Tensor Product

Let

∣v,w⟩≜∣v⟩⊗∣w⟩\left\lvert v, w \right\rangle \triangleq \left\lvert v \right\rangle\otimes\left\lvert w \right\rangle

Given Vector Space HAH_A and HBH_B, the tensor product HA⊗HBH_A\otimes H_B is a new vector space

Properties

  1. ∣v⟩∈HA, ∣w⟩∈HB  ⟹  ∣v⟩⊗∣w⟩∈HAB|v\rangle \in {H}_A, \ |w\rangle \in {H}_B \implies |v\rangle \otimes |w\rangle \in {H}_{AB}

  2. Distributive law:

∣v⟩⊗(c0∣w0⟩+c1∣w1⟩)=c0(∣v⟩⊗∣w0⟩)+c1(∣v⟩⊗∣w1⟩)(c0∣v0⟩+c1∣v1⟩)⊗∣w⟩=c0(∣v0⟩⊗∣w⟩)+c1(∣v1⟩⊗∣w⟩) \begin{align*} |v\rangle \otimes (c_0 |w_0\rangle + c_1 |w_1\rangle) & = c_0 (|v\rangle \otimes |w_0\rangle) + c_1 (|v\rangle \otimes |w_1\rangle) \\ (c_0 |v_0\rangle + c_1 |v_1\rangle) \otimes |w\rangle & = c_0 (|v_0\rangle \otimes |w\rangle) + c_1 (|v_1\rangle \otimes |w\rangle) \end{align*}
  1. Inner product
⟨v1,w1∣v2,w2⟩=(⟨v1∣⊗⟨w1∣)(∣v2⟩⊗∣w2⟩)=⟨v1∣v2⟩⟨w1∣w2⟩ \begin{align*} \langle v_1, w_1 \mid v_2, w_2 \rangle & = (\langle v_1 | \otimes \langle w_1 |)(|v_2\rangle \otimes |w_2\rangle) \\ & = \langle v_1 | v_2 \rangle \langle w_1 | w_2 \rangle \end{align*}
  1. ∀ ∣ψ⟩∈HAB, ∣ψ⟩=∑μcμ∣vμ⟩⊗∣wμ⟩\forall \, |\psi\rangle \in \mathcal{H}_{AB}, \ |\psi\rangle = \sum_{\mu} c_\mu |v_\mu\rangle \otimes |w_\mu\rangle
    for some sequence of (cμ,∣vμ⟩,∣wμ⟩)(c_\mu, |v_\mu\rangle, |w_\mu\rangle)

Note

  1. ∣v⟩⊗∣w⟩≠∣w⟩⊗∣v⟩\left\lvert v \right\rangle\otimes\left\lvert w \right\rangle\neq\left\lvert w \right\rangle\otimes\left\lvert v \right\rangle

  2. a∣w⟩⊗∣v⟩=∣w⟩⊗a∣v⟩=a(∣w⟩⊗∣v⟩)a\left\lvert w \right\rangle\otimes\left\lvert v \right\rangle=\left\lvert w \right\rangle\otimes a\left\lvert v \right\rangle=a(\left\lvert w \right\rangle\otimes\left\lvert v \right\rangle)

  3. 0⊗∣v⟩=∣w⟩⊗0=00\otimes\left\lvert v \right\rangle=\left\lvert w \right\rangle\otimes 0 = 0

  4. ⟨vi,wj∣vk,wl⟩=⟨vi∣vk⟩⟨wj∣wl⟩=δikδjl\left\langle v_i, w_{j} | v_k, w_l \right\rangle=\left\langle v_i | v_k \right\rangle\left\langle w_j | w_l \right\rangle=\delta_{ik}\delta_{jl}

The dimensions are multiplied

dim(HA⊗HB)=dim(HA)×dim(HB)dim(H_A\otimes H_B)=dim(H_A)\times dim(H_B) dim⁡(C2⊗C2⊗⋯⊗C2⏟n)=2n(dim of n-qubit Hilbert space)\dim(\underbrace{\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \cdots \otimes \mathbb{C}^2}_{n}) = 2^n \quad \text{(dim of } n\text{-qubit Hilbert space)} {∣vi⟩⊗∣wj⟩;1≤i≤dim(HA),1≤j≤dim(HB)}\{\left\lvert v_i \right\rangle\otimes\left\lvert w_j \right\rangle; 1\leq i \leq dim(H_A), 1\leq j \leq dim(H_B)\}

is an orthonormal basis for HA⊗HBH_A\otimes H_B.