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Direct product of operators

When analyzing composite quantum systems, such as entangled particles or quantum computing registers, we must understand how operators act on the combined state space. This concept is formalized through the Direct Product (or Tensor Product) of Operators, which allows us to extend local operators to the full global Hilbert space.

[Image of Tensor product matrix construction kronecker product]

Mathematical Formalism of Operator Direct Products

1. Definition and Action

Let \( \mathcal{H}_1 \) and \( \mathcal{H}_2 \) be two Hilbert spaces. If \( A \) is an operator on \( \mathcal{H}_1 \) and \( B \) is an operator on \( \mathcal{H}_2 \), their direct product \( A \otimes B \) is a linear operator acting on the composite space \( \mathcal{H}_1 \otimes \mathcal{H}_2 \).

The action of this operator is defined by how it transforms a product state (a basis vector of the combined space):

\[ (A \otimes B) (|u\rangle \otimes |v\rangle) = (A|u\rangle) \otimes (B|v\rangle) \]

Here, \( A \) acts exclusively on the state \( |u\rangle \) from the first space, and \( B \) acts exclusively on \( |v\rangle \) from the second space.

2. Matrix Representation: The Kronecker Product

In matrix mechanics, if \( A \) is an \( N \times N \) matrix and \( B \) is an \( M \times M \) matrix, the direct product \( A \otimes B \) corresponds to the Kronecker Product. The resulting matrix has dimensions \( (NM) \times (NM) \).

It is constructed as a block matrix where every element \( a_{ij} \) of matrix \( A \) is multiplied by the entire matrix \( B \):

\[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B & \dots \\ a_{21}B & a_{22}B & \dots \\ \vdots & \vdots & \ddots \end{pmatrix} \]

3. Operators Acting on Single Subsystems

In physics, we often deal with operators that only affect one part of the system (e.g., a magnetic field acting on only one electron). To represent this on the total space, we tensor the operator with the Identity Operator (\( \mathbb{1} \)) of the other space.

Operator on System 1

\[ \hat{O}_{total} = \hat{O}_1 \otimes \mathbb{1}_2 \] \[ (\hat{O}_1 \otimes \mathbb{1}_2) (|u\rangle \otimes |v\rangle) = (\hat{O}_1|u\rangle) \otimes |v\rangle \]

Operator on System 2

\[ \hat{O}_{total} = \mathbb{1}_1 \otimes \hat{O}_2 \] \[ (\mathbb{1}_1 \otimes \hat{O}_2) (|u\rangle \otimes |v\rangle) = |u\rangle \otimes (\hat{O}_2|v\rangle) \]

4. Key Algebraic Properties

The direct product follows strict algebraic rules, distinct from standard matrix multiplication. These are essential for derivations.

  • Mixed-Product Rule: When multiplying two direct product operators, we multiply the components of the respective spaces separately:
    \[ (A \otimes B)(C \otimes D) = (AC) \otimes (BD) \]
  • Trace Rule: The trace of a direct product is the product of the traces:
    \[ \text{Tr}(A \otimes B) = \text{Tr}(A) \cdot \text{Tr}(B) \]
  • Hermitian Adjoint:
    \[ (A \otimes B)^\dagger = A^\dagger \otimes B^\dagger \]

5. Example: Pauli Matrices

Consider applying a Pauli-X gate (\( \sigma_x \)) to the first qubit of a two-qubit system. The operator is \( \sigma_x \otimes I \).

\[ \sigma_x \otimes I = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \] \[ = \begin{pmatrix} 0\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} & 1\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ 1\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} & 0\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \]