The tensor product: putting qubits together

The joint state of independent qubits is found by multiplying their amplitudes. Learn to read bitstrings correctly and see why an entangled state cannot be written as any product.

The joint state of two qubits

Let the state of q0 be (a₀, a₁) and that of q1 be (b₀, b₁). If the two qubits are independent, the four amplitudes of the joint state are products: b₀·a₀ for |00⟩, b₀·a₁ for |01⟩, b₁·a₀ for |10⟩ and b₁·a₁ for |11⟩. This operation is called the tensor product and is written ⊗.

It is the amplitude version of the rule from the probability lesson that independent events multiply. Multiply the amplitudes and square them, and you get the product of the probabilities.

Read bitstrings from the right

On this site and in Qiskit, bitstrings are read from right to left: the rightmost bit is q0, the one to its left q1. The result 01 means “q1 = 0, q0 = 1”.

Example: apply X to q0 and H to q1. Then q0 = (0, 1) and q1 = (0.707, 0.707). The products are 0.707 for |01⟩ and |11⟩ and 0 for the others. Measured, only 01 and 11 appear: the right bit is always 1, the left one is a coin toss.

Entanglement: a state that is not a product

In the Bell state only |00⟩ and |11⟩ have amplitudes, both 0.707. Try to write it as a product of two separate qubits. If b₀·a₀ for |00⟩ is not zero, then a₀ and b₀ are not zero; if b₁·a₁ for |11⟩ is not zero, then a₁ and b₁ are not zero. But then b₀·a₁, the amplitude of |01⟩, cannot be zero either. In the Bell state it is. Contradiction!

So the Bell state cannot be written as any product: the qubits have no separate states of their own, only a joint one. That is the mathematical definition of entanglement. Quantum computers draw their power from being able to use such non-product states.

Try it yourself

In the product state example see that only 01 and 11 appear. Then switch to the Bell example and open the state vector: |00⟩ and |11⟩ are 0.707, |01⟩ and |10⟩ are 0. There is no way to write these four numbers as a product of two qubits.

Open the circuit in your browser

The circuit playground is in Turkish. Button guide: “1000 kez ölç” = measure 1000 times, “Durum vektörünü göster” = show the state vector, “Bu devrenin Qiskit kodu” = this circuit’s Qiskit code, “Geri al” = undo, “Temizle” = clear. Click a gate in the circuit to remove it.

Key points

The joint state of independent qubits is the tensor product (⊗) of their amplitudes.
Bitstrings are read from the right: the rightmost bit is q0.
A state that cannot be written as a product is called entangled.
The Bell state is entangled: the qubits have no separate states.

Test yourself

If q0 = |1⟩ and q1 = |0⟩, which bitstring do the two qubits give when measured? (The rightmost bit is q0.)

Show the answer

01. Written left to right as q1 q0: q1 = 0, q0 = 1, so 01.

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