Tensor Products and Entanglement

Introduction

In our previous lesson, we established a rigorous framework for describing single-particle states, accommodating both discrete and continuous spectra through the concept of the Rigged Hilbert Space. We saw that the position and momentum representations are simply different "views" of the same abstract state vector, connected by the Fourier transform. Now, we will expand our formalism to address a fundamental question: how do we describe systems composed of more than one particle?

This lesson will guide you through the construction of state spaces for composite quantum systems. You will learn that the correct mathematical tool for this is not a simple Cartesian product, as in classical mechanics, but the tensor product of the individual Hilbert spaces. This construction has profound physical consequences. It not only leads to an exponential increase in the size of the state space but also gives rise to entanglement—a uniquely quantum phenomenon where subsystems are correlated in ways that defy classical intuition. By the end of this lesson, you will be able to construct these composite spaces and formally distinguish between states that are entangled and those that are not.

The Tensor Product Postulate

When we combine two classical systems, the state of the composite system is just a pair of states from the individual phase spaces. For instance, the state of two particles is described by the set of their positions and momenta $(q_{1}, p_{1}, q_{2}, p_{2})$ , which lives in the Cartesian product of their individual phase spaces.

In quantum mechanics, this simple "listing" of states is insufficient because of the principle of superposition. If particle 1 can be in state $∣ ψ_{1} ⟩$ and particle 2 in $∣ ϕ_{1} ⟩$ , that's one possibility. But if particle 1 can also be in $∣ ψ_{2} ⟩$ and particle 2 in $∣ ϕ_{2} ⟩$ , the total system must also allow for superpositions like $c_{1} (∣ ψ_{1} ⟩, ∣ ϕ_{1} ⟩) + c_{2} (∣ ψ_{2} ⟩, ∣ ϕ_{2} ⟩)$ .

This leads to the fundamental postulate for composite systems:

The state space $H$ of a composite quantum system consisting of two subsystems with state spaces $H_{1}$ and $H_{2}$ is the tensor product $H = H_{1} \otimes H_{2}$ .

This is a crucial departure from the classical picture and from other simple ways of combining vector spaces, like the direct sum $H_{1} \oplus H_{2}$ . A state in the direct sum, like $(∣ ψ ⟩, 0)$ , would fail to assign any state to the second subsystem, contradicting the basic requirements of the theory.

The following video from MIT OpenCourseWare provides an excellent introduction to the tensor product, motivating its necessity and outlining its core properties.

18. Two State Systems (continued), Multiparticle States and Tensor Products

This lecture, "Two State Systems (continued), Multiparticle States and Tensor Products", by Professor Barton Zwiebach, builds the concept of the tensor product from the ground up.

First, watch the introduction to multi-particle states to understand why a new mathematical structure is needed. Next, focus on the section introducing the tensor product notation, v \otimes w, and the idea that the composite space allows for superpositions beyond simple pairs of states. Pay close attention to the axioms and properties of the tensor product, especially the linearity rules. These define how to manipulate elements in the new space. Finally, see how a basis is constructed for the tensor product space. This will clarify the most important consequence: the dimensions of the spaces multiply, \dim(\mathcal{H}_1 \otimes \mathcal{H}_2) = \dim(\mathcal{H}_1) \cdot \dim(\mathcal{H}_2).

The video provides the physical and operational intuition. For a more formal mathematical grounding, which your background will allow you to appreciate, we can define the algebraic tensor product more rigorously.

[PDF] 12. The formalism of quantum mechanics - OU Math Department

This document by Christian Remling provides a mathematically precise treatment of the tensor product.

First, read the paragraph explaining axiom (Comp) on page 148. This short section formally states the postulate and reiterates why the direct sum is not a viable alternative. Next, read the detailed discussion starting with "Let’s first discuss..." on pages 148-150. This section provides the rigorous definition of the algebraic tensor product as a quotient of a free vector space, which underpins the operational rules you saw in the video. It also introduces the universal property, which is central to the mathematical definition of a tensor product.

To summarize, a state in $H_{1} \otimes H_{2}$ is not just a pair $(∣ ψ ⟩, ∣ ϕ ⟩)$ . The elements of this new space are linear combinations of pure tensors of the form $∣ ψ ⟩ \otimes ∣ ϕ ⟩$ . A general vector $∣Ψ ⟩$ in the composite space is a superposition:

∣Ψ ⟩ = i, j \sum c_{ij} ∣ u_{i} ⟩ \otimes ∣ v_{j} ⟩

where ${∣ u_{i} ⟩}$ and ${∣ v_{j} ⟩}$ are bases for $H_{1}$ and $H_{2}$ , respectively. This exponential scaling of the state space is a key reason for the computational power of quantum systems.

This diagram illustrates the fundamental difference in how information capacity scales in classical versus quantum systems. Classical systems combine additively, while quantum systems combine via the tensor product, leading to an exponential growth in the dimension of the state space.

Structure of the Tensor Product Space

Now that we have a space, we need to define its essential structures: the inner product and the action of operators.

Inner Product

The inner product on $H_{1} \otimes H_{2}$ is defined by its action on pure tensors and then extended by (sesqui)linearity to all other vectors. For two pure tensors, we define:

⟨ ψ_{1} \otimes ϕ_{1} ∣ ψ_{2} \otimes ϕ_{2} ⟩ = ⟨ ψ_{1} ∣ ψ_{2} ⟩_{H_{1}} ⟨ ϕ_{1} ∣ ϕ_{2} ⟩_{H_{2}}

The inner product of two composite states is simply the product of the inner products of their constituent parts.

Operators

An observable $A^$ associated with the first subsystem must be "upgraded" to an operator that acts on the entire composite space $H_{1} \otimes H_{2}$ . This is done by taking its tensor product with the identity operator on the second space: $A^\mapsto A^\otimes I_{2}$ . Its action on a pure tensor is:

(A^\otimes I_{2}) (∣ ψ ⟩ \otimes ∣ ϕ ⟩) = (A^∣ ψ ⟩) \otimes ∣ ϕ ⟩

It acts on the first part and leaves the second part unchanged. Similarly, an operator $B^$ on the second subsystem becomes $I_{1} \otimes B^$ .

A crucial consequence of this definition is that observables pertaining to different subsystems commute:

[A^\otimes I_{2}, I_{1} \otimes B^] = 0

This makes sense physically: a measurement on one particle should not depend on the order in which you perform a measurement on another, spatially separated particle.

The following video lecture continues the discussion from the previous one, formalizing the inner product and the action of operators.

19. Multiparticle States and Tensor Products (continued)

This lecture, "Multiparticle States and Tensor Products (continued)", continues the formal development of composite systems.

Watch the segment from the beginning until the definition of the inner product. This section carefully defines the inner product for general superpositions of tensor product states.

18. Two State Systems (continued), Multiparticle States and Tensor Products

Returning to the previous video, we can now see how operators are defined and used.

Watch the segment on operators on the tensor product. This explains how to "upgrade" an operator from a single subsystem to the full space and proves that operators on different subsystems commute. Then, follow the concrete example of two spin-1/2 particles. This demonstrates the construction of the 4D composite space and shows how the total S_z operator, S_z^{\text{tot}} = S_z^{(1)} \otimes \mathbb{I} + \mathbb{I} \otimes S_z^{(2)}, acts on a general two-particle state.

For infinite-dimensional Hilbert spaces, there is a subtlety. The algebraic tensor product is not a complete space. The true Hilbert space tensor product is the completion of the algebraic one with respect to the norm induced by the inner product. This is a fine point, but one worth noting given your background.

[PDF] 12. The formalism of quantum mechanics - OU Math Department

This reading from Remling's notes formalizes the construction of the Hilbert space tensor product.

Read the section on page 150-151, starting from "If we now have...". This part explains how the inner product is properly defined and demonstrates through an example (\ell^2(Z) \otimes \ell^2(Z)) why the algebraic tensor product is not complete, motivating the formal definition of the Hilbert space tensor product as its completion.

Separable vs. Entangled States

The true richness of the tensor product structure becomes apparent when we classify the states it contains.

A state $∣Ψ ⟩ \in H_{1} \otimes H_{2}$ is called separable (or a product state) if it can be written as a single tensor product of states from the constituent subsystems:

∣Ψ ⟩ = ∣ ψ ⟩ \otimes ∣ ϕ ⟩, where ∣ ψ ⟩ \in H_{1}, ∣ ϕ ⟩ \in H_{2}

Physically, a separable state is one where each subsystem has its own definite quantum state. This is the closest quantum mechanics gets to the classical idea of independent systems.

However, most states in the tensor product space are not separable. A state that cannot be factored in this way is called entangled.

∣Ψ ⟩ = i, j \sum c_{ij} ∣ u_{i} ⟩ \otimes ∣ v_{j} ⟩ \neq = ∣ ψ ⟩ \otimes ∣ ϕ ⟩ for any ∣ ψ ⟩, ∣ ϕ ⟩

In an entangled state, the composite system as a whole is in a definite pure state, but the individual subsystems are not. Their properties are inextricably linked. For example, in the famous singlet state of two spin-1/2 particles, $∣ Ψ^{-} ⟩ = \frac{1}{2} (∣ ↑ ⟩_{1} \otimes ∣ ↓ ⟩_{2} - ∣ ↓ ⟩_{1} \otimes ∣ ↑ ⟩_{2})$ , neither particle has a definite spin orientation, but if you measure particle 1 to be spin-up, you are guaranteed to measure particle 2 to be spin-down, no matter how far apart they are.

19. Multiparticle States and Tensor Products (continued)

Professor Zwiebach provides a clear definition of entanglement and a practical test for it in a simple system.

Watch the segment on entangled states. It formally defines entanglement as non-separability and derives a simple, powerful criterion for a 2-qubit system: a state |\Psi\rangle = \sum c_{ij} |e_i\rangle \otimes |f_j\rangle is separable if and only if the determinant of the coefficient matrix C = (c_{ij}) is zero.

The Schmidt Decomposition: A General Test for Entanglement

The determinant test is specific to $2 \times 2$ systems. A more general and powerful tool for analyzing bipartite pure states is the Schmidt decomposition. It is a result analogous to the singular value decomposition (SVD) for matrices.

Theorem (Schmidt Decomposition): Any pure state $∣ ψ ⟩$ in a composite Hilbert space $H_{A} \otimes H_{B}$ can be written as:

∣ ψ ⟩ = i = 1 \sum k d_{i} ∣ u_{i} ⟩_{A} \otimes ∣ v_{i} ⟩_{B}

where ${∣ u_{i} ⟩_{A}}$ and ${∣ v_{i} ⟩_{B}}$ are orthonormal sets of states in $H_{A}$ and $H_{B}$ respectively, and the $d_{i}$ are positive real coefficients called Schmidt coefficients satisfying $\sum_{i} d_{i}^{2} = 1$ . The number of non-zero coefficients, $k$ , is called the Schmidt rank.

The Schmidt decomposition expresses a bipartite pure state as a sum over a single index, linking orthonormal bases in each subsystem. The Schmidt coefficients $d_i$ and the Schmidt rank $k$ are key to understanding the state's entanglement.

The Schmidt decomposition provides an elegant and definitive test for entanglement:

A state $∣ ψ ⟩$ is separable if and only if its Schmidt rank is 1.
A state $∣ ψ ⟩$ is entangled if and only if its Schmidt rank is greater than 1.

The Schmidt coefficients not only tell us if a state is entangled, but also how much. A state with Schmidt coefficients $(1/ 2, 1/ 2, 0, \dots)$ is considered "maximally entangled", while a state with coefficients $(0.99, 0.1, 0, \dots)$ is only weakly entangled.

[PDF] An Introduction to Quantum Entanglement - Ion Nechita

This text by Ion Nechita provides concise, formal definitions that nicely summarize the concepts of entanglement and its characterization.

Read Section 2.1, Pure state entanglement (p. 11-12). This section gives the formal definition of separability and provides examples, including the singlet state. Next, read Section 2.2, Schmidt decomposition (p. 12) and Section 2.3, Measure of entanglement (p. 12). These sections introduce the Schmidt decomposition theorem and directly relate the Schmidt coefficients to the separability of the state. The concept of entropy of entanglement is also introduced, providing a quantitative measure based on the Schmidt coefficients.

Conclusion

This lesson has laid the mathematical and conceptual groundwork for describing multi-particle quantum systems. We have moved from single-particle states to the vast, complex landscapes of composite state spaces.

Key Takeaways:

The state space of a composite system is the tensor product of the individual Hilbert spaces, $H = H_{1} \otimes H_{2}$ .
The dimension of the composite space grows exponentially: $dim (H) = dim (H_{1}) \cdot dim (H_{2})$ .
Observables on a subsystem are represented by operators of the form $A^\otimes I$ and commute with observables on other subsystems.
A state is separable if it can be written as a single product state, $∣ ψ ⟩_{1} \otimes ∣ ϕ ⟩_{2}$ .
A state is entangled if it is not separable. Entangled states exhibit non-classical correlations, and their subsystems do not possess definite individual states.
The Schmidt decomposition provides a general method to analyze a bipartite pure state, with the Schmidt rank serving as a definitive test for entanglement.

We have focused here on pure states, which are described by vectors in a Hilbert space. However, what if a system is entangled with an environment that we can't access? Or what if we only have statistical knowledge about how a state was prepared? To handle these situations, we need a more general object. In our next lesson, we will introduce the density matrix, which allows us to describe both pure and "mixed" states and provides a complete framework for describing any quantum system.

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