Suppose we have two qubits in the state α|00⟩+β|11⟩.

(a) If we measure the first qubit in the sign basis, what is the probability of seeing a +?

(b) What is the resulting state of the second qubit in that case? Use a and b to denote α and β respectively.(b) What is the resulting state of the second qubit in that case? Use a and b to denote α and β respectively.

(a) 1/2

pls answer part c

(a) is 1 not 1/2

1/2 is correct

To answer these questions, we will use the concept of measuring qubits in different bases and the resulting state after measurement.

(a) To find the probability of seeing a + when measuring the first qubit in the sign basis, we need to determine the coefficients of the resulting state after measurement.

In the sign basis, the states |+⟩ and |-⟩ are defined as:
|+⟩ = 1/√2 (|0⟩ + |1⟩)
|-⟩ = 1/√2 (|0⟩ - |1⟩)

Given the initial state α|00⟩ + β|11⟩, we can rewrite it as:
α|0⟩ ⊗ |0⟩ + β|1⟩ ⊗ |1⟩

To measure the first qubit in the sign basis, we need to calculate the projection of the initial state onto the |+⟩ basis. This can be done by taking the inner product of the initial state with the |+⟩ state.

Let's denote the resulting state after measurement as |m⟩, which can be either |+⟩ or |-⟩. The probability of observing |+⟩ is the square of the absolute value of the inner product of |m⟩ and |+⟩.

Performing the inner product:
<m|+⟩ = (α|0⟩ ⊗ |0⟩ + β|1⟩ ⊗ |1⟩) · (1/√2 (|0⟩ + |1⟩))
= α/√2 (<0|⊗<0| + <1|⊗<1|) · (|0⟩ + |1⟩)
= α/√2 (<0|0⟩ + <1|1⟩)
= α/√2

The probability of observing |+⟩ is |α/√2|^2, which simplifies to |α|^2/2. Therefore, the probability of seeing a + is |α|^2/2.

(b) After measuring the first qubit in the sign basis and finding |+⟩, the resulting state of the second qubit will be determined. To analyze this, we need to calculate the conditional state given that the observed outcome is |+⟩.

The conditional state can be obtained by normalizing the projection of the initial state onto the subspace spanned by the |+⟩ basis state. Let's denote the resulting conditional state as |c⟩.

To calculate |c⟩, we need to project the initial state onto the |+⟩ basis and then normalize it.

|m⟩ = |+⟩ = 1/√2 (|0⟩ + |1⟩)
|c⟩ = Projection of the initial state onto |+⟩ basis / || Projection of the initial state onto |+⟩ basis ||

Performing the projection:
|m⟩ = |+⟩ = 1/√2 (|0⟩ + |1⟩)
Projection of the initial state onto |+⟩ basis = <m| (α|0⟩ ⊗ |0⟩ + β|1⟩ ⊗ |1⟩)
= (α/√2 <0|⊗<0| + β/√2 <1|⊗<1|) (1/√2 (|0⟩ + |1⟩))
= 1/2 (α<0|0⟩ + α<0|1⟩ + β<1|0⟩ + β<1|1⟩)
= 1/2 (α|00⟩ + α|01⟩ + β|10⟩ + β|11⟩)

Normalizing the projection:
|| Projection of the initial state onto |+⟩ basis || = √(<m| Projection of the initial state onto |+⟩ basis) ⊗ (Projection of the initial state onto |+⟩ basis |m>)
= √(1/2 (α|00⟩ + α|01⟩ + β|10⟩ + β|11⟩)·(1/2 (α|00⟩ + α|01⟩ + β|10⟩ + β|11⟩))

The resulting state of the second qubit, denoted as |c⟩, can now be calculated as:
|c⟩ = (1/2 (α|00⟩ + α|01⟩ + β|10⟩ + β|11⟩)) / √(1/2 (α|00⟩ + α|01⟩ + β|10⟩ + β|11⟩)·(1/2 (α|00⟩ + α|01⟩ + β|10⟩ + β|11⟩))

Simplifying |c⟩ will depend on the specific values of α and β.