Quantum Mechanics Course

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← Part I/Tensor Products

5. Tensor Products of Hilbert Spaces

Reading time: ~35 minutes | Pages: 7

When describing composite quantum systems (e.g., two particles, or spin and orbital degrees of freedom), we need to combine their individual Hilbert spaces. The tensor product is the fundamental mathematical construction for this purpose.

Motivation: Composite Systems

Consider two quantum systems:

  • System A: Lives in Hilbert space $\mathcal{H}_A$ with basis $\{|a_i\rangle\}$
  • System B: Lives in Hilbert space $\mathcal{H}_B$ with basis $\{|b_j\rangle\}$

How do we describe the combined system? We cannot simply add the spaces or take their Cartesian product. Instead, we form the tensor product $\mathcal{H}_A \otimes \mathcal{H}_B$.

Definition: Tensor Product

Definition:

The tensor product $\mathcal{H}_A \otimes \mathcal{H}_B$ is the Hilbert space whose basis consists of all pairs $|a_i\rangle \otimes |b_j\rangle$ (often written $|a_i, b_j\rangle$ or $|a_i\rangle|b_j\rangle$).

$$|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B \quad \Rightarrow \quad |\psi\rangle = \sum_{i,j} c_{ij} |a_i\rangle \otimes |b_j\rangle$$

where $c_{ij} \in \mathbb{C}$ are complex coefficients satisfying $\sum_{i,j} |c_{ij}|^2 = 1$.

Dimension

If $\dim(\mathcal{H}_A) = n$ and $\dim(\mathcal{H}_B) = m$, then:

$$\dim(\mathcal{H}_A \otimes \mathcal{H}_B) = n \times m$$

Product States vs. Entangled States

Product States (Separable)

A state $|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$ is called a product state(or separable) if it can be written as:

$$|\psi\rangle = |\phi\rangle_A \otimes |\chi\rangle_B$$

where $|\phi\rangle_A \in \mathcal{H}_A$ and $|\chi\rangle_B \in \mathcal{H}_B$.

Example: Two spin-1/2 particles

The state $|\uparrow\rangle \otimes |\downarrow\rangle$ means particle 1 is spin-up and particle 2 is spin-down. This is a product state.

Entangled States

A state that cannot be written as a product is called entangled.

Example: Bell State (EPR Pair)

$$|\Phi^+\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow\uparrow\rangle + |\downarrow\downarrow\rangle\right)$$

This state cannot be factored into $|\phi\rangle_A \otimes |\chi\rangle_B$. Measuring one particle instantaneously determines the state of the other, regardless of distance—this is quantum entanglement.

Inner Product in Tensor Product Spaces

The inner product on $\mathcal{H}_A \otimes \mathcal{H}_B$ is defined by:

$$\langle \phi_1 \otimes \chi_1 | \phi_2 \otimes \chi_2 \rangle = \langle \phi_1 | \phi_2 \rangle_A \cdot \langle \chi_1 | \chi_2 \rangle_B$$

For general states:

$$\left\langle \sum_{i,j} c_{ij} |a_i, b_j\rangle \middle| \sum_{k,\ell} d_{k\ell} |a_k, b_\ell\rangle \right\rangle = \sum_{i,j,k,\ell} c_{ij}^* d_{k\ell} \langle a_i|a_k\rangle \langle b_j|b_\ell\rangle$$

Operators on Tensor Product Spaces

Product Operators

If $\hat{A}$ acts on $\mathcal{H}_A$ and $\hat{B}$ acts on $\mathcal{H}_B$, we can define:

$$(\hat{A} \otimes \hat{B})(|\phi\rangle \otimes |\chi\rangle) = (\hat{A}|\phi\rangle) \otimes (\hat{B}|\chi\rangle)$$

Example:

The operator $\hat{\sigma}_z \otimes \mathbb{I}$ measures the z-component of spin for particle 1 only, leaving particle 2 unaffected.

General Operators

A general operator on $\mathcal{H}_A \otimes \mathcal{H}_B$ can be expanded as:

$$\hat{O} = \sum_{\alpha,\beta} c_{\alpha\beta} \, \hat{A}_\alpha \otimes \hat{B}_\beta$$

where $\{\hat{A}_\alpha\}$ and $\{\hat{B}_\beta\}$ are operator bases for $\mathcal{H}_A$ and $\mathcal{H}_B$.

Partial Trace and Reduced Density Matrices

When we cannot observe subsystem B, we describe subsystem A using the reduced density matrix, obtained via the partial trace:

$$\hat{\rho}_A = \text{Tr}_B(\hat{\rho}_{AB})$$

For a pure state $|\psi\rangle = \sum_{ij} c_{ij} |a_i\rangle \otimes |b_j\rangle$:

$$\hat{\rho}_A = \text{Tr}_B(|\psi\rangle\langle\psi|) = \sum_{i,i'} \left(\sum_j c_{ij} c_{i'j}^*\right) |a_i\rangle\langle a_{i'}|$$

Key Insight:

Even if the composite system is in a pure state, the reduced density matrix $\hat{\rho}_A$ may describe a mixed state—this is a signature of entanglement!

Schmidt Decomposition

Any pure state $|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$ can be written in the Schmidt form:

$$|\psi\rangle = \sum_{i=1}^r \lambda_i |\alpha_i\rangle \otimes |\beta_i\rangle$$

where:

  • $\{|\alpha_i\rangle\}$ are orthonormal in $\mathcal{H}_A$
  • $\{|\beta_i\rangle\}$ are orthonormal in $\mathcal{H}_B$
  • $\lambda_i > 0$ are called Schmidt coefficients, with $\sum_i \lambda_i^2 = 1$
  • $r$ is the Schmidt rank (number of non-zero $\lambda_i$)

Entanglement Criterion:

  • $r = 1$: Product state (not entangled)
  • $r > 1$: Entangled state

Physical Examples

Example 1: Position and Spin

An electron's state space is $L^2(\mathbb{R}^3) \otimes \mathbb{C}^2$ (position ⊗ spin). A general state:

$$|\psi\rangle = \int d^3r \left[\psi_\uparrow(\vec{r}) |\vec{r}, \uparrow\rangle + \psi_\downarrow(\vec{r}) |\vec{r}, \downarrow\rangle\right]$$

Example 2: Two Particles

Two distinguishable particles: $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$

For identical particles, we must further impose symmetrization (bosons) or antisymmetrization (fermions).

Example 3: Quantum Computing

$n$ qubits live in $(\mathbb{C}^2)^{\otimes n}$, a $2^n$-dimensional Hilbert space. Entanglement between qubits is the resource for quantum computation!

Summary

  • ✓ Tensor products $\mathcal{H}_A \otimes \mathcal{H}_B$ describe composite quantum systems
  • ✓ Product states are separable; entangled states are not
  • ✓ Inner products and operators extend naturally to tensor spaces
  • ✓ Partial trace gives reduced density matrices for subsystems
  • ✓ Schmidt decomposition reveals entanglement structure
  • ✓ Essential for quantum information, many-body physics, and identical particles
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