CALCU-1. A single qubit is prepared in the state\[

Solution by Steps

step 1

For question 1(a), the possible outcomes in the computational basis are

|0\rangle

and

|1\rangle

. The probabilities are calculated as follows:

P(0) = |\langle 0 | \psi \rangle|^2

and

P(1) = |\langle 1 | \psi \rangle|^2

step 2

The state

|\psi\rangle = \frac{3}{5}|+\rangle - \frac{4i}{5}|-\rangle

can be expressed in the computational basis as

|\psi\rangle = \frac{3}{5}\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) - \frac{4i}{5}\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)

step 3

Simplifying this gives

|\psi\rangle = \left(\frac{3}{5\sqrt{2}} - \frac{4i}{5\sqrt{2}}\right)|0\rangle + \left(\frac{3}{5\sqrt{2}} + \frac{4i}{5\sqrt{2}}\right)|1\rangle

step 4

The probabilities are then calculated as

P(0) = \left|\frac{3 - 4i}{5\sqrt{2}}\right|^2

and

P(1) = \left|\frac{3 + 4i}{5\sqrt{2}}\right|^2

step 5

For question 2, the probability of measuring the qubit in state

|1\rangle

is given by

P(1) = \left|\frac{\sqrt{3}}{2}\right|^2 = \frac{3}{4}

step 6

For question 3, we verify that

P_0 + P_1 = E

by showing that the projection operators

P_0 = |0\rangle\langle 0|

and

P_1 = |1\rangle\langle 1|

sum to the identity matrix

E = |0\rangle\langle 0| + |1\rangle\langle 1|

step 7

For question 4(a), applying the Hadamard gate

H

to the first qubit of

|01\rangle

results in

H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)

, giving the state

\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)|1\rangle

step 8

For question 4(b), applying the Hadamard gate

H

to the second qubit results in

|0\rangle H|1\rangle = |0\rangle \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)

, giving the state

|0\rangle \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)

[1] Answer

C

[2] Answer

C

[3] Answer

A

[4] Answer

A

Key Concept

Quantum Measurement and Probability

Explanation

The probabilities of measuring a qubit in a specific state are derived from the squared magnitudes of the coefficients in its state representation. The Hadamard gate transforms qubit states, affecting measurement outcomes.

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