The general idea behind using C^*–algebra is to find a succinct way of studying commuting systems. One way to do this is to find a description of a system in terms of its Hilbert space. Properties of commuting C^*–algebras allow for a fine–grained partition of local Hilbert space. This partitions allow for the independent study of the components of said local system.

A local Hilbert space $\mathcal{H}_j$ (in certain circumstances) admits a partition into several smaller Hilbert spaces that are independent. For example $$ \mathcal{H}_j = \bigotimes_{k} \mathcal{H}_{j,k}, $$ where each $\mathcal{H}_{j,k}$ is independent of the others and where the algebra $\mathfrak{A}_{j,k} = \mathbf{L}(\mathcal{H}_{j,k})\otimes\mathbb{I}$. It is possible to find a set of operators $\{A_\alpha^{({j,k})}\}$ that generates the algebra $\mathfrak{A}_{j,k}$ meaning any operator in $\mathbf{L}(\mathcal{H}_j)$ admits a decomposition of the form $$ A = \sum_{\alpha} \bigotimes_k A_\alpha^{({j,k})}. $$

For commuting operators $A$ and $B$, it is well known that both share a common eigenbasis hence each operator can be simultaneously diagonalised. For a set of operators $\{A_i\}$ that pairwise commute, an eigenvalue $\ket{\lambda}$ hosts an eigenvalue set $\{\lambda_i : A_i\ket{\lambda} = \lambda_i \ket{\lambda}\}$. The energy for such a state in then $\bra{\lambda}{H}\ket{\lambda} = \sum_i \lambda_i$.

Definitions

The main focus of this section is the study of Hilbert spaces and their associated operators. We restrict our attention to the study of operators that are bounded and linear; we denote the set of all such operators as $\mathbf{L}(\mathcal{H})$ where $\mathcal{H}$ is the Hilbert space. An important idea central to the use of C^*–algebras is the notion of a commutant. For example, two operators $A$ and $B$ are said to commute if $AB = BA$. The commutator of two operators $A$ and $B$ is defined as $[A,B] := AB - BA$. Within all sets of bounded linear operators lies the identity operator $\mathbb{I}$. The identity is such that $\mathbb{I}^2 = \mathbb{I}$, $AA^{-1} = A^{-1}A = \mathbb{I}$ and $[\mathbb{I},A]=0$ for any $A \in \mathbf{L}(\mathcal{H})$. A common transformation on operators on a Hilbert space include the Hermitian adjoint $\dagger : A \rightarrow A^\dagger$, which maps an operator to its conjugate transpose. Those operators that satisfy $A = A^\dagger$ are called Hermitian (or self–adjoint). Note that the Hermitian adjoint is a linear transformation that is an involution, i.e. $(A^\dagger)^\dagger = A$. Another interesting operation is the direct sum which takes two operations $A$ and $B$ to a block diagonal matrix $A\oplus B$. We can ''combine'' operators from two different Hilbert spaces to form new operators. The tensor product is a bilinear map $V \times W \rightarrow V \otimes W$ that maps a pair $(v,w)$ for $v\in V$ and $w\in W$ to an element $v\otimes w \in V\otimes W$.

Within the algebra of quantum theory lies ^*–algebras, particularly C^*–algebras. These are complex Banach algebras with an involution operation $^*$. The formal definition of a ^*–algebra is:

A C^*–algebra is a ^*–algebra that satisfies the C^* condition: $$ ||a^*a|| = ||a||^2 \quad \text{for all}\;\, a\in\mathfrak{A}. $$

The C^*–algebra can be seen as a way to generalise the set of complex numbers. Moreover, the set of bounded linear operators on a Hilbert space coupled with the Hermitian adjoint operation is a C^*–algebra. Note that an algebra may not necessarily contain an identity element. Those that do are called unital algebras.

A subalgebra $\mathfrak{B}$ of an algebra $\mathfrak{A}$ is a subset of $\mathfrak{A}$ that is closed under the operations of $\mathfrak{A}$. Within the framework of algebraic structures we can discuss a particular subalgebra — the center of an algebra $Z(\mathfrak{A})$. Essentially the center of an algebra is the set of all elements that commute with every element in the algebra. This is clearly an algebra in itself. $$ Z(\mathfrak{A}) := \{A\in\mathfrak{A} : [A,B]=0 \;\text{for all}\; B\in\mathfrak{A}\}. $$ If the center of an algebra is trivial, i.e. $Z(\mathfrak{A}) = \mathbb{I}$, then the algebra is said to be irreducible.

These tools are crucial for understanding the structure of commuting Hilbert spaces in problems such as the local commuting Hamiltonian problem. Finally, when we say an algebra $\mathfrak{A}$ is generated by a set of operators $\{A_\alpha\}$, we mean $$ \mathfrak{A} = \text{Gen}(\{A_\alpha\}) = \text{span}\{A_{1}^{i}A_{2}^{j}\cdots A_{n}^{l} : i,j,\dots,l \in \mathbb{Z}\}. $$ A commutative C^*–algebra is one such that the underlying associative algebra is a commutative algebra. It is known that a commutative C^*–algebra $\mathfrak{A}$ is isomorphic to the algebra of continuous complex–valued functions on a compact Hausdorff space $X$. This is known as the Gelfand–Naimark theorem.

The Structure Lemma

The structure lemma is at the heart of studying the complexity of the commuting local Hamiltonian problem. A high–level overview of the lemma is to say that it is possible to perform a fine–grained partition of the local Hilbert space. The partition (as the name suggests) decomposes the Hilbert space into smaller, independent Hilbert spaces. Studying these independent spaces allows us to solve the whole problem. Of course, this partition is not always possible and even when it is, it can be a difficult process. Due to the nature of commuting local Hamiltonians in certain circumstances and on particular geometries, the structure lemma can be applied.

To see how a local Hilbert space on a given particle can be decomposed into smaller Hilbert spaces, consider the following scenario. Let a system be composed of a set of particles. The full Hilbert space of the system is then the tensor product of the local Hilbert spaces of each particle, i.e. $$ \mathcal{H} = \bigotimes_{p} \mathcal{H}_p. $$ Let $\mathcal{H}_p$ be a local Hilbert space on a particle $p$ where the dimension of $\mathcal{H}_p$ is $d$. Said particle interacts non–trivially with an arbitrary non–zero number of other particles; we label other particles as $q,r,s,\dots$. All interactions in this system are $2$–local, i.e. the interaction between particle $p$ and particle $q$ is described by a Hamiltonians operator $h_{pq} \in \mathbf{L}(\mathcal{H}_p \otimes \mathcal{H}_q)$. The full local Hamiltonian admits an interaction graph $G = (V,E)$ where $V = [p,q,r,\dots]$ is the set of particles and $E = \{(p,q) : h_{pq} \neq \mathbb{I}\}$ is the set of non–trivial interactions between particles. The goal is to find a fine–grained partition of the local Hilbert space $\mathcal{H}_p$. This analysis closely follows the ideas of [BV03].

Define a set of operator for a local Hilbert space $\mathcal{H}_p$ as $$ \mathcal{S}_p := \{O_p \in \mathbf{L}(\mathcal{H}_p) : [{O_p},{h_{pq}}]=0 \quad \forall (p,q)\in E\}, $$ i.e. the set of local operators on a particle that commute with all non–trivial interactions for that given particle. Before proceeding we state key lemmas in showing the structure lemma.

We can define a C^*–subalgebra $\mathfrak{A}_{p.p} = \mathcal{S}_p \subseteq \mathbf{L}(\mathcal{H}_p)$. By the first Lemma above we know that $\mathfrak{A}_{p.p}$ is equivalent to a span of mutually orthogonal projections. Let $\alpha$ index said projectors. We can employ the direct sum decomposition Lemma, using $\mathcal{H}_p$ and $\mathfrak{A}_{p.p}$ to find $$ \begin{align} \mathcal{H}_p &= \bigoplus_\alpha \mathcal{H}_{p}^{(\alpha)}, \\ \mathfrak{A}_{p.p} &= \bigoplus_\alpha \mathfrak{A}_{p.p}^{(\alpha)}. \end{align} $$ It is common to refer to each $\mathcal{H}_{p}^{(\alpha)}$ as a slice. Recall from the Lemma that each $\mathfrak{A}_{p.p}^{(\alpha)}$ has a trivial center. We can then use the tensor product decomposition Lemma to find a decomposition of the form $$ \begin{align} \mathcal{H}_p &= \bigoplus_\alpha \left( \mathcal{H}_{p.p}^{(\alpha)} \otimes \mathcal{V}_p^{(\alpha)} \right), \\ \mathfrak{A}_{p.p} &= \bigoplus_\alpha \left( \mathbf{L}(\mathcal{H}_{p.p}^{(\alpha)}) \otimes \mathbb{I}\right). \end{align} $$ The Hilbert spaces $\mathcal{V}_p^{(\alpha)}$ are left undefined for the moment. So each subalgebra $\mathfrak{A}_{p.p}^{(\alpha)}$ acts non–trivially on the Hilbert space slice $\mathcal{H}_{p.p}^{(\alpha)}$ and trivially on the Hilbert space slice $\mathcal{V}_p^{(\alpha)}$. To investigate further we now look at the non–trivial $2$–local interactions.

Consider a pair $(p,q)$, i.e. a non–trivial interaction between particle $p$ and particle $q$; there exists an operator $h_{pq}$ acting over such particles. We want to construct two C^*–subalgebras $\mathfrak{A}_{p.q}$ and $\mathfrak{A}_{q.p}$ where

1. $\mathfrak{A}_{p.q} \subseteq \mathbf{L}(\mathcal{H}_p)$ and $\mathfrak{A}_{q.p} \subseteq \mathbf{L}(\mathcal{H}_q)$,
2. $h_{pq} \in \mathfrak{A}_{p.q} \otimes \mathfrak{A}_{q.p}$.

This allows for the independent study of the components of a local system. Moreover, we decompose particle $p$ into a set of subparticles $\{p.q\}$ such that the local interaction between $p$ and $q$ only affects the subparticle $p.q$ (and $q.p$ in the respective $q$ Hilbert space). In this way we can understand the possible structure of the eigenstates of the Hamiltonian.

Decompose the operator $h_{pq}$ into a sum of tensor products of operators acting on the local Hilbert spaces of $p$ and $q$, $$ h_{pq} := \sum_{j} A_j^{(p)} \otimes B_j^{(q)}. $$ Let the family of operators $\{A_j^{(p)}\}$ belong to $\mathbf{L}(\mathcal{H}_p)$ and the family of operators $\{B_j^{(q)}\}$ belong to $\mathbf{L}(\mathcal{H}_q)$. For brevity we drop the superscript. We can define spaces that are spanned by $\{A_j\}$ and $\{B_j\}$. Since this particular case in considering the interaction between $p$ and $q$, we define the spaces $\mathcal{S}_{p.q}$ and $\mathcal{S}_{q.p}$ as the linear spaces spanned by $\{A_j\}$ and $\{B_j\}$ respectively.

A simple observation shows that $\mathcal{S}_{p.q}$ and $\mathcal{S}_{q.p}$ are closed under Hermitian conjugation. We can define two algebras, $$ \begin{align*} \mathfrak{A}_{p.q} \subseteq \mathbf{L}(\mathcal{H}_p),\\ \mathfrak{A}_{q.p} \subseteq \mathbf{L}(\mathcal{H}_q), \end{align*} $$ as the minimal C^*–algebras such that $\mathcal{S}_{p.q} \subseteq \mathfrak{A}_{p.q}$ and $\mathcal{S}_{q.p} \subseteq \mathfrak{A}_{q.p}$. Hence it can be seen $$ \begin{align*} \mathfrak{A}_{p.q} \subseteq \text{Gen}(\{A_j\} \cup \mathbb{I}),\\ \mathfrak{A}_{q.p} \subseteq \text{Gen}(\mathbb{I} \cup \{B_j\}). \end{align*} $$ Consider now another non–trivial interaction between particle $p$ and particle $r$. Given the pieces above, we can consider how the algebras $\mathfrak{A}_{p.q}$ and $\mathfrak{A}_{p.r}$ interact. Recall that the goal of this analysis was to find algebras allowing us to partition and isolate the Hilbert space local to particle $p$. It turns out the two algebras commute, i.e. $[\mathfrak{A}_{p.q},\mathfrak{A}_{p.r}] = 0$. This becomes clear from the following analysis.

The triplet of particles $p,q,r$ form a Hilbert space $\mathcal{H}_p \otimes \mathcal{H}_q \otimes \mathcal{H}_r$. Non–trivial operators that act over $(p,q)$ and $(p,r)$ can be expressed as $h_{pq} \otimes \mathbb{I}_r$ and $\mathbb{I}_q \otimes h_{pr}$. We decompose them as, $$ \begin{align*} h_{pq} \otimes \mathbb{I}_r &= \sum_{j} A_j^{(p)} \otimes B_j^{(q)} \otimes \mathbb{I}_r,\\ \mathbb{I}_q \otimes h_{pr} &= \sum_{k} \mathbb{I}_q \otimes C_k^{(p)} \otimes D_k^{(r)}. \end{align*} $$ The commutative property dictates that $[{O_{pq} \otimes \mathbb{I}_r },{\mathbb{I}_q \otimes O_{pr}}] = 0$, hence $$ \sum_{j,k} \big( A_j^{(p)}C_k^{(p)} - C_k^{(p)}A_j^{(p)}\big) \otimes B_j^{(q)} \otimes D_k^{(r)} = 0. $$ This is only true if $[{A_j^{(p)}},{C_k^{(p)}}] = 0$ for all $j,k$. Recall that the algebras $\mathfrak{A}_{p.q}$ is generated by $\{A_j\}$ and $\mathfrak{A}_{p.r}$ will be generated by $\{C_k\}$ by analogy. Therefore the algebras $\mathfrak{A}_{p.q}$ and $\mathfrak{A}_{p.r}$ commute.

Let us investigate the relationship between the algebras $\mathfrak{A}_{p.q}$ and $\mathfrak{A}_{p.p}$. A simple observation shows that $[{\mathfrak{A}_{p.q}},{\mathfrak{A}_{p.p}}] = 0$ for any $(p,q)\in E$. Recall that the definition of $\mathfrak{A}_{p.p}$ is the set of operators that commute with all non–trivial interactions for particle $p$. It is also straightforward to see that any operator in $\mathfrak{A}_{p.q}$ preserves $\mathcal{H}_p^{(\alpha)}$. This means each algebra $\mathfrak{A}_{p.q}$ can be decomposed as $$ \mathfrak{A}_{p.q} = \bigoplus_\alpha \mathfrak{A}_{p.q}^{(\alpha)}, $$ such that $\mathfrak{A}_{p.q}^{(\alpha)} \subseteq \mathbf{L}(\mathcal{H}_p^{(\alpha)})$ and $Z(\mathfrak{A}_{p.q}^{(\alpha)}) = \mathbb{C}\mathbb{I}$. Hence $\mathfrak{A}_{p.q}^{(\alpha)} \subseteq \mathbf{L}(\mathcal{H}_{p.p}^{(\alpha)} \otimes \mathcal{V}_p^{(\alpha)})$. Recall that any operator $A \in \mathfrak{A}_{p.p}^{(\alpha)}$ is of the form $A = P \otimes \mathbb{I}$ for some $P \in \mathbf{L}(\mathcal{H}_{p.p}^{(\alpha)})$. Let $B \in \mathfrak{A}_{p.q}^{(\alpha)}$ be an operator of the form $B = X \otimes Q$ such that $X \in \mathbf{L}(\mathcal{H}_{p.p}^{(\alpha)})$ is non–trivial, and $Q \in \mathbf{L}(\mathcal{V}_p^{(\alpha)})$. Then it is clear that $[{A},{B}] = 0$ for any $A,B \in \mathfrak{A}_{p.p}^{(\alpha)}, \mathfrak{A}_{p.q}^{(\alpha)}$, thus, $$ [{A},{B}] = [{P},{X}] \otimes Q = 0. $$ By contradiction we conclude that any operator $B \in \mathfrak{A}_{p.q}^{(\alpha)}$ acts trivially on $\mathcal{H}_{p.p}^{(\alpha)}$. This means the operators $B$ are of the form $$ B = \mathbb{I} \otimes Q, $$ where $Q \in \mathbf{L}(\mathcal{V}_p^{(\alpha)})$. Before we make the final conclusions on this decomposition we consider how the Hilbert spaces for the algebras $\mathfrak{A}_{p.q}$ and $\mathfrak{A}_{p.r}$ relate. In the event that the centers of each $\mathfrak{A}_{p.q}$ are trivial, we can use the contradiction argument to show derive how operators from algebra should look. Since we make no assumptions on the centers of the algebras we must go one layer further and turn to the subalgebra slices. Recall that each $\mathfrak{A}_{p.q}$ is decomposed as a direct sum of subalgebra slices where the subalgebras have trivial centers. The commutation between algebras will hold for the subalgebra slices. Hence we can iteratively apply the tensor product decomposition to the Hilbert space $\mathcal{V}_p^{(\alpha)}$ to form a tensor product partition into component subparticle Hilbert spaces.

To elaborate consider two pairs of particles $(p,q)$ and $(p,r)$. We know that any operator from $\mathfrak{A}_{p.q}^{(\alpha)}$ acts only non–trivially on $\mathcal{V}_p^{(\alpha)}$. Use the tensor product decomposition with $\mathcal{V}_p^{(\alpha)}$ and $\mathfrak{A}_{p.q}^{(\alpha)}$ to find $$ \begin{align} \mathcal{V}_p^{(\alpha)} &= \mathcal{H}_{p.q}^{(\alpha)} \otimes \mathcal{V}_p'^{(\alpha)},\\ \mathfrak{A}_{p.q}^{(\alpha)} &= \textbf{L}(\mathcal{H}_{p.q}^{(\alpha)}) \otimes \mathbb{I}. \end{align} $$ Then since an operator from $\mathfrak{A}_{p.r}^{(\alpha)}$ must be of the form $C = \mathbb{I} \otimes R$ for some $R \in \mathbf{L}(\mathcal{V}_p'^{(\alpha)})$, we again employ the tensor product Lemma to find the Hilbert space becomes $$ \mathcal{V}_p^{(\alpha)} = \mathcal{H}_{p.q}^{(\alpha)} \otimes \mathcal{H}_{p.r}^{(\alpha)} \otimes \mathcal{V}_p''^{(\alpha)}. $$ Iteratively applying this idea we find that the Hilbert space $\mathcal{V}_p^{(\alpha)}$ can be decomposed into a tensor product of subparticle Hilbert spaces. We can now put all the pieces together to find the final decomposition of the local Hilbert space $\mathcal{H}_p$.

Recap!

Let us take a quick recap of the ideas above. The abstract–ness of the mathematics can be daunting, but the ideas are relatively straightforward.

• We defined an algebra based on commuting operators local to some particle $p$.
• This in turn allowed for a direct sum decomposition of the local Hilbert space $\mathcal{H}_p$ into slices.
• A consequence of this decomposition was a subsequent tensor product decomposition of each slice.
• We then studied the algebras generated by the non–trivial $2$–local interaction between particles centered at $p$.
• Demanding a certain tensor product decomposition of the operators over disjoint edges, motivated by a goal of finding a fine–grained partition of the local Hilbert space, we found that commuting algebras over subparticles.
• The subalgebras of the subparticles admitted a similar direct sum decomposition.
• It was then possible to iteratively apply the tensor product decomposition to the subparticle Hilbert spaces.
• This allowed for a fine–grained partition of the local Hilbert space.

Completing the Puzzle

The local Hilbert space $\mathcal{H}_p$ can be partitioned in the following way, $$ \mathcal{H}_p = \bigoplus_{\alpha} \left( \bigotimes_q \mathcal{H}_{p.q}^{(\alpha)}\right) \otimes \mathcal{H}_{p.p}^{(\alpha)}. $$ $$ \mathfrak{A}_p = \bigoplus_{\alpha} \left( \bigotimes_q \mathfrak{A}_{p.q}^{(\alpha)}\right) \otimes \mathfrak{A}_{p.p}^{(\alpha)}. $$ The total Hilbert space for a system of particles is then the tensor product of the local Hilbert spaces of each particle. Moreover $$ \mathcal{H} = \mathcal{H}_p \otimes \mathcal{H}_q \otimes \mathcal{H}_r \otimes \cdots = \bigotimes_{p} \mathcal{H}_p. $$ The structure Lemmas then follows as:

Note that this lemma cannot be used in all circumstances. We showed a fine–grained partition of a local Hilbert space for a specific example. Using this prescription can fail for $3$–local commuting Hamiltonians over qubits. The reason we showed this application was to demonstrate the power of the C^*–algebra language and provided some intuitive context for these abstract ideas.

Technically this Lemma can be gleaned from the tensor product decomposition Lemma and the direct sum decomposition Lemma. The figure below gives a pictorial representation of the structure Lemma and the underlying mechanics of the partitioning of the local Hilbert space.