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The Two Qubit States and Their Transformation

The Two Qubit States and Their Transformation

Get introduced to the two-qubit states and their transformation.

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Two qubit states

Let’s say we have two qubits. We can call them a|a\rangle and b|b\rangle. Each of the two qubits has its own probability amplitudes: a=a00+a11=[a0a1]|a\rangle=a_0|0\rangle+a_1|1\rangle=\begin{bmatrix}a_0 \\ a_1\end{bmatrix} and b=b00+b11=[b0b1]|b\rangle=b_0|0\rangle+b_1|1\rangle=\begin{bmatrix}b_0 \\ b_1\end{bmatrix}. When we look at these two qubits concurrently, there are four different combinations of the basis states. Each of these combinations has its probability amplitude. These are the products of the probability amplitudes of the two corresponding states.

  • a00b00a_0|0\rangle b_0 |0\rangle
  • a00b11a_0|0\rangle b_1 |1\rangle
  • a11b00a_1|1\rangle b_0 |0\rangle
  • a11b11a_1|1\rangle b_1 |1\rangle

These four states form a quantum system on their own. Therefore, we can represent them in a single equation. While we are free to choose an arbitrary name for the state, we use ab|ab\rangle because this state is the collective quantum state of a|a\rangle and b|b\rangle.

ab=ab=a0b000+a0b101+a1b010+a1b111|ab\rangle=|a\rangle\otimes|b\rangle=a_0 b_0|0\rangle|0\rangle+a_0 b_1|0\rangle|1\rangle+a_1 b_0|1\rangle|0\rangle+a_1 b_1|1\rangle|1\rangle

In this equation, ab|ab\rangle is an arbitrary name. The last term is the four combinations reordered to have the amplitudes at the beginning. But what does ab|a\rangle\otimes|b\rangle mean?

The term ab|a\rangle\otimes|b\rangle is the tensor product of the two vectors a|a\rangle and b|b\rangle.

The tensor product (denoted by the symbol \otimes) is the mathematical way of calculating the amplitudes. In general, the tensor product of two vectors vv and ww is a vector of all combinations. Like this:

With v=[v0v1 ...