mathematics

Tensor product of Hilbert spaces

Tensor product of Hilbert spaces is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Tensor product of Hilbert spaces rather than just read about it. In short: In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product.

Key takeaways

Tensor product of Hilbert spaces belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Tensor product of Hilbert spaces to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tensor product of Hilbert spaces from memory before moving on to harder problems.

Reference excerpt

In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.

Definition Since Hilbert spaces have inner products, one would like to introduce an inner product, and thereby a topology, on the tensor product that arises naturally from the inner products on the factors. Let H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} be two Hilbert spaces with inner products ⟨ ⋅ , ⋅ ⟩ 1 {\displaystyle \langle \cdot ,\cdot \rangle _{1}} and ⟨ ⋅ , ⋅ ⟩ 2 , {\displaystyle \langle \cdot ,\cdot \rangle _{2},} respectively. Construct the tensor product of H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner product space by defining

⟨ ϕ 1 ⊗ ϕ 2 , ψ 1 ⊗ ψ 2 ⟩ = ⟨ ϕ 1 , ψ 1 ⟩ 1 ⟨ ϕ 2 , ψ 2 ⟩ 2 {\displaystyle \left\langle \phi _{1}\otimes \phi _{2},\psi _{1}\otimes \psi _{2}\right\rangle =\left\langle \phi _{1},\psi _{1}\right\rangle _{1}\,\left\langle \phi _{2},\psi _{2}\right\rangle _{2}}

for all ϕ 1 , ψ 1 ∈ H 1 {\displaystyle \phi _{1},\psi _{1}\in H_{1}} ϕ 2 , ψ 2 ∈ H 2 {\displaystyle \phi _{2},\psi _{2}\in H_{2}} and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on H 1 × H 2 {\displaystyle H_{1}\times H_{2}} and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of H 1 {\displaystyle H_{1}} and H 2 . {\displaystyle H_{2}.}

Explicit construction The tensor product can also be defined without appealing to the metric space completion. If H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} are two Hilbert spaces, one associates to every simple tensor product x 1 ⊗ x 2 {\displaystyle x_{1}\otimes x_{2}} the rank one operator from H 1 ∗ {\displaystyle H_{1}^{*}} to H 2 {\displaystyle H_{2}} that maps a given x ∗ ∈ H 1 ∗ {\displaystyle x^{*}\in H_{1}^{*}} as

x ∗ ↦ x ∗ ( x 1 ) x 2 . {\displaystyle x^{*}\mapsto x^{*}(x_{1})\,x_{2}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Tensor product of Hilbert spaces

Start with the simplest possible case. Write down what Tensor product of Hilbert spaces claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tensor product of Hilbert spaces before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tensor product of Hilbert spaces ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tensor product of Hilbert spaces

In research: Tensor product of Hilbert spaces appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Tensor product of Hilbert spaces in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Tensor product of Hilbert spaces is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functional analysis, Hilbert spaces, Linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Tensor product of Hilbert spaces outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Tensor product of Hilbert spaces” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Tensor product of Hilbert spaces in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tensor product of Hilbert spaces means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Tensor product of Hilbert spaces out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tensor product of Hilbert spaces in simple terms?

Why does Tensor product of Hilbert spaces matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tensor product of Hilbert spaces?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tensor product of Hilbert spaces.