{B}(x\cdot
w_{1},P_{w_{2}}(y,z))$
$\displaystyle-\varepsilon(x,y+z+w_{1})\mathfrak{B}(T(y,z)w_{1},x\cdot w_{2})$
$\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x}(y,z)w_{2})+\varepsilon(x,w_{1})\varepsilon(y+z,w_{1}+w_{2})\mathfrak{B}(w_{1},x\cdot
P_{w_{2}}(y,z))$
$\displaystyle-\varepsilon(x,y+z+w_{1})\varepsilon(y+z,x+w_{1}+w_{2})\mathfrak{B}(w_{1},P_{x\cdot
w_{2}}(y,z))$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x}(y,z)w_{2})+\varepsilon(x,w_{1})\varepsilon(y+z,w_{1}+w_{2})\mathfrak{B}(w_{1},x\cdot
P_{w_{2}}(y,z))$
$\displaystyle-\varepsilon(x,w_{1})\varepsilon(y+z,w_{1}+w_{2})\mathfrak{B}(w_{1},P_{x\cdot
w_{2}}(y,z))$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x}(y,z)w_{2}+\varepsilon(y+z,w_{2})x\cdot
P_{w_{2}}(y,z)-\varepsilon(y+z,w_{2})P_{x\cdot w_{2}}(y,z))$ $\displaystyle=$
$\displaystyle\varepsilon(x+y+z,w_{1})\mathfrak{B}(w_{1},P_{x}(y,z)w_{2}+\varepsilon(y+z,w_{2})x\cdot
P_{w_{2}}(y,z)$ $\displaystyle-(P_{x}(y,z)w_{2}+\varepsilon(y+z,w_{2})x\cdot
P_{w_{2}}(y,z)))$ $\displaystyle=$ $\displaystyle 0.$

By the assumptions that $A$ is an $F$-manifold color algebra and
$\mathfrak{B}$ is nondegenerate, the conclusion is obtained. ∎

##  4\. Pre-$F$-manifold color algebras

In this section, we introduce the notion of a pre-$F$-manifold color algebra,
and then construct an $F$-manifold color algebra from a pre-$F$-manifold color
algebra.

######  Definition 4.1.

A triple $(A,\diamond,\varepsilon)$ is called a Zinbiel color algebra if $A$
is a G-graded vector space and $\diamond:A\otimes A\longrightarrow A$ is a
bilinear multiplication satisfying

  1. (1)

$A_{a}\diamond A_{b}\subseteq A_{a+b},$

  2. (2)

$x\diamond(y\diamond z)=(x\diamond y)\diamond z+\varepsilon(x,y)(y\diamond
x)\diamond z,$

for all $x\in A_{a},y\in A_{b},z\in A_{c}$, and $a,b,c\in G$.

Let $(A,\diamond,\varepsilon)$ be a Zinbiel color algebra. For any homogeneous
elements $x,y\in A$, define

$x\cdot y=x\diamond y+\varepsilon(x,y)y\diamond x,$

and define $\mathfrak{L}:A\longrightarrow\mathfrak{gl}(A)$ by

(5) $\mathfrak{L}_{x}y=x\diamond y.$

######  Lemma 4.2.

With the above notations, we have that $(A,\cdot,\varepsilon)$ is an
$\varepsilon$-commutative associative algebra, and $(A,\mathfrak{L})$ is its
representation.

###### Proof.

First, for any homogeneous elements $x,y\in A$, we have

$\displaystyle\varepsilon(x,y)y\cdot x=\varepsilon(x,y)(y\diamond
x+\varepsilon(y,x)x\diamond y)=x\diamond y+\varepsilon(x,y)y\diamond x=x\cdot
y.$

Thus, the $\varepsilon$-commutativity follows.

Secondly, for any homogeneous elements $x,y,z\in A$, we have

$\displaystyle(x\cdot y)\cdot z=(x\diamond y+\varepsilon(x,y)y\diamond x)\cdot
z$ $\displaystyle=$ $\displaystyle(x\diamond y)\diamond
z+\varepsilon(x+y,z)z\diamond(x\diamond y)+\varepsilon(x,y)\\{(y\diamond
x)\diamond z+\varepsilon(x+y,z)z\diamond(y\diamond x)\\}$ $\displaystyle=$
$\displaystyle(x\diamond y)\diamond z+\varepsilon(x,y)(y\diamond x)\diamond
z+\varepsilon(x,y)\varepsilon(x+y,z)z\diamond(y\diamond
x)+\varepsilon(x+y,z)z\diamond(x\diamond y)$ $\displaystyle=$
$\displaystyle(x\diamond y)\diamond z+\varepsilon(x,y)(y\diamond x)\diamond
z+\varepsilon(x,y)\varepsilon(x+y,z)\\{(z\diamond y)\diamond
x+\varepsilon(z,y)(y\diamond z)\diamond x\\}$
$\displaystyle+\varepsilon(x+y,z)\\{(z\diamond x)\diamond
y+\varepsilon(z,x)(x\diamond z)\diamond y\\}$ $\displaystyle=$
$\displaystyle\\{(x\diamond y)\diamond z+\varepsilon(x,y)(y\diamond x)\diamond
z\\}+\\{\varepsilon(x,y+z)(y\diamond z)\diamond
x+\varepsilon(x,y+z)\varepsilon(y,z)(z\diamond y)\diamond x\\}$
$\displaystyle+\varepsilon(y,z)\\{(x\diamond z)\diamond
y+\varepsilon(x,z)(z\diamond x)\diamond y\\}$ $\displaystyle=$ $\displaystyle
x\diamond(y\diamond z)+\varepsilon(x,y+z)(y\diamond z)\diamond
x+\varepsilon(y,z)\varepsilon(x,y+z)(z\diamond y)\diamond
x+\varepsilon(y,z)x\diamond(z\diamond y),$

and

$\displaystyle x\cdot(y\cdot z)=x\cdot(y\diamond z+\varepsilon(y,z)z\diamond
y)$ $\displaystyle=$ $\displaystyle x\diamond(y\diamond
z)+\varepsilon(x,y+z)(y\diamond z)\diamond
x+\varepsilon(y,z)\\{x\diamond(z\diamond y)+\varepsilon(x,y+z)(z\diamond
y)\diamond x\\}$ $\displaystyle=$ $\displaystyle x\diamond(y\diamond
z)+\varepsilon(x,y+z)(y\diamond z)\diamond
x+\varepsilon(y,z)\varepsilon(x,y+z)(z\diamond y)\diamond
x+\varepsilon(y,z)x\diamond(z\diamond y).$

Thus, the associativity follows.

From the definition of $\mathfrak{L}$, it follows that

$\mathfrak{L}_{xy}z=(x\cdot y)\diamond z=(x\diamond
y+\varepsilon(x,y)(y\diamond x))\diamond z=x\diamond(y\diamond
z)=\mathfrak{L}_{x}\mathfrak{L}_{y}(z).$

Hence, $(A,\mathfrak{L})$ is a representation of the $\varepsilon$-commutative
associative algebra $(A,\cdot,\varepsilon)$. ∎

######  Definition 4.3.

$(A,\diamond,\ast,\varepsilon)$ is called a pre-$F$-manifold color algebra if
$(A,\diamond,\varepsilon)$ is a Zinbiel color algebra and
$(A,\ast,\varepsilon)$ is a pre-Lie color algebra, such that, for all
homogeneous elements $x,y,z,w\in A$,

$F_{1}(x\cdot y,z,w)=x\diamond F_{1}(y,z,w)+\varepsilon(x,y)y\diamond
F_{1}(x,z,w),$
$\displaystyle(F_{1}(x,y,z)+\varepsilon(y,z)F_{1}(x,z,y)+\varepsilon(x,y+z)F_{2}(y,z,x))\diamond
w$ $\displaystyle=$ $\displaystyle\varepsilon(x,y+z)F_{2}(y,z,x\diamond
w)-x\diamond F_{2}(y,z,w).$

where $F_{1},F_{2}:\otimes^{3}A\longrightarrow A$ are defined by

$\displaystyle F_{1}(x,y,z)$ $\displaystyle=$ $\displaystyle x\ast(y\diamond
z)-\varepsilon(x,y)y\diamond(x\ast z)-[x,y]\diamond z,$ $\displaystyle
F_{2}(x,y,z)$ $\displaystyle=$ $\displaystyle x\diamond(y\ast
z)+\varepsilon(x,y)y\diamond(x\ast z)-(x\cdot y)\ast z,$

and the operation $\cdot$ and bracket $[,]$ are defined by

$x\cdot y=x\diamond y+\varepsilon(x,y)y\diamond x,\quad[x,y]=x\ast
y-\varepsilon(x,y)y\ast x.$

It is well known that $(A,[,],\varepsilon)$ is a Lie color algebra, and
$(A,L)$ is a representation of the Lie color algebra $(A,[,],\varepsilon)$
where $L:A\longrightarrow\mathfrak{gl}(A)$ is defined by

$L_{x}y=x\ast y,$

for any homogeneous elements $x,y\in A$.

######  Theorem 4.4.

Let $(A,\diamond,\ast,\varepsilon)$ be a pre-$F$-manifold color algebra. Then

  * $\rm(1)$

$(A,\cdot,[,],\varepsilon)$ is an $F$-manifold color algebra, where the
operation $\cdot$ and bracket $[,]$ are defined in Definition 4.3.

  * $\rm(2)$

$(A;L,\mathfrak{L})$ is a representation of $(A,\cdot,[,],\varepsilon)$, where
$L,\mathfrak{L}:A\longrightarrow\mathfrak{gl}(A)$ are defined by

$L_{x}y=x\ast y,\ \ \ \mathfrak{L}_{x}y=x\diamond y,$

for any homogeneous elements $x,y\in A$.

###### Proof.

(1) By Lemma 4.2, $(A,\cdot,\varepsilon)$ is an $\varepsilon$-commutative
associative algebra. It is known that $(A,[,],\varepsilon)$ is a Lie color
algebra. Thus we only need to prove that the color Hertling-Manin relation
holds.

Assume that $x,y,z,w\in A$ are all homogeneous elements. We claim the
following identity:

(6)
$P_{x}(y,z)=F_{1}(x,y,z)+\varepsilon(y,z)F_{1}(x,z,y)+\varepsilon(x,y+z)F_{2}(y,z,x).$

In fact, we have

$\displaystyle P_{x}(y,z)=[x,y\cdot z]-[x,y]\cdot
z-\varepsilon(x,y)y\cdot[x,z]$ $\displaystyle=$ $\displaystyle x\ast(y\cdot
z)-\varepsilon(x,y+z)(y\cdot z)\ast x-[x,y]\diamond
z-\varepsilon(x+y,z)z\diamond[x,y]$
$\displaystyle-\varepsilon(x,y)\\{y\diamond[x,z]+\varepsilon(y,x+z)[x,z]\diamond
y\\}$ $\displaystyle=$ $\displaystyle x\ast(y\diamond
z)-\varepsilon(x,y)y\diamond(x\ast z)-[x,y]\diamond
z+\varepsilon(y,z)\\{x\ast(z\diamond y)-\varepsilon(x,z)z\diamond(x\ast
y)-[x,z]\diamond y\\}$ $\displaystyle+\varepsilon(x,y+z)\\{y\diamond(z\ast
x)+\varepsilon(y,z)z\diamond(y\ast x)-(yz)\ast x\\}$ $\displaystyle=$
$\displaystyle
F_{1}(x,y,z)+\varepsilon(y,z)F_{1}(x,z,y)+\varepsilon(x,y+z)F_{2}(y,z,x).$

With the above identity, we have

$\displaystyle P_{x\cdot y}(z,w)-x\cdot P_{y}(z,w)-\varepsilon(x,y)y\cdot
P_{x}(z,w)$ $\displaystyle=$ $\displaystyle F_{1}(x\cdot
y,z,w)+\varepsilon(z,w)F_{1}(x\cdot
y,w,z)+\varepsilon(x+y,z+w)F_{2}(z,w,x\cdot y)$
$\displaystyle-x\cdot\\{F_{1}(y,z,w)+\varepsilon(z,w)F_{1}(y,w,z)+\varepsilon(y,z+w)F_{2}(z,w,y)\\}$
$\displaystyle-\varepsilon(x,y)y\cdot\\{F_{1}(x,z,w)+\varepsilon(z,w)F_{1}(x,w,z)+\varepsilon(x,z+w)F_{2}(z,w,x)\\}$
$\displaystyle=$ $\displaystyle\big{\\{}F_{1}(x\cdot y,z,w)-x\diamond
F_{1}(y,z,w)-\varepsilon(x,y)y\diamond F_{1}(x,z,w)\big{\\}}$
$\displaystyle+\big{\\{}\varepsilon(z,w)F_{1}(x\cdot
y,w,z)-\varepsilon(z,w)x\diamond
F_{1}(y,w,z)-\varepsilon(x,y)\varepsilon(z,w)y\diamond F_{1}(x,w,z)\big{\\}}$
$\displaystyle+\big{\\{}\varepsilon(x+y,z+w)F_{2}(z,w,x\diamond
y)-\varepsilon(x,y)\varepsilon(y,x+z+w)F_{1}(x,z,w)\diamond y$
$\displaystyle-\varepsilon(x,y)\varepsilon(z,w)\varepsilon(y,x+w+z)F_{1}(x,w,z)\diamond
y-\varepsilon(x,y)\varepsilon(x,z+w)\varepsilon(y,x+w+z)F_{2}(z,w,x)\diamond
y$ $\displaystyle-\varepsilon(y,z+w)x\diamond
F_{2}(z,w,y)\big{\\}}+\big{\\{}\varepsilon(x+y,z+w)\varepsilon(x,y)F_{2}(z,w,y\diamond
x)$ $\displaystyle-\varepsilon(x,y+z+w)F_{1}(y,z,w)\diamond
x-\varepsilon(z,w)\varepsilon(x,y+z+w)F_{1}(y,w,z)\diamond x$
$\displaystyle-\varepsilon(y,z+w)\varepsilon(x,z+w+y)F_{2}(z,w,y)\diamond
x-\varepsilon(x,y)\varepsilon(x,z+w)y\diamond F_{2}(z,w,x)\big{\\}}$
$\displaystyle=$
$\displaystyle\varepsilon(x+y,z+w)\big{\\{}F_{2}(z,w,x\diamond
y)-\varepsilon(z+w,x)F_{1}(x,z,w)\diamond y$
$\displaystyle-\varepsilon(z,w)\varepsilon(z+w,x)F_{1}(x,w,z)\diamond
y-F_{2}(z,w,x)\diamond y-\varepsilon(z+w,x)x\diamond F_{2}(z,w,y)\big{\\}}$
$\displaystyle+\varepsilon(x,y+z+w)\big{\\{}\varepsilon(y,z+w)F_{2}(z,w,y\diamond
x)-F_{1}(y,z,w)\diamond x-\varepsilon(z,w)F_{1}(y,w,z)\diamond x$
$\displaystyle-\varepsilon(y,z+w)F_{2}(z,w,y)\diamond x-y\diamond
F_{2}(z,w,x)\big{\\}}$ $\displaystyle=$
$\displaystyle\varepsilon(y,z+w)\big{\\{}\varepsilon(x,z+w)F_{2}(z,w,x\diamond
y)-F_{1}(x,z,w)\diamond y$ $\displaystyle-\varepsilon(z,w)F_{1}(x,w,z)\diamond
y-\varepsilon(x,z+w)F_{2}(z,w,x)\diamond y-x\diamond F_{2}(z,w,y)\big{\\}}$
$\displaystyle=$ $\displaystyle 0.$

Hence, $(A,\cdot,[,],\varepsilon)$ is an $F$-manifold color algebra.

(2) $(A,\mathfrak{L})$ is a representation of the $\varepsilon$-commutative
associative algebra $(A,\cdot,\varepsilon)$ by Lemma 4.2. It is known that
$(A,L)$ is a representation of the Lie color algebra $(A,[,],\varepsilon)$.
Note that $F_{1}(x,y,z)=R_{L,\mathfrak{L}}(x,y)(z)$, thus the equation

$F_{1}(x\cdot y,z,w)=x\diamond F_{1}(y,z,w)+\varepsilon(x,y)y\diamond
F_{1}(x,z,w)$

implies

$R_{L,\mathfrak{L}}(x\cdot
y,z)=\mathfrak{L}_{x}R_{L,\mathfrak{L}}(y,z)+\varepsilon(x,y)\mathfrak{L}_{y}R_{L,\mathfrak{L}}(x,z).$

On the other hand, $F_{2}(x,y,z)=S_{L,\mathfrak{L}}(x,y)(z)$, thus combining
(6), the equation

$\displaystyle(F_{1}(x,y,z)+\varepsilon(y,z)F_{1}(x,z,y)+\varepsilon(x,y+z)F_{2}(y,z,x))\diamond
w$ $\displaystyle=$ $\displaystyle\varepsilon(x,y+z)F_{2}(y,z,x\diamond
w)-x\diamond F_{2}(y,z,w)$

implies

$\mathfrak{L}_{P_{x}(y,z)}=\varepsilon(x,y+z)S_{L,\mathfrak{L}}(y,z)\mathfrak{L}_{x}-\mathfrak{L}_{x}S_{L,\mathfrak{L}}(y,z).$

Hence, $(A;L,\mathfrak{L})$ is a representation of
$(A,\cdot,[,],\varepsilon)$. ∎

##  5\. Acknowledgements

ZC was partially supported by National Natural Science Foundation of China
(11931009) and Natural Science Foundation of Tianjin (19JCYBJC30600). J. Li
was partially supported by National natural Science Foundation of China
(11771331).

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#  Feature Map for Quantum Data: Probabilistic Manipulation

Hyeokjea Kwon  School of Electrical Engineering, Korea Advanced Institute of
Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141,
Republic of Korea.  Hojun Lee  School of Electrical Engineering, Korea
Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-
gu, Daejeon 34141, Republic of Korea.  Joonwoo Bae  School of Electrical
Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291
Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea.

###### Abstract

The kernel trick in supervised learning signifies transformations of an inner
product by a feature map, which then restructures training data in a larger
Hilbert space according to an endowed inner product. A quantum feature map
corresponds to an instance with a Hilbert space of quantum states by fueling
quantum resources to ML algorithms. In this work, we point out that the
quantum state space is specific such that a measurement postulate
characterizes an inner product and that manipulation of quantum states
prepared from classical data cannot enhance the distinguishability of data
points. We present a feature map for quantum data as a probabilistic
manipulation of quantum states to improve supervised learning algorithms.

In supervised learning, one aims to construct a model that makes predictions
based on training data. Recently, the framework has begun to apply the laws of
quantum mechanics, Quantum Machine Learning, to fuel nonclassical properties
such as entanglement and superposition to machine learning (ML) algorithms for
further advantages, see e.g., Biamonte  _et al._ (2017); Maria Schuld (2018);
Dunjko  _et al._ (2016). One way to apply Quantum Information Theory (QIT) in
ML is to process ML algorithms with quantum states prepared according to
classical data.

From the view of QIT, the state preparation rephrases embedding classical data
to quantum systems. Technically, the space of quantum states is described by a
Hilbert space where the measurement postulate, called the Born rule, specifies
an inner product von Neumann (1932). From the view of ML, embedding data in a
Hilbert space corresponds to a feature map: its quantum application is called
a quantum feature map Schuld and Killoran (2019); Schuld (2021). Then, the
resulting quantum states and the Hilbert space are referred to as feature
vectors and a feature space, respectively.

On the one hand, a quantum feature map allows one to exploit quantum resources
such as entanglement and superposition existing in quantum states to enhance
ML algorithms. As a result, one may envisage quantum advantages over classical
counterparts. On the other hand, one notices that the quantum state space is a
specific and restricted object. It is a Hilbert space entirely characterized
by the postulates of quantum theory von Neumann (1932).

The consequences show that once a feature map prepares quantum states, quantum
operations are contractive, i.e., the norm of feature vectors does not
increase Pérez-García  _et al._ (2006). Moreover, a mathematical space
describing quantum states is not hypothetical: the measurement postulate
defines an inner product uniquely in the space, known as the Gleason theorem
Gleason (1957). The uniqueness implies limitations on the so-called kernel
tricks in the quantum feature space. Apart from the fact the quantum state
preparation corresponds to a feature map per se, little is known about how
quantum principles can be incorporated to feature vectors to enhance ML
algorithms.

In this work, we show that once quantum states are prepared for an ML
algorithm, their distinguishability does not increase by a feature map. In
contrast to classical ML algorithms, quantum data cannot be manipulated such
that their distinguishability is enhanced. We then present a general
manipulation of quantum data, namely a feature map for quantum data, by
relaxing the trace-preserving condition, as a versatile tool to improve ML
algorithms. We also develop a circuit construction of a feature map for
quantum data and demonstrate its advantages in supervised learning for binary
classification.

We first summarize a feature map as a general process of embedding data into a
larger Hilbert space. For sample data $D=\\{x_{m}\in\mathbbm{R}^{N}\\}$, a
feature map $\phi:\mathbbm{R}^{N}\rightarrow\mathcal{H}$ for
$\dim(\mathcal{H})\geq N$ induces an inner product,
$\kappa(x_{m},x_{n})=\langle{\phi}(x_{m}),{\phi}(x_{n})\rangle_{\mathcal{H}}$
on a resulting Hilbert space, where $\kappa:(x_{m},x_{n})\mapsto\mathbbm{R}$
denotes a kernel function via an inner product on the feature space
$\mathcal{H}$. Thus, a structure is introduced to data in terms of an inner
product.

A quantum feature space is a Hilbert space of quantum states such that $n$
qubits define a $2^{n}$-dimensional space on which a quantum state could
contain superposition or entanglement. A quantum feature map devises a quantum
circuit $U_{\phi}$ that prepares states from sample data as follows,

$\displaystyle\phi~{}:~{}x_{m}\in\mathbb{R}^{N}\mapsto\rho_{\phi}(x_{m})\in
S(\mathcal{H})$ (1)

where $\rho_{\phi}(x_{m})$ is a quantum state on a Hilbert space $\mathcal{H}$
and $S(\mathcal{H})$ denotes the set of states. Consequently, a kernel is a
mapping from a pair of datapoints to an inner product in the feature space via
a feature map,

$\displaystyle\kappa(x_{m},x_{n})=\mathrm{tr}[\rho_{\phi}(x_{m})\rho_{\phi}(x_{n})]~{}~{}$
(2)

which is in fact given by the measurement postulate. For instance, Eq. (2) may
be obtained as a probability of obtaining an outcome on positive-operator-
valued-measure $\alpha\rho_{\phi}(x_{n})$ with some $\alpha>0$ for a state
$\rho_{\phi}(x_{m})$. Or, it corresponds to the estimation of the visibility
of two states in an interferometric setup, e.g., the Hong-Ou-Mandel fringe for
photonic qubits Hong  _et al._ (1987); Horodecki and Ekert (2002).

The Hilbert space of quantum systems is specific in the sense that an inner
product is introduced by a measurement postulate Gleason (1957); von Neumann
(1932). It immediately implies limitations of a quantum feature map in that,
contrasting to feature spaces by classical systems, quantum data cannot be
repeatedly embedded in some other feature space such that they are structured
with a higher distinguishability.

Proposition. Quantum data cannot be embedded in a feature space with enhanced
distinguishability.

Proof. We first consider cases that mapping quantum data directly back to a
classical feature space by measurements. It is clear that non-orthogonal
states cannot be perfectly distinguished, and thus the mapping introduces
either an error Helstrom (1969) or ambiguous outcomes Ivanovic (1987); Dieks
(1988); Peres (1988), see also Bergou (2007); Bae and Kwek (2015). There are
fundamental limitations in the bounds Croke  _et al._ (2008); Bae  _et al._
(2011); Hwang and Bae (2010)

Or, one can consider a feature map for quantum data as embedding quantum data
to high-dimensional quantum systems and then a measurement. A quantum channel,
denoted by $\Lambda:\mathcal{H}\rightarrow\widetilde{H}$ for
$\mathrm{dim}\mathcal{H}\leq\mathrm{dim}\widetilde{\mathcal{H}}$, corresponds
to a positive and completely positive map for quantum states. Such a map does
not increase pairwise distinguishability: for two states $\rho_{1}$ and
$\rho_{2}$ appearing with probabilities $q_{1}$ and $q_{2}$, it holds that

$\displaystyle\|\Lambda[X]\|_{1}\leq\|X\|_{1},~{}\mathrm{for}~{}X=q_{1}\rho_{1}-q_{2}\rho_{2}\ngeq
0$ (3)

where $\|\cdot\|_{1}$ denotes the $L_{1}$ norm, i.e.,
$\|A\|_{1}=\mathrm{tr}\sqrt{A^{\dagger}A}$. $\Box$

Figure 1:  A quantum feature map transforms classical data (yellow) to quantum
states (blue), where the mathematical structure is provided by a Hilbert space
where an inner product is realized by a measurement axiom. A feature map
defined by Kraus operators ($K,K_{0}$) for quantum states introduces two
ensembles. One is by $K$ in which states are manipulated beyond the
contractive properties (green), i.e., states can be even better distinguished.
The other Kraus operator $K_{0}$ collects states whenever $K$ is unsuccessful.

In the above, one can consider other measures of distinguishability such as
von Neumann entropy or min-entropy and find the same conclusion Jozsa and
Schlienz (2000); Bae (2013).

We put a step forward to relaxing quantum channels to quantum filtering
operations and present a general form of a feature map for quantum data to
enhance ML algorithms with quantum states. To this end, we recall that a
quantum channel can be described as a dynamics of a subsystem that interacts
with an ancilla through a unitary transformation. When a state and an ancilla
initialized in $\rho_{S}\otimes|a\rangle_{A}\langle a|$ result in
$U_{SA}\rho_{S}\otimes|a\rangle_{A}\langle a|U_{SA}^{\dagger}$ for some
interaction $U_{SA}$, a measurement on an ancilla in an orthonormal basis
$\\{|i\rangle_{A}\\}$ finds the probability of having an outcome $i$,

$\displaystyle p_{A}(i)=\mathrm{tr}[U_{SA}\rho_{S}\otimes|a\rangle_{A}\langle
a|U_{SA}^{\dagger}|i\rangle_{A}\langle i|].$

Then, the resulting state of a system can be described by

$\displaystyle\rho_{S}(i)=\frac{1}{p_{A}(i)}{K_{i}}\rho
K_{i}^{\dagger}~{}~{}\mathrm{where}~{}K_{i}=_{A}\langle i|U_{SA}|a\rangle_{A}$
(4)

where $\\{K_{i}\\}$ are called Kraus operators, satisfying the relation
$\sum_{i}K_{i}^{\dagger}K_{i}=\mathbbm{I}$.

It is worth mentioning that probabilistic manipulations of quantum states by
exploiting Kraus operators have been used to resolve non-trivial problems in
various contexts of quantum information applications. In entanglement theory,
two-qubit entangled states can be transformed to a more entangled one with
some probability by local operations and classical operations, called local
filtering Verstraete  _et al._ (2001); Verstraete and Wolf (2002). The
protocol for distilling entanglement can be rephrased as a sequence of local
filtering operations Bennett  _et al._ (1996); Deutsch  _et al._ (1996). In
fact, local filtering operations can also reveal hidden nonlocality existing
in some entangled states Popescu (1995). In experiments, a post-selection
technique can be described by Kraus operators. For instance, one way to
demonstrate a quantum gate with photonic qubits which hardly interact with
each other is to select particular measurement outcomes whenever photon-photon
interactions were successful, see e.g., Lim  _et al._ (2011).

We now present a feature map for quantum data via Kraus operators as a
probabilistic strategy of manipulating quantum states to enhance ML
algorithms, see Fig. 1. Let $K$ and $K_{0}$ denote two Kraus operators such
that $K$ describes a desired transformation of quantum data and
$K_{0}=(\mathbb{I}-K^{\dagger}K)^{1/2}$ otherwise. Then, quantum data
previously prepared by a quantum feature map $\phi$ denoted by
$D_{\phi}=\\{\rho_{\phi}(x_{m})\in S(\mathcal{H})\\}$ can be transformed to
$\widetilde{D}_{\phi}=\\{\widetilde{\rho}_{\phi}(x_{m})\in
S(\widetilde{\mathcal{H}})\\}$ such that

$\displaystyle
K~{}:~{}{\rho}_{\phi}(x_{m})\mapsto\widetilde{\rho}_{\phi}(x_{m})~{}\mathrm{where}~{}$
(5)
$\displaystyle\widetilde{\rho}_{\phi}(x_{m})=\frac{K\rho_{\phi}(x_{m})K^{\dagger}}{p(x_{m})}~{}\mathrm{with}~{}p(x_{m})=\mathrm{tr}[K^{\dagger}K\rho_{\phi}(x_{m})].$

From Proposition, we can safely restrict the consideration to the case
$\dim(\widetilde{\mathcal{H}})=\dim({\mathcal{H}})$ due to no advantage of
utilizing a larger Hilbert space of quantum states. It is straightforward to
see that, once the transformation is successful, a Kraus operator also leads
to a kernel:

$\displaystyle\kappa(\rho_{\phi}(x_{m}),\rho_{\phi}(x_{n}))=\mathrm{tr}[\widetilde{\rho}_{\phi}(x_{m})\widetilde{\rho}_{\phi}(x_{n})]$
(6)

Therefore, the task to enhance ML algorithms for quantum data is to identify a
Kraus operator $K$ for manipulating quantum sample data beyond unitary
transformations. If no enhancement occurs, one returns a trivial choice
$K=\mathbbm{I}$.

One may assert the weakness that a feature map in Eq. (5) is probabilistic so
that for a large number of datapoints, the probability of its realization
$p(x_{1})\times p(x_{2})\times\cdots\times p(x_{m})\times\cdots$ quickly falls
to zero. A collective interaction over blocks of quantum data can offer a
prescription. That is, a Kraus operator is constructed for quantum states
$D_{\phi}=\\{\rho_{\phi}(x_{m})\in S(\mathcal{H})\\}$,

$\displaystyle
K:\bigotimes_{m}{\rho}_{\phi}(x_{m})\mapsto\bigotimes_{m}\widetilde{\rho}_{\phi}(x_{m})$
(7)

that prevents the overall success probability from dropping out to zero. Note
that the success probability is given by
$p(x_{1},\cdots,x_{m})=\mathrm{tr}[K^{\dagger}K\bigotimes_{m}{\rho}_{\phi}(x_{m})]$,
which could be low but does not depend on the size of data. Thus, the price to
pay for a non-vanishing probability is to build a Kraus operator $K$ for
interaction among all datapoints, see $V(\theta)$ in Fig. 2.

In the following, we revisit supervised learning for binary classification: a
training dataset is given as follows,

$\displaystyle
D=\\{(x_{1},y_{1}),\dots,(x_{M},y_{M}):x_{m}\in\mathbbm{R}^{N},~{}y_{m}=\pm
1\\}.$ (8)

A linear hyperplane is sought to separate the set ${D}$ into two, one with
$y_{m}=+1$ and the other with $y_{m}=-1$. A kernelized support vector machine
applies a kernel $\phi:\mathbbm{R}^{N}\rightarrow\mathbbm{R}^{L}$ for $L\geq
N$ so that the decision boundary can be characterized in a higher-dimensional
Hilbert space by a normal vector $w\in\mathbbm{R}^{L}$ such that
$f(x)=w^{\top}\phi(x)$. A decision function is given as the sign of $f(x)$.
The classification function can be obtained as,

$\displaystyle f(x)=\sum_{m}y_{m}\kappa(x_{m},x)$ (9)

where uniform weights are assumed and the kernel $\kappa$ depends on the
choice of a feature map $\phi$.

For simplicity, let us consider an amplitude encoding as a quantum feature map
that prepares quantum states from sample data $D$,

$\displaystyle\rho_{\phi}(x_{m})=|\psi_{x_{m}}\rangle\langle\psi_{x_{m}}|~{}\mathrm{where}~{}|\psi_{x_{m}}\rangle=\sum_{i=0}^{N-1}x_{m,i}|i\rangle.$
(10)

To apply a SWAP-test based classifier Schuld  _et al._ (2017), we write by $U$
a quantum circuit for the state preparation,

$\displaystyle U|0\rangle^{\otimes
l}=\frac{1}{\sqrt{M}}\sum_{m=1}^{M}|m\rangle_{L}|\psi_{x_{m}}\rangle_{T}|\psi_{x}\rangle_{t}|0\rangle_{S}|s_{m}\rangle_{C},~{}~{}~{}$
(11)

where $l=(2+\log M+2\log N)$, see also Fig. 2. The first register (L) labels
input data, the second one (T) collects training quantum data, the third one
(t) contains a test state, the fourth one (S) is an ancilla needed for a SWAP
test, and the last one (C) denotes a binary classification of training data:
$|s_{m}\rangle=|0\rangle$ for $y_{m}=-1$ and $|s_{m}\rangle=|1\rangle$ for
$y_{m}=1$. Registers (S) and (C) contain single qubits.

Figure 2:  (A) A quantum feature map $U$ prepares quantum states from data,
see the main text. (B) A feature map for quantum data implements a
probabilistic transformation on both training and test data, i.e., when an
outcome $0$ only is obtained in the register $F$. An interaction $V(\theta)$
may be trained to build an efficient feature map. (C) A Hadamard classifier is
implemented with a SWAP test.

Then, a Hadamard classifier, see the dotted box (C) in Fig. 2, applies a SWAP
test for states in registers $T$ and $t$. Qubits in registers in $S$ and $C$
are measured in the computational basis $\\{|0\rangle_{z},|1\rangle_{z}\\}$,
respectively. The classifier in Eq. (9) can be constructed from outcomes in
the registers. When an outcome $0$ $(1)$ is obtained in register $C$, states
in registers $T$ are given by $\rho$ $(\sigma)$ where

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{M_{A}}\sum_{y_{m}=-1}\rho_{\phi}(x_{m})~{}\mathrm{and}~{}\sigma=\frac{1}{M_{B}}\sum_{y_{m}=1}\rho_{\phi}(x_{m}),~{}~{}~{}$
(12)

where $M_{A}=|\\{x_{m}:y_{m}=-1\\}|$ and $M_{B}=|\\{x_{m}:y_{m}=1\\}|$. Given
an outcome $0$ in register $C$, a measurement is performed in register $S$:
probabilities of $0$ and $1$ are given by
$(1\pm\langle\psi_{x}|\rho|\psi_{x}\rangle)/2$, respectively. When an outcome
in register $C$ is $1$, probabilities of $0$ and $1$ in register $S$ are given
by $(1\pm\langle\psi_{x}|\sigma|\psi_{x}\rangle)/2$. Hence, from the
probabilities, one can find the classification function in Eq. (9) as

$\displaystyle f(x)=\langle\psi_{x}|\rho-\sigma|\psi_{x}\rangle$ (13)

The empirical risk with the loss function by $L(f(x),y)=yf(x)$ is given by

$\displaystyle R_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{m}L(f(x_{m}),y_{m})=-\mathrm{tr}[(\rho-\sigma)^{2}],$
(14)

which also quantifies a classifier. Hence, a quantum feature map that
maximizes the Hilbert-Schmidt distance between two ensembles $\rho$ and
$\sigma$ builds a classifier with the least empirical risk Lloyd  _et al._
(2021).

Figure 3:  An interaction that realizes a feature map for quantum data, see
Fig. 2, is designed as
$V_{TFt}(\theta)=\widetilde{V}_{Ft}(\theta)\widetilde{V}_{TF}(\theta)$ and
trained. A parameterized quantum circuit $\widetilde{V}(\theta)$ is chosen
such that it contains arbitrary single qubit rotations and controlled-NOT
gates.

A feature map in Eq. (7) applies to quantum data in Eq. (11) to enhance
quantum supervised learning for binary classification. The map is facilitated
by a unitary transformation $V(\theta)$ over training and test data as well as
an ancilla $F$,

$\displaystyle
V(\theta)\sum_{m=1}^{M}|\psi_{x_{m}}\rangle_{T}|\psi_{x}\rangle_{t}|0\rangle_{F}=$
$\displaystyle\sum_{m=1}^{M}K(\theta)|\psi_{x_{m}}\rangle_{T}|\psi_{x}\rangle_{t}|0\rangle_{F}+\sum_{m=1}^{M}K_{0}(\theta)|\psi_{x_{m}}\rangle_{T}|\psi_{x}\rangle_{t}|1\rangle_{F}.~{}~{}~{}$

An interaction $V(\theta)$ is to be trained to minimize the empirical risk in
Eq. (14). Once a desired one $K(\theta)$ is realized with an ancilla state
$|0\rangle_{F}$, ensembles of training data are transformed, denoted by
$\widetilde{\rho}$ and $\widetilde{\sigma}$, respectively, similarly to
ensembles in Eq. (12), and the classifier is given by,
${f}(x)=\langle\widetilde{\psi}_{x}|\widetilde{\rho}-\widetilde{\sigma}|\widetilde{\psi}_{x}\rangle$.
Resulting quantum data with $|1\rangle_{F}$ that is not of interest will be
discarded. In practice, an efficient unitary $V(\theta)$ may be designed as a
parameterized quantum circuit and trained to minimize the cost function
${C}(\theta)={R}_{L}({f}_{\mathrm{}})$, as it is shown in Fig. 3. If a feature
map by a Kraus operator $K(\theta)$ cannot make any advantage, an optimal
parameter after training would return $V(\theta)=\mathbbm{I}$.

Let us illustrate the usefulness of a feature map for quantum data, for
instance, the Iris dataset and the modified national institute of standard and
technology (MNIST) dataset. In the former, Iris versicolor and Iris setosa are
set as class $A$ and $B$, respectively. Then, sepal lengths and sepal widths
are given as input data. We pick the $36$-th data in the Iris versicolor and
the $34$-th data in the Iris setosa as the training dataset, and the $29$-th
data in the Iris versicolor for the test one,

$\displaystyle\mathrm{Training~{}data}~{}$ $\displaystyle:$
$\displaystyle\\{((0,1),-1),((0.796,0.607),1)\\},$
$\displaystyle\mathrm{Test~{}data}$ $\displaystyle:$
$\displaystyle((-0.557,0.83),-1)$

For the latter, we consider the images of $3$ and $6$ for binary
classification. For convenience, we rescale each image from $28\times 28$
pixels to $4\times 4$ pixels Farhi and Neven (2018). In both cases, we use a
classical optimizer COBYLA to train the parameters.

| Iris Dataset | MNIST Dataset
---|---|---
$V(\theta)=\mathbbm{I}$ | $V(\theta)\neq\mathbbm{I}$ | $V(\theta)=\mathbbm{I}$ | $V(\theta)\neq\mathbbm{I}$
Empirical Risk | $-1.251$ | $-2.666$ | $-1.534$ | $-1.963$
Classifier | $-0.708$ | $-0.996$ | $0.121$ | $0.056$
Probability | $-$ | $0.475$ | $-$ | $0.647$
Decision | $-1$ | $-1$ | $1$ | $1$
Table 1:  A feature map for quantum data, denoted by $V(\theta)$, see Fig. 3,
is trained for binary classification with datasets from Iris and MNIST. The
case that a feature is not applied to quantum data is denoted by
$V(\theta)=\mathbbm{I}$. The figure of merit is the empirical risk in Eq.
(14), which is smaller whenever $V(\theta)$ is trained for both datasets. The
success probabilities are maintained as $0.457$ and $0.647$.

Finally, we reiterate that a feature map for quantum data presents a strategy
for enhancing an ML algorithm for quantum states prepared according to a
quantum feature map. It could be compared with the quantum embedding of
classical data shown in Ref. Lloyd  _et al._ (2021), which formulates training
the state preparation to improve ML algorithms with quantum states. It is
worth mentioning that, with some probability, our results always improve a
quantum feature map or the quantum embedding, since a unitary transformation
is an instance of a Kraus operator. Further enhancements of an ML algorithm
may be envisaged by combining both strategies, i.e., with optimal embedding
followed by filtering operations: in Fig. 2 both $U$ and $V(\theta)$ are
trained. Hence, a feature map for quantum data is a versatile tool to enhance
existing ML algorithms.

In conclusion, we have established a general framework for manipulating
quantum data and then shown its applications to an ML algorithm for binary
classification. We have also developed a circuit construction for a feature
map for quantum data. In particular, our results shed light on the opportunity
of restructuring quantum data on a Hilbert space beyond the constraint of
contractivity. With some probability, quantum states can be distributed in a
quantum feature space such that they are better distinguishable. We have
applied the results to supervised learning for binary classification and
demonstrated enhancements in the empirical risk. Our results present a
versatile tool readily applicable to improve existing quantum ML algorithms.

##  I Acknowledgement

This work is supported by National Research Foundation of Korea
(NRF-2019M3E4A1080001) and the ITRC (Information Technology Research Center)
Program (IITP-2022-2018-0-01402).

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##  II Appendix : MNIST data

The MNIST dataset contains pixelized $28\times 28$ hand-writting images of
numbers from $0$ to $9$. We rescale the $28\times 28$ images to $4\times 4$ to
encode input data into quantum states Farhi and Neven (2018). A pixel data
with a value $0$ to $255$ are normalized such that they are mapped to
amplitudes of a quantum state. We pick numbers $3$ and $6$. Then, the training
dataset

$\displaystyle\mathcal{D}=\\{(x_{1},y_{1}),\cdots,(x_{M},y_{M}):x_{m}\in[0,1]^{16},y_{m}=\pm
1\\}$

are prepared by a quantum state
$|\psi_{x}\rangle=\sum_{i=0}^{15}x_{i}|i\rangle$.

Unlike Iris dataset, a parameterized quantum circuit for the rescaled MNIST
dataset requires $5$ qubits. We use single quantum layer, which means there
are $10$ parameters to optimize and $10$ CNOT gates. COBYLA, a classical
optimizer for the training, is used.

Figure 4:  (a) The original images of numbers $3$ and $6$ have $28\times 28$
pixels. Each pixel has a value between $0$ to $255$. (b) The original images
are rescaled to $4\times 4$ pixels and then normalized. Thus, $16$ amplitudes
are generated and used to prepare quantum states.