On multi-variable Zassenhaus formula

TL;DR

This paper introduces a recursive algorithm for computing the multivariable Zassenhaus formula, enabling efficient calculation of exponential decompositions in Lie algebra contexts.

Contribution

It presents a novel recursive method and an effective recursion formula for the multivariable Zassenhaus expansion.

Findings

Developed a recursive algorithm for multivariable Zassenhaus formula

Derived an effective recursion formula for the terms

Facilitates efficient computation of exponential decompositions

Abstract

In this paper, we give a recursive algorithm to compute the multivariable Zassenhaus formula $$e^{X_1+X_2+\cdots +X_n}=e^{X_1}e^{X_2}\cdots e^{X_n}\prod_{k=2}^{\infty}e^{W_k}$$ and derive an effective recursion formula of $W_k$.

Equations

$$e^{X_1+X_2+\cdots +X_n}=e^{X_1}{e}^{X_2}\cdots e^{X_n}\prod _{k=2}^{\infty }{e}^{W_k}$$

$$e^X{e}^Y=e^{X+Y+{\sum }_{k=2}^{\infty }{Z}_k(X,Y)},$$

$$Z_2=\frac{1}{2}[X,Y],Z_3=\frac{1}{12}([X,[X,Y]]-[Y,[X,Y]]),$$

$$Z_4=\frac{1}{24}[X,[Y,[Y,X]]].$$

$$e^{X+Y}$$

$$=e^X{e}^Y{e}^{W_2(X,Y)}{e}^{W_3(X,Y)}\cdots e^{W_k(X,Y)}\cdots ,$$

$$W_2=-\frac{1}{2}[X,Y],W_3=\frac{1}{3}[Y,[X,Y]]+\frac{1}{6}[X,[X,Y]],$$

$$W_4=-\frac{1}{24}[X,[X,[X,Y]]]-\frac{1}{8}([Y,[X,[X,Y]]]+[Y,[Y,[X,Y]]]).$$

$$e^{X_1}{e}^{X_2}\cdots e^{X_n}=e^{X_1+X_2+\cdots +X_n+{\sum }_{m=2}^{\infty }{Z}_m(X_1,X_2,\cdots ,X_n)},$$

$$e^{X_1+X_2+\cdots +X_n}=e^{X_1}{e}^{X_2}\cdots e^{X_n}\prod _{k=2}^{\infty }{e}^{W_k}$$

$$e^{-X_{n-1}}{e}^{X_{n-1}+X_n+W_2^{(n-3)}+W_3^{(n-4)}+\cdots +W_{n-2}^{\prime }+D_{n-3}}$$

$$=e^{X_n+W_2+W_3^{(n-3)}+\cdots +W_{n-2}^{\prime \prime }+W_{n-1}^{\prime }+D_{n-2}}$$

$$e^{t(X_1+X_2+\cdots +X_n)}=e^{t{X}_1}{e}^{t{X}_2}\cdots e^{t{X}_n}{e}^{t^2{W}_2}{e}^{t^3{W}_3}\cdots .$$

$$R_1(t)$$

$$R_m(t)$$

$$R_m(t)=e^{t^{m+1}{W}_{m+1}}{e}^{t^{m+2}{W}_{m+2}}\cdots ,m\ge 1.$$

$$F_m(t)=R_m^{\prime }(t)R_m(t{)}^{-1}m\ge 1.$$

$$F_1(t)$$

$$+e^{-ta{d}_{X_n}}{e}^{-ta{d}_{X_{n-1}}}\cdots e^{-ta{d}_{X_1}}(X_1+X_2+\cdots +X_n)$$

$$=e^{-ta{d}_{X_n}}(e^{-ta{d}_{X_{n-1}}}\cdots e^{-ta{d}_{X_2}}{e}^{-ta{d}_{X_1}}-I)X_n$$

$$+e^{-ta{d}_{X_n}}{e}^{-ta{d}_{X_{n-1}}}(e^{-ta{d}_{X_{n-2}}}\cdots e^{-ta{d}_{X_2}}{e}^{-ta{d}_{X_1}}-I)X_{n-1}$$

$$+e^{-ta{d}_{X_n}}{e}^{-ta{d}_{X_{n-1}}}{e}^{-ta{d}_{X_{n-2}}}(e^{-ta{d}_{X_{n-3}}}\cdots e^{-ta{d}_{X_2}}{e}^{-ta{d}_{X_1}}-I)X_{n-2}+\cdots$$

$$+e^{-ta{d}_{X_n}}{e}^{-ta{d}_{X_{n-1}}}\cdots e^{-ta{d}_{X_3}}(e^{-ta{d}_{X_2}}{e}^{-ta{d}_{X_1}}-I)X_3$$

$$+e^{-ta{d}_{X_n}}{e}^{-ta{d}_{X_{n-1}}}\cdots e^{-ta{d}_{X_2}}(e^{-ta{d}_{X_1}}-I)X_2$$

$$=\sum _{k=1}^{\infty }(-t{)}^k\sum _{i=2}^n\sum _{\begin{array}{c}{j}_1+\cdots +j_{i-1}\ge 1\\ j_1+\cdots +j_n=k\end{array}}\frac{a{d}_{X_n}^{j_n}\cdots a{d}_{X_2}^{j_2}a{d}_{X_1}^{j_1}}{j_1!j_2!\cdots j_n!}{X}_i,$$

$$e^{A}B{e}^{-A}=e^{a{d}_A}B=\sum _{n\ge 0}\frac{1}{n!}a{d}_A^{n}B,$$

$$F_1(t)=\sum _{k=1}^{\infty }{f}_{1,k}{t}^k,$$

$$f_{1,k}=(-1{)}^k\sum _{i=2}^n\sum _{\begin{array}{c}{j}_1+\cdots +j_{i-1}\ge 1\\ j_1+\cdots +j_n=k\end{array}}\frac{a{d}_{X_n}^{j_n}\cdots a{d}_{X_2}^{j_2}a{d}_{X_1}^{j_1}}{j_1!j_2!\cdots j_n!}{X}_i.$$

$$F_m(t)$$

$$=-m{W}_m{t}^{m-1}+e^{-t^{m}a{d}_{W_m}}{F}_{m-1}(t)$$

$$=e^{-t^{m}a{d}_{W_m}}(F_{m-1}(t)-m{W}_m{t}^{m-1}).$$

$$f_{m,k}=\sum _{j=o}^{[\frac{k}{m}]-1}\frac{(-1{)}^j}{j!}a{d}_{W_m}^j{f}_{m-1,k-mj},k\ge m$$

$$F_m(t)=(m+1)W_{m+1}{t}^m+\sum _{j=m+2}^{\infty }j{t}^{j-1}{e}^{t^{m+1}a{d}_{W_{m+1}}}\cdots e^{t^{j-1}a{d}_{W_{j-1}}}{W}_j.$$

$$f_{1,1}=2W_2,f_{1,2}=3W_3,f_{1,3}=4W_4,f_{1,4}=5W_5+3[W_2,W_3],$$

$$W_2=\frac{1}{2}{f}_{1,1},W_3=\frac{1}{3}{f}_{1,2},W_4=\frac{1}{4}{f}_{1,3},W_5=\frac{1}{5}{f}_{1,4}-\frac{1}{10}[f_{1,1},f_{1,2}].$$

$$f_{m,m}=(m+1)W_{m+1},$$

$$W_{m+1}=\frac{1}{m+1}{f}_{m,m}=\frac{1}{m+1}{f}_{[\frac{m}{2}],m},m\ge 4,$$

Full text

[TABLE]

and derive an effective recursion formula of $W_{k}$.

$\dagger$ Corresponding author: jing@math.ncsu.edu

MSC (2010): Primary: 16W25; Secondary: 22E05, 16S20

Keywords: Baker-Campbell-Hausdorff formula, Zassenhaus formula

1. Introduction

The celebrated Baker-Campbell-Hausdorff (BCH) is a fundamental identity in Lie theory [1, 2, 3] connecting Lie algebra with Lie group. The BCH says that for any linear operators $X,Y$ in a bounded Hilbert space one has the formula

[TABLE]

where $\exp$ is defined in the usual sense and $Z_{k}(X,Y)$ is a degree $k$ homogeneous Lie polynomial in the noncommutative variables $X$ and $Y$. The first few terms are

[TABLE]

and the general expressions of $Z_{k}(X,Y)$ can be explicitly computed by combinatorial formulas.

The dual form of the BCH is the famous Zassenhaus formula which establishes that the exponential $e^{X+Y}$ can be uniquely decomposed as

[TABLE]

where $W_{k}(X,Y)$ is a homogeneous Lie polynomial in $X$ and $Y$ of degree $k$ [4]. The first few terms are

[TABLE]

There are several methods to compute $W_{k}$ [5, 6, 7, 8]. In particular, a recursive algorithm has been proposed in [9] to express directly $W_{k}$ with the minimum number of independent commutators required at each degree $k$.

Similar to the BCH formula, the Zassenhaus formula is useful in many different fields: $q$-analysis in quantum groups [10], quantum nonlinear optics [13], the Schrödinger equation in the semiclassical regime [12], and splitting methods in numerical analysis [11], etc.

We now consider the multivariate BCH and Zassenhaus formulas. It is easy to obtain the multivariable BCH formula by repeatedly using the usual BCH:

[TABLE]

where $Z_{m}$ is a Lie polynomial in the $X_{i}$ of degree $m$. On the hand, we also have the multivariable Zassenhaus formula

[TABLE]

where the product is ordered and $W_{k}$ is a homogeneous Lie polynomial in the $X_{i}$ of degree $k$. However, it is more complicated to express $W_{k}$ in terms of $X_{i}^{\prime}s$.

The existence of the formula (1.4) is a consequence of Eq. (1.3). In fact, it is clear that $e^{-X_{1}}e^{X_{1}+X_{2}+\cdots+X_{n}}=e^{X_{2}+X_{3}+\cdots+X_{n}+D}$, where $D$ involves Lie polynomials of degree $>1$. Then $e^{-X_{2}}e^{X_{2}+X_{3}+\cdots+X_{n}+D}=e^{X_{3}+X_{4}+\cdots+X_{n}+W_{2}^{\prime}+D_{1}}$, where $D_{1}$ is an infinite Lie power series in the $X_{i}$ with minimum degree $>2$. Note that $W_{2}^{\prime}\neq W_{2}$, we need to repeat the process $(n-1)$ times to determine $W_{2}$, i.e.

[TABLE]

where $D_{n-2}$ involves Lie polynomials of degree $>(n-1)$. Finally, we can get the formula (1.4) by repeating the process.

In this paper, we will give a new recursive algorithm to compute $W_{k}$ in (1.4). Our method is inspired by the recent algorithm in [9], and our formula is based on a new formula for $f_{1,k}$ using compositions of integers.

The paper is organized as follows. In Section 2, we give our recursive algorithm and a concrete procedure to compute $W_{k},k=1,2,3,4,5$. In Section 3. we establish a combinatorial formula of $f_{1,k}$ (see Theorem 3.1). We will show that our formula can give a slightly better recursion formula of $W_{k}$ when $k>5$ in Theorem 3.2. Finally we use examples to show how the $f_{1,k}$ are used to derive Lie polynomial formulas of $W_{k}$ in terms of the operators $X_{1},\ldots,X_{n}$. The latter set of formulas are expected be useful in the quantum control problem.

2. Multivariable Zassenhaus terms

2.1. A recurrence.

For the operators $X_{1},\ldots,X_{n}$ we consider the following function of $t$:

[TABLE]

where the $W_{k}$ can be determined by differential equations step by step, and it is easy to see that $W_{k}$ is a polynomial of degree $k$ in the $X_{i}$. Note that the multivariable Zassenhaus formula (1.4) is the case when $t=1$.

First we consider the iterated system of equations

[TABLE]

It follows from (2.3) that

[TABLE]

We then take the logarithmic differentiation

[TABLE]

For $m=1$, we have that

[TABLE]

where $ad_{A}B=[A,B]$ and we have used the well-known formula

[TABLE]

as well as the fact that $e^{ad_{X}}X=X$. Write

[TABLE]

then

[TABLE]

A similar expansion can be obtained for $F_{m}(t)$, $m\geq 2$, by using $R_{m}(t)$ in (2.3). More specifically,

[TABLE]

Writing $F_{m}(t)=\sum_{k=m}^{\infty}f_{m,k}t^{k}$, we immediately get that

[TABLE]

where $[\frac{k}{m}]$ denotes the integer part of $\frac{k}{m}$.

On the other hand, if we take the logarithmic derivative of $R_{m}(t)$ using the expression (2.4), we arrive at

[TABLE]

Comparing the coefficients of the terms $t$, $t^{2}$, $t^{3}$ and $t^{4}$ in (2.6) and (2.9) for $F_{1}(t)$, we get that

[TABLE]

so that

[TABLE]

Similarly, comparing (2.1) and (2.9), we get

[TABLE]

therefore

[TABLE]

i.e.

[TABLE]

2.2. Examples of $W_{k}$.

When $k=1$ in the expression (2.7), the summation of the first $i-1$ terms is already at least $1$, so we have the formula

[TABLE]

Thus

[TABLE]

Similarly for $k=2$ in (2.7), we have

[TABLE]

where $m_{i}$ is the multiplicity of $j_{i}$.

Therefore

[TABLE]

We list the first few other terms as follows:

[TABLE]

so that

[TABLE]

[TABLE]

[TABLE]

3. Iteration Formulas

To reveal the explicit rule for $f_{1,k}(k\geq 1)$ based on the computations we gave in Section 2, we recall the definition of partitions and compositions [14].

3.1. Formulation in terms of partitions.

A partition of a positive integer $m$ is an integral unordered decomposition $m=\lambda_{1}+\cdots+\lambda_{l}$ such that $\lambda_{1}\geq\cdots\geq\lambda_{l}>0$, denoted by $\lambda=(\lambda_{1}\lambda_{2}\dots\lambda_{l})$ and $\lambda\vdash m$. Here $\lambda_{i}$ are called the parts and $l$ is the length of the partition. A composition is an ordered integral decomposition of $m$: $m=\lambda_{1}+\cdots+\lambda_{l}$ such that $\lambda_{i}>0$ and denoted by $\lambda\vDash m$, in another words, compositions of $m$ are obtained by permuting the unequal parts of the associated partition of $m$. The set of partitions of $m$ is denoted by $\mathcal{P}(m)$ and the cardinality is denoted by $p(m)$.

For example, the partitions of $4$ are:

[TABLE]

Therefore, $p(4)=5$. The associated compositions are distinct permutations of the partitions: $(4),(3,1),(1,3),(2,2),(2,1,1),(1,2,1),(1,1,2),(1,1,1,1)$.

Accordingly the formulas of $f_{1,k}$ go as follows. For $f_{1,1}$, $p(1)=1$,

[TABLE]

For $f_{1,2}$, $p(2)=2$,

[TABLE]

For $f_{1,3}$, $p(3)=3$,

[TABLE]

For $f_{1,4}$, $p(4)=5$,

[TABLE]

We define the long commutator inductively as follows.

[TABLE]

Fix a partition $\lambda=(\lambda_{1}\lambda_{2}\cdots\lambda_{l})$ of $k$, and for each composition out of $\lambda$: $(k_{1}k_{2}\cdots k_{l})\models k$ which is a rearrangement of $\lambda$ by permuting its parts, we associate the commutator

[TABLE]

where the multiplicity of $i_{s}$ is $k_{s}$ for $1\leq s\leq l$. For this reason, we will write (3.1) as $[X_{j}X_{i_{1}}^{k_{1}}X_{i_{2}}^{k_{2}}\cdots X_{i_{l}}^{k_{l}}]$. Then we have the following result.

Theorem 3.1**.**

For each $k$, the following formula holds

[TABLE]

3.2. Determination of $W_{m}(m\geq 6)$.

We have given the formulas of $W_{k}$ for $1\leq k\leq 5$ in terms of $f_{1,k}$ (2.10). We now give the next a few terms as follows.

[TABLE]

We postpone the verification of these formulas till the general result. The following result gives the general iterative formula for the multivariable Zassenhaus formula.

Theorem 3.2**.**

*For each $k\geq 2$ the exponents $W_{m}$ in the multi-variable Zassenhaus formula (1.4) for $m=6k+i$, where $i=0,1,2,3,4,5$ are given by

[TABLE]

[TABLE]

Proof.

As we know that $W_{m}=\frac{1}{m}f_{[\frac{m-1}{2}],m-1}$, $m\geq 5$ in Section 2, we divide $m$ into even and odd integers.

When $m=2a+1,a\geq 2$,

[TABLE]

if $a=2$, we stop the computation since we reach $f_{1,k}$. Otherwise,

[TABLE]

[TABLE]

so that if $a=3$,

[TABLE]

we stop the computation. If $a\geq 4$,

[TABLE]

Repeating the procedure, we obtain (3.3) and (3.5) as well as (3.8), (3.10), (3.12) in the theorem by using induction.

Similarly, when $m=2a,a\geq 3$,

[TABLE]

if $a=3$, we stop the computation. Otherwise,

[TABLE]

[TABLE]

so that if $a=4$,

[TABLE]

we stop the computation. If $a=5$,

[TABLE]

If $a\geq 6$

[TABLE]

Repeating the procedure, we obtain (3.2), (3.4) and (3.6) as well as (3.7), (3.9), (3.11) in Theorem 3.2 using induction. ∎

According to Theorem 3.2, we know that $W_{m}(m\geq 5)$ can be expressed as a linear combination of $f_{1,k}(k\geq 1)$ in the end, then we use $f_{1,k}(k\geq 1)$ given in Theorem 3.1 to obtain $W_{m}(m\geq 5)$. To explain how this works, we give the explicit formulas of $W_{5},W_{6}$ according to (2.10) and (3.2):

[TABLE]

[TABLE]

Acknowledgments

N. Jing’s work was partially supported by the National Natural Science Foundation of China (Grant No.11531004) and Simons Foundation (Grant No. 523868).