1] Given qubit 1 in the state 35|0>+45|1> and qubit 2 in the state 12√|0>+i2√|1>, what is the state of the composite system?
in form of a│00> + b│01> + c│10> + d│11>
find a b c d.
2] Which of the following states is entangled?
a) |++>
b) 45|01>−35|11>]
c) 1/√3|01> +1/√3|10>−1/3|11>
d) 1+i/√22|++> − 1+i/√22|+−>
+ 1−i/√22|−+> − 1−i/√22|−−>
To find the state of the composite system in question 1, we need to multiply the states of the individual qubits together. Let's start with the first qubit:
Qubit 1: 35|0> + 45|1>
Now, let's move on to the second qubit:
Qubit 2: 12√|0> + i2√|1>
To find the state of the composite system, we need to multiply these two states together. Here's how to do it:
a) Calculate the outer product of the two qubits:
(35|0> + 45|1>) ⊗ (12√|0> + i2√|1>)
b) Distribute the multiplication across all terms:
35 * 12√|00> + 35 * i2√|01> + 45 * 12√|10> + 45 * i2√|11>
c) Simplify the expressions:
420√|00> + 35i2√|01> + 540√|10> + 45i2√|11>
The state of the composite system can be written as:
a│00> + b│01> + c│10> + d│11>
Where:
a = 420√
b = 35i2√
c = 540√
d = 45i2√
Answer:
a = 420√
b = 35i2√
c = 540√
d = 45i2√
Now, let's move on to question 2. To determine which of the given states is entangled, we need to check if the states can be expressed as a pure tensor product of individual qubit states.
a) |++>
This state can be written as |+> ⊗ |+>, which is not an entangled state.
b) 45|01>−35|11>
This state cannot be written as the tensor product of two individual qubit states, so it is an entangled state.
c) 1/√3|01> +1/√3|10>−1/3|11>
This state also cannot be expressed as the tensor product of two individual qubit states, so it is an entangled state.
d) 1+i/√22|++> − 1+i/√22|+−> + 1−i/√22|−+> − 1−i/√22|−−>
This state can be expressed as (1+i/√22)(|+⟩ |+⟩) − (1+i/√22)(|+⟩ |−⟩) + (1−i/√22)(|−⟩ |+⟩) − (1−i/√22)(|−⟩ |−⟩), which is not an entangled state.
Answer:
The entangled states are:
b) 45|01>−35|11>
c) 1/√3|01> +1/√3|10>−1/3|11>