1. Alice and Bob’s Excellent Adventure
Presented by:
Lacie Zimmerman
Adam Serdar
Jacquie Otto
Paul Weiss

2. What’s to Come… Brief Review of Quantum Mechanics
Quantum Circuits/Gates
No-Cloning
Distinguishability of Quantum States
Superdense Coding
Quantum Teleportation

3. Dirac Bra-Ket Notation
Inner Products
Outer Products

4. Bra-Ket Notation Involves
Vector Xn can be represented two ways
Ket
|n>
Bra
<n| = |n>t
is the complex conjugate of m.

5. Inner Products An Inner Product is a Bra multiplied by a Ket
<a| |b> can be simplified to <a|b>
<a|b> = =

6. An Outer Product is a Ket multiplied by a Bra
Outer Products
An Outer Product is a Ket multiplied by a Bra
|a><b| = =
By Definition

7. Postulates of
Quantum Mechanics

8. Postulate 1
State Space: The inner product space associated with an isolated quantum system.
The system at any given time is described by a “state”, which is a unit vector in V.

9. Postulate 1 Simplest state space - (Qubit) If and form a basis for ,
then an arbitrary qubit state has the form , where a and b in
have
Qubit state differs from a bit because “superpositions” of a qubit state are possible.
Reminiscent of the title, not clear if in either. a squared and b squared represent prob of being in either state.

10. Postulate 2
The evolution of an isolated quantum system is described by a unitary operator on its state space.
The state is related to the state by a unitary operator
i.e.,

11. Postulate 3
Quantum measurements are described by a finite set of projections, {Pm}, acting on the state space of the system being measured.
The index m refers to the measurement outcomes that may occur. The projections satisfy the completeness equation If the state of the system is
immediately before the measurement, then the probability that result m occurs is given by ; if the result m occurs, then the state of the system immediately after the measurement is

12. Postulate 3
If is the state of the system immediately before the measurement.
Then the probability that the result m occurs is given by .
The index m refers to the measurement outcomes that may occur. The projections satisfy the completeness equation If the state of the system is
immediately before the measurement, then the probability that result m occurs is given by ; if the result m occurs, then the state of the system immediately after the measurement is

13. Postulate 3
If the result m occurs, then the state of the system immediately after the measurement is

14. Postulate 4
The state space of a composite quantum system is the tensor product of the state of its components.
If the systems numbered 1 through n are prepared in states , i = 1,…, n, then the joint state of the total composite system is

15. Quantum Uncertainty and Quantum Circuits
Classical Circuits vs. Quantum Circuits
Hadamard Gates
C-not Gates
Bell States
Other Important Quantum Circuit Items

16. Classical Circuits vs. Quantum Circuits
Classical Circuits based upon bits, which are represented with on and off states. These states are usually alternatively represented by 1 and 0 respectively.
The medium of transportation of a bit is a conductive material, usually a copper wire or something similar. The 1 or 0 is represented with 2 different levels of current through the wire.

17. Circuits Continued…
Quantum circuits use electron “spin” to hold their information, instead of the conductor that a classical circuit uses.
While a classical circuit uses transistors to perform logic, quantum circuits use “quantum gates” such as the Hadamard Gates.

18. Hadamard Gates
Hadamard Gates can perform logic and are usually used to initialize states and to add random information to a circuit.
Hadamard Gates are represented mathematically by the Hadamard Matrix which is below.

19. Circuit Diagram of a Hadamard Gate
When represented in a Quantum Circuit Diagram, a Hadamard Gate looks like this: H
Where the x is the input qubit and the y is the output qubit.

20. C-Not Gates
C-not Gates are one of the basic 2-qubit gates in quantum computing. C-not is short for controlled not, which means that one qubit (target qubit) is flipped if the other qubit (control qubit) is |1>, otherwise the target qubit is left alone.
The mathematical representation of a C-Not Gate is below.

21. Circuit Diagram of a C-Not Gate
When represented in a Quantum Circuit Diagram, a C-Not Gate looks like this:
Where x is the control qubit and y is the target qubit.

22. Bell States Bell States are sets of qubits that are entangled.
They can be created with the following Quantum Circuit called a Bell State Generator: H
With H being a Hadamard Gate and x and y being the input qubits. is the Bell State.

23. Bell State Equations
The following equations map the previous Bell State Generator:
So we can write:

24. Other Important Quantum Circuit Items
Controlled U-Gates
Measurement Devices

25. Controlled U-Gate
A Controlled U-Gate is an extension of a C-Not Gate. Where a C-Not Gate works on one qubit based upon a control qubit, a U-Gate works on many qubits based upon a control qubit.
A Controlled U-Gate can be represented with the following diagram: U n
Where n is the number of qubits the gate is acting on.

26. Measurement Devices
These devices convert a qubit state into a probabilistic classical bit.
It can be represented in a diagram with the following: M x
A measurement with x possible outcomes has x wires coming from the device that measures it.

27. Cloning of a Quantum State

28. Can copying of an unknown qubit state really happen?
Cloning
Can copying of an unknown qubit state really happen?
By copy we mean:
Take a quantum state
Perform an operation
End with an exact copy of

29. Using a Classical Idea
A classical CNOT gate can be used for an unknown bit x
Let x be the control bit and 0 be the target
Send x0  xx where  is a CNOT gate
Yields an exact copy of x in the classical setting

30. Move the Logic to Quantum States
Given a qubit in an unknown quantum state
such that
Through a CNOT gate we take 
Note if indeed we copied we would thus end up with which would equal

31. Limits on Copying
Note that:
only at ab=0 and for a and b being or

32. Proving the difficulty of cloning
Suppose there was a copying machine
Such that can be copied with a standard state
This gives an initial state which when the unitary operation U is applied yields

33. …difficulty cloning Let By taking inner products of both sides:
From this we can see that: = 0 or 1
Therefore this must be true: or
Thus if the machine can successfully copy it is highly unlikely that the machine will copy an arbitrary unknown state unless is orthogonal to

34. Final cloning summary Cloning is improbable.
Basically all that can be accomplished is what we know as a cut-n-paste.
Original data is lost.
The process of this will be shown in the teleportation section soon to follow.

35. Distinguishability To determine the state of an element in the set:
This must be true: -
Finding the probability of observing a specific state , let be the measurement such that

36. Distinguishability cont.
Then the probability that m will be observed is: -
Which yields
Because the set is orthogonal
If the set was not orthogonal we couldn’t know for certain that m will be observed.

37. Cloning and Distinguishability
Take some quantum information
Send it from one place to another
Original is destroyed because it can’t just be cloned (copied)
Basically it must be combined with some orthogonal group or distinguishing the quantum state with absolute certainty is impossible.

38. Superdense Coding Pauli Matrices Alice & Bob The Conditions
How it Works

39. Pauli Matrices

40. Superdense Coding THE CONDITIONS…
Alice and Bob are a long way from one another.
Alice wants to transmit some classical information in the form of a 2-bit to Bob.

41. Superdense Coding HOW IT WORKS…
Alice and Bob initially share a 2-qubit in the entangled Bell state
which is just a pair of quantum particles.

42. Superdense Coding HOW IT WORKS…
is a fixed state and it is not necessary for Alice to send any qubits to Bob to prepare this state.
For example, a third party may prepare the entangled state ahead of time, sending one of the qubits to Alice and the other to Bob.

43. Superdense Coding HOW IT WORKS…
Alice keeps the first qubit (particle).
Bob keeps the second qubit (particle).
Bob moves far away from Alice.

44. Superdense Coding HOW IT WORKS…
The 2-bit that Alice wishes to send to Bob determines what quantum gate she must apply to her qubit before she sends it to Bob.

45. Superdense Coding
The four resulting states are:

46. Superdense Coding HOW IT WORKS…
Since Bob is in possession of both qubits, he can perform a measurement on this Bell basis and reliably determine which of the four possible 2-bits Alice sent.

47. What is it used for? Teleportation Circuit

48. Teleportation
Teleportation is sending unknown quantum information not classical information.
Teleportation starts just like Superdense coding.
Alice and Bob each take half of the 2-qubit Bell state:
Alice takes the first qubit (particle) and Bob moves with the other particle to another location.

49. Teleportation Alice wants to teleport to Bob:
She combines the qubit with her half of the Bell state and sends the resulting 3-qubit (the 2 qubits-Alice & 1 qubit-Bob) through the Teleportation circuit (shown on the next slide):

50. Teleportation Circuit{
Single line denotes quantum information being transmitted
Double line denotes classical info being transmitted
Top 2 wires represent Alice's system
Bottom wire represents Bob’s system

51. Teleportation Circuit{
Initial State

52. Teleportation Circuit
C-Not gate {
After Applying the C-Not gate to Alice’s bits:

53. Teleportation Circuit
Hadamard gate {
After applying the Hadamard gate to the first qubit:

54. Teleportation Circuit{
Measurement devices
After Alice observes/measures her 2 qubits, she sends the resulting classical information to Bob:

55. Teleportation Circuit{
Bob applies the appropriate quantum gate to his qubit based on the classical information from Alice:

56. Bob finally recovers the initial qubit that Alice teleported to him.
Teleportation
Bob finally recovers the initial qubit that Alice teleported to him.

57. Conclusion Brief Review of Quantum Mechanics Quantum Circuits/Gates
Classical Gates vs. Quantum Gates
Hadamard Gates
C-not Gates
Bell States

58. Conclusion, cont. No-Cloning Distinguishability of Quantum States
Superdense Coding
Pauli Matrices
The Conditions
How it Works

59. Conclusion, cont. Quantum Teleportation What is it used for?
Teleportation Circuit

60. Special Thanks to: Dr. Steve Deckelman Bibliography
Gudder, S. (2003-March). Quantum Computation. American Mathmatical Monthly. 110, no. 3,
Special Thanks to:
Dr. Steve Deckelman