A conceptual introduction to Quantum Mechanical Systems, their properties and qubits. Research focused on Chapters 4 – 7 of * Quantum Computing for Everyone by Chris Bernhardt*. The weekly content here is representative of assignments and exercises related to my coursework for ECE 802 – 730 | Quantum Sensor and System Engineering at Michigan State University.

#### No-Cloning Theorem

The “no cloning theorem” states that creating identical copies(clones) of quantum particles is impossible. Quantum particles can exist in superpositions of several states at once over which the sum of probable quantum states must equal one whole of the possible states. Any change to one state of the quantum particle in superposition will have cascading impacts on all other states in order to rebalance to 1 whole superposition. Cloning would require that the sum and product of any set of superposition states be equal through the mechanism of some change/gate in the system. This is impossible though because there is no mechanism for which to make

#### Super-Dense Coding

In Super-Dense Coding, a sender is able to transmit two classical bits of information to a receiver by sending a single qubit. The receiver does not know what the resultant data should be until the qubit is reverse processed through identical gates as occurred in the transmission process by action of the sender. This gate configuration of course, must be predetermined and conveyed to receiver by sender out-of-band.

#### Exercise 1

Show that the two qubit system given by,

from Chapter 4 gives the same probabilities if (1) both qubits are measured at the same time, or (2) one is measured before the other, or (3) the other is measured first. This is the case for a two qubit system not entangled.

What if this were a quantum cryptographic system and we measured on Alice’s side first (assuming Alice’s qubits are and ). What would the probability be that Alice’s value is a 1 or a 0? What about Bob’s probability of the measured value being a 1 or a 0?

Let

… but does ?

Since the values on both sides of the equation are equal, then the qubits must not be equal. Simultaneous measurements only impact the qubits independently. What happens on one qubit has no impact on any of the others.

First, we pull out Alice’s common factors and rewrite the equation as a tensor product:

And then continue to factor out values:

Which can be rewritten as:

Therefore measuring for Alice first has no impact on Bob, where Alice has a 50% probability of being a 1 or 0. Bob measuring first has no impact on Alice, where Bob’s probability of being 1 is 75% and 0 is 25%.

#### Exercise 2

For the Bell Circuit (page 127 of Quantum Computing for Everyone) show that:

Input | Input | Output | Output |

For

Post Hadamard Gate:

Input to CNOT:

After the data has completed passing through the CNOT Gate:

– This Checks Out

For

Post Hadamard Gate:

Input to CNOT:

After the data has completed passing through the CNOT Gate:

– This Checks Out

For

Post Hadamard Gate:

Input to CNOT:

After the data has completed passing through the CNOT Gate:

– This Checks Out

#### Exercise 3

Show that for the circuit shown below the output is the same as the input. As an example input

Input | Input | Output | Output |

Pre CNOT:

CNOT Output:

Fed back through H Gate:

Pre CNOT:

Output: