Quantum Mathematics Learning Guide 3: Tensor Products, Qubits and Entanglement

The mathematics of a single qubit is two-dimensional. The mathematics of several qubits is built by tensor products—and this is where quantum state spaces begin to grow in a way that has no simple classical analogue.

Guide 1 introduced complex state vectors. Guide 2 introduced operators and measurement. This third guide asks a new question: how do we represent several quantum systems together? The answer is the tensor product.

Tensor products are not merely a notation trick. They determine the dimension of composite state spaces, define how local operations extend to larger systems and make entanglement mathematically possible. A learner who understands the tensor product can see exactly where multi-qubit quantum mathematics becomes richer than a collection of independent single-qubit calculations.

Single system: vector space. Composite system: tensor-product space. Entanglement: a joint state that cannot be separated into individual state vectors.

1. Why ordinary vector addition is not enough for composite systems

If system A has state space of dimension m and system B has state space of dimension n, the joint system has dimension mn, not m+n. The mathematical construction that produces this joint space is the tensor product.

For two qubits, each individual state space has dimension two. The joint space therefore has dimension 2×2=4. A convenient basis is

• |00⟩
• |01⟩
• |10⟩
• |11⟩

These are shorthand for tensor products such as |0⟩⊗|1⟩ = |01⟩.

2. Computing a tensor product

If a=[a₁,a₂]ᵀ and b=[b₁,b₂]ᵀ, then

a⊗b = [a₁b₁, a₁b₂, a₂b₁, a₂b₂]ᵀ.

The ordering convention matters. In this guide we use the standard computational ordering |00⟩,|01⟩,|10⟩,|11⟩.

Worked example: |0⟩ tensor |1⟩

|0⟩=[1,0]ᵀ and |1⟩=[0,1]ᵀ. Therefore

|0⟩⊗|1⟩ = [0,1,0,0]ᵀ = |01⟩.

Worked example: two superposed qubits

Let |+⟩=(|0⟩+|1⟩)/√2. Then

|+⟩⊗|+⟩ = (1/2)(|00⟩+|01⟩+|10⟩+|11⟩).

Each computational-basis outcome has amplitude 1/2 and therefore probability 1/4.

3. The general two-qubit pure state

A general pure state of two qubits can be written

|ψ⟩ = a|00⟩ + b|01⟩ + c|10⟩ + d|11⟩

with complex amplitudes satisfying

|a|²+|b|²+|c|²+|d|²=1.

The measurement probabilities in the computational basis are the four squared magnitudes. But, just as in the one-qubit case, these probabilities alone do not contain all the phase information stored in the state.

4. Product states

A two-qubit state is a product state if it can be written as

|ψ⟩ = |u⟩⊗|v⟩

for single-qubit states |u⟩ and |v⟩. Product states describe joint pure states that are separable into individual pure-state descriptions.

Let |u⟩=α|0⟩+β|1⟩ and |v⟩=γ|0⟩+δ|1⟩. Then

|u⟩⊗|v⟩ = αγ|00⟩ + αδ|01⟩ + βγ|10⟩ + βδ|11⟩.

The four amplitudes therefore have a special factorised structure.

5. A fast separability test for two pure qubits

For a two-qubit pure state a|00⟩+b|01⟩+c|10⟩+d|11⟩, arrange the coefficients into

M=[[a,b],[c,d]].

The state is a product state exactly when this coefficient matrix has rank one, equivalently when

ad-bc=0.

If ad-bc ≠ 0, the pure state is entangled.

Worked example: a separable state

Consider (1/2)(|00⟩+|01⟩+|10⟩+|11⟩). Here a=b=c=d=1/2, so ad-bc=1/4-1/4=0. The state is separable; indeed it equals |+⟩⊗|+⟩.

Worked example: an entangled state

Consider

|Φ+⟩=(|00⟩+|11⟩)/√2.

Here a=d=1/√2 and b=c=0. Thus ad-bc=1/2, which is non-zero. The state cannot be factored into a tensor product of two single-qubit pure states.

6. What entanglement means mathematically

Entanglement is often described dramatically as “spooky action” or as particles “knowing” what happens to each other. Those phrases can distract from the mathematical definition.

For pure states, the clean statement is:

A composite pure state is entangled if it cannot be written as a tensor product of pure states for its subsystems.

This non-factorisation creates joint correlations that cannot be described by assigning each subsystem its own independent pure-state vector.

IBM Quantum Learning develops composite systems using tensor-product state spaces and uses this framework to distinguish product states from entangled states. See IBM Quantum Learning: Multiple Systems.

7. Bell states

Four standard maximally entangled two-qubit states are

• |Φ+⟩=(|00⟩+|11⟩)/√2
• |Φ−⟩=(|00⟩-|11⟩)/√2
• |Ψ+⟩=(|01⟩+|10⟩)/√2
• |Ψ−⟩=(|01⟩-|10⟩)/√2

They form an orthonormal basis for the two-qubit state space. Notice how phase again matters: the plus and minus versions have the same computational-basis probabilities but are different quantum states.

8. Measurement correlations in a Bell state

For |Φ+⟩=(|00⟩+|11⟩)/√2, a computational-basis measurement of both qubits gives only 00 or 11, each with probability 1/2. Outcomes 01 and 10 have probability zero.

Each individual qubit, considered only through those measurement statistics, looks random: either local result is equally likely to be 0 or 1. Yet the pair is perfectly correlated in this basis.

This is a useful reminder that “random individually” does not mean “independent jointly.” Probability theory already distinguishes independence from marginal randomness; quantum entanglement adds a stronger structure because the joint state itself need not factor into subsystem states.

9. Conditional statements after measurement

Suppose the two-qubit system is in |Φ+⟩ and the first qubit is measured in the computational basis. If the outcome is 0, the corresponding joint branch is |00⟩; if the outcome is 1, the branch is |11⟩.

It is important not to turn this into a faster-than-light communication claim. Correlation and communication are different concepts. Local outcomes remain probabilistic, and quantum theory’s no-signalling structure prevents entanglement alone from being used to transmit controllable information instantaneously.

10. Local operators on a composite system

If operator A acts on the first subsystem and operator B acts on the second, the joint operator is A⊗B.

To apply X only to the first qubit of a two-qubit system, use X⊗I. To apply X only to the second, use I⊗X.

Worked example: X on the first qubit

(X⊗I)|01⟩ = |11⟩, because the first bit flips from 0 to 1 while the second remains 1.

Similarly, (I⊗X)|01⟩=|00⟩.

11. Tensor products of matrices

The tensor product also combines operators. If

A=[[a,b],[c,d]] and B is a matrix, then

A⊗B = [[aB,bB],[cB,dB]].

Each scalar entry of A is replaced by a scaled copy of B. This construction is called the Kronecker product when working with matrix representations.

Worked example: Z tensor I

With Z=[[1,0],[0,-1]] and I=[[1,0],[0,1]],

Z⊗I = diag(1,1,-1,-1).

It multiplies basis states beginning with 0 by +1 and those beginning with 1 by -1.

12. Controlled operations

Some two-qubit operations cannot be decomposed into independent single-qubit operations. The controlled-NOT gate, CNOT, is a standard example. In the computational basis it maps

• |00⟩→|00⟩
• |01⟩→|01⟩
• |10⟩→|11⟩
• |11⟩→|10⟩

The first qubit acts as a control; when it is 1, the second qubit is flipped.

Creating a Bell state mathematically

Start from |00⟩. Apply H to the first qubit:

(H⊗I)|00⟩ = (|00⟩+|10⟩)/√2.

Now apply CNOT:

CNOT[(|00⟩+|10⟩)/√2] = (|00⟩+|11⟩)/√2 = |Φ+⟩.

A product state has become entangled through a joint unitary operation.

13. Dimension growth

An n-qubit pure state is represented by a vector with 2ⁿ complex amplitudes before accounting for normalisation and global phase constraints. The basis states correspond to all binary strings of length n.

• 1 qubit → 2 amplitudes
• 2 qubits → 4 amplitudes
• 3 qubits → 8 amplitudes
• 10 qubits → 1,024 amplitudes
• 20 qubits → 1,048,576 amplitudes

This exponential state-space growth helps explain both the expressive power of multi-qubit quantum systems and the difficulty of simulating arbitrary large pure quantum states on classical computers. It does not mean a quantum computer automatically reveals all 2ⁿ amplitudes in one measurement.

14. Why “quantum parallelism” needs careful wording

A quantum state can contain amplitudes across many basis states, and a unitary operation acts linearly on the whole vector. But measurement does not simply print every branch. Quantum algorithms are designed so that amplitudes interfere, increasing useful outcomes and suppressing others before measurement.

The mathematical resource is therefore not “try every answer and read them all.” It is controlled manipulation of a high-dimensional amplitude structure.

15. Schmidt decomposition: a deeper view of bipartite pure states

For a pure state of two finite-dimensional subsystems, there exist orthonormal sets {|uₖ⟩} and {|vₖ⟩} such that

|ψ⟩ = Σₖ sₖ |uₖ⟩⊗|vₖ⟩,

where sₖ≥0 and Σₖ sₖ²=1. This is the Schmidt decomposition.

A bipartite pure state is separable exactly when only one Schmidt coefficient is non-zero. More than one non-zero Schmidt coefficient signals entanglement.

This result is a powerful example of how the right basis can reveal hidden structure. An apparently complicated coefficient table can be reorganised into a canonical sum of paired basis states.

16. Local descriptions and mixed states

Entangled pure states expose a limitation of state vectors for subsystems. If the joint system is in |Φ+⟩, neither qubit by itself can be assigned a pure state vector that reproduces all local statistics. The correct local description uses a density operator.

For |Φ+⟩, each individual qubit has reduced density matrix I/2, corresponding to a maximally mixed local state. The joint system is pure while each part is mixed.

This series does not develop density matrices fully, but the point is important: entanglement forces us to distinguish the state of the whole from the state description available to each part.

17. Entanglement is not ordinary correlation

A classical pair of bits can also be perfectly correlated—for example, a random process may produce 00 half the time and 11 half the time. That classical mixture and the Bell state |Φ+⟩ have the same computational-basis outcome probabilities.

They are nevertheless different physical states because the Bell state preserves coherent phase relationships that become visible under other measurement bases. The classical mixture does not.

This is another example of the principle from Guide 1: matching probabilities in one basis does not imply identical quantum states.

18. Common misconception: entanglement means each particle has the same hidden value

That description is too classical. Entangled states are defined by the non-factorisable joint state, and experiments involving different measurement settings reveal correlations that cannot generally be reproduced by simple local hidden-variable models. The full physical story belongs to quantum foundations; the mathematical starting point is the tensor-product structure and the state’s failure to factor.

19. Common misconception: tensor product means multiplication of probabilities

No. Tensor products combine state spaces and amplitudes. For independent product states, some resulting probabilities factor, but the underlying operation is on vectors. Entangled states demonstrate why reducing the construction to probability multiplication misses the essential structure.

20. Worked synthesis problem

Consider

|ψ⟩=(1/√3)|00⟩+(1/√3)|01⟩+(1/√3)|11⟩.

Step 1: Check normalisation. Three amplitudes each have squared magnitude 1/3, so the total is one.

Step 2: Measurement probabilities. P(00)=P(01)=P(11)=1/3 and P(10)=0.

Step 3: Test separability. The coefficient matrix is [[1/√3,1/√3],[0,1/√3]]. Its determinant is 1/3, non-zero. Therefore the state is entangled.

Step 4: Interpret. The state’s joint information cannot be represented as one pure state for qubit A tensor one pure state for qubit B.

21. Practice set

1. Compute |1⟩⊗|0⟩ as a four-component vector.
2. Expand |+⟩⊗|0⟩.
3. Show that |+⟩⊗|+⟩ is normalised.
4. Determine whether (|00⟩+|11⟩)/√2 is separable.
5. Determine whether (|00⟩+|01⟩)/√2 is separable.
6. Apply X⊗I to |10⟩.
7. Apply I⊗X to |10⟩.
8. List the four Bell states.
9. How many amplitudes are required for a general 6-qubit pure state vector?
10. Explain why equal computational-basis probabilities do not make a Bell state identical to a classical correlated mixture.

Answers

1. [0,0,1,0]ᵀ=|10⟩.
2. (|00⟩+|10⟩)/√2.
3. Four amplitudes of magnitude 1/2 give total probability one.
4. No; it is entangled.
5. Yes; it equals |0⟩⊗|+⟩.
6. |00⟩.
7. |11⟩.
8. |Φ+⟩,|Φ−⟩,|Ψ+⟩,|Ψ−⟩.
9. 2⁶=64.
10. The Bell state contains coherent relative phase and produces different correlations in other bases.

22. Series navigation

• Guide 1: Complex Numbers, State Vectors and Probability Amplitudes
• Guide 2: Matrices, Operators, Eigenvalues and Measurement
• Guide 3: Tensor Products, Qubits and Entanglement
• Guide 4: Unitary Evolution, Schrödinger’s Equation and Quantum Dynamics
• Return to the BTT Mathematics Learning Hub

Educational note: this guide uses finite-dimensional qubit systems to teach tensor-product structure. Quantum mechanics also describes systems with larger or infinite-dimensional state spaces, identical particles, continuous variables and quantum fields.