If the first qubit is in the state √2/√3√|0>+1/√3|1> and the second qubit is in the state 1/√3|0>+√2/√3|1>, what is the state of the composite system?
To find the state of the composite system, we need to take the tensor product of the individual qubit states.
The tensor product of two qubits is represented by the symbol ⊗. So, the state of the composite system can be found by taking the tensor product of the two qubits.
Let's calculate it step-by-step:
First qubit state: √2/√3√|0> + 1/√3|1>
Second qubit state: 1/√3|0> + √2/√3|1>
To take the tensor product, multiply each term of the first qubit state by each term of the second qubit state:
= (√2/√3√|0> ⊗ 1/√3|0>) + (√2/√3√|0> ⊗ √2/√3|1>) + (1/√3|1> ⊗ 1/√3|0>) + (1/√3|1> ⊗ √2/√3|1>)
Now, simplify each term by distributing:
= (√2/√3 * 1/√3) (√|0> ⊗ |0>) + (√2/√3 * √2/√3) (√|0> ⊗ √|1>) + (1/√3 * 1/√3) (|1> ⊗ |0>) + (1/√3 * √2/√3) (|1> ⊗ √|1>)
= (√2/3) (√|0> ⊗ |0>) + (2/3) (√|0> ⊗ √|1>) + (1/3) (|1> ⊗ |0>) + (1/3) (√|1> ⊗ √|1>)
= (√2/3 √|00>) + (2/3 √|01>) + (1/3 |10>) + (1/3 √|11>)
Therefore, the state of the composite system is:
√2/3 √|00> + 2/3 √|01> + 1/3 |10> + 1/3 √|11>