Coursework focused on Chapter 3 of * Quantum Computing: A Gentle Introduction*.

#### Exercise 1

Show that the state:

is entangled, with respect to the decomposition into the qubits, for every .

,

∴ for every element of in the set

AND for every element of in the set

Since the coefficients of all states that don’t appear in the tensor products of must be zero,

then there is a contradiction of values in the state – hence entanglement.

#### Exercise 2

Is the state entangled?

Since this cannot be due to the necessity of opposite signs for sets and . Since this is a contradiction the state of is entangled.

#### Exercise 3

Write the following states in terms of the Bell basis.

The Bell basis for a 2-qubit system is:

**a.**

**b.**

**c.**

#### Exercise 4

Give an example of a two-qubit state that is a superposition with respect to the standard basis but that is not entangled.

#### Exercise 5

**a.** Show that the four-qubit state of example 3.2.3 is entangled with respect to the decomposition into two two-qubit subsystems consisting of the first and second qubits and the third and fourth qubits.

values of and are both respectively non-zero (must equate )

this is a logical impossibility and therefore entangled.

**b.** For the four decompositions into two subsystems consisting of one and three qubits, say whether is entangled or unentangled with respect to each of these decompositions.

IF written in terms of two subsystems, such that:

,

is not zero,

,

is not zero

This is a logical impossibility with the composition of these subsystems within this configuration or any of the other three as well

∴ must be entangled, regardless of subsystem combination

#### Exercise 6

**a.** For the standard basis, the Hadamard basis, and the basis , determine the probability of each outcome when the second qubit of a two qubit system in the state of is measured in each of these bases.

Hadamard basis state of

∴ When measured w/ Hadamard basis the second qubit is equivalent to and with equal probability of .

For the basis

∴

Such that,

Where there is a probability of the second bit equaling or

**b.** Determine the probability of each outcome when the second qubit of the state is first measured in the Hadamard basis and then in the basis *B* of part a).

IF measured with Hadamard basis we have:

The second bit will be measured as:

represented as * *OR represented as * *with equal probability.

IF

,

AND

THEN if case 1 is the first measurement,

∴ the second measurement in this case would yield with a probability of ,

AND with a probability of

OR

IF

,

AND

THEN if case 1 is the first measurement,

∴ the second measurement in this case would yield with a probability of

AND

with a probability of

**b.** Determine the probability of each outcome when the second qubit of the state is first. Measured in the Hadamard basis and then in the standard basis.

Exactly the same as the last scenario and results in equal probability of .