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Alice and Bob’s Excellent Adventure. Presented by : Lacie Zimmerman Adam Serdar Jacquie Otto Paul Weiss. What’s to Come…. Brief Review of Quantum Mechanics Quantum Circuits/Gates No-Cloning Distinguishability of Quantum States Superdense Coding Quantum Teleportation.

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Presentation Transcript
slide1

Alice and Bob’s Excellent Adventure

Presented by:

Lacie Zimmerman

Adam Serdar

Jacquie Otto

Paul Weiss

slide2

What’s to Come…

  • Brief Review of Quantum Mechanics
  • Quantum Circuits/Gates
  • No-Cloning
  • Distinguishability of Quantum States
  • Superdense Coding
  • Quantum Teleportation
dirac bra ket notation

Dirac Bra-Ket Notation

Notation

Inner Products

Outer Products

bra ket notation involves

Bra

<n| = |n>t

Ket

|n>

is the complex conjugate of m.

Bra-Ket Notation Involves

Vector Xn can be represented two ways

inner products

<a|b> =

Inner Products

An Inner Product is a Bra multiplied by a Ket

<a| |b> can be simplified to <a|b>

=

outer products

|a><b| =

=

By Definition

Outer Products

An Outer Product is a Ket multiplied by a Bra

slide7

Postulates of

Quantum Mechanics

slide8

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.

Postulate 1

slide9

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.
slide10

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.,

slide11

Postulate 3

Quantum measurements are described by a finite set of projections, {Pm}, acting on the state space of the system being measured.

slide12

Postulate 3

  • If is the state of the system immediately before the measurement.
  • Then the probability that the result m occurs is given by .
if the result m occurs then the state of the system immediately after the measurement is
If the result m occurs, then the state of the system immediately after the measurement is

Postulate 3

slide14

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 .
quantum uncertainty and quantum circuits
Quantum Uncertainty and Quantum Circuits

Classical Circuits vs. Quantum Circuits

Hadamard Gates

C-not Gates

Bell States

Other Important Quantum Circuit Items

classical circuits vs quantum circuits
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.

circuits continued
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.

hadamard gates
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.

circuit diagram of a hadamard gate

H

Circuit Diagram of aHadamard Gate

When represented in a Quantum Circuit Diagram, a Hadamard Gate looks like this:

Where the x is the input qubit and the y is the output qubit.

c not gates
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.

circuit diagram of a c not gate
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.

bell states

H

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:

With H being a Hadamard Gate and x and y being the input qubits. is the Bell State.

other important quantum circuit items
Other Important Quantum Circuit Items
  • Controlled U-Gates
  • Measurement Devices
controlled u gate

n

n

U

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:

Where n is the number of qubits the gate is acting on.

measurement devices

These devices convert a qubit state into a probabilistic classical bit.

It can be represented in a diagram with the following:

x

M

A measurement with x possible outcomes has x wires coming from the device that measures it.

Measurement Devices
cloning

By copy we mean:

    • Take a quantum state
    • Perform an operation
    • End with an exact copy of

Cloning

Can copying of an unknown qubit state really happen?

using a classical idea
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
move the logic to quantum states
Move the Logic to Quantum States
  • Given a qubit in an unknown quantum state

such that

  • Through a CNOT gate we take 

such that

  • Note if indeed we copied we would thus end up with which would equal
limits on copying
Limits on Copying

Note that:

only at ab=0 and for a and b being or

proving the difficulty of cloning
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
difficulty cloning
…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
final cloning summary
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.
distinguishability
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
distinguishability cont
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.
cloning and distinguishability
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.
slide38

Superdense Coding

  • Pauli Matrices
  • Alice & Bob
  • The Conditions
  • How it Works
slide40

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.
slide41

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.

slide42

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.
slide43

Superdense Coding

HOW IT WORKS…

  • Alice keeps the first qubit (particle).
  • Bob keeps the second qubit (particle).
  • Bob moves far away from Alice.
slide44

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.
slide45

Superdense Coding

The four resulting states are:

slide46

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.
teleportation

Teleportation

What is it used for?

Teleportation Circuit

teleportation48
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.
teleportation49
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):
teleportation circuit

Single line denotes quantum information being transmitted

Double line denotes classical info being transmitted

{

Teleportation Circuit

Top 2 wires represent Alice's system

Bottom wire represents Bob’s system

teleportation circuit52

C-Not gate

TeleportationCircuit

{

After Applying the C-Not gate to Alice’s bits:

teleportation circuit53

Hadamard gate

TeleportationCircuit

{

After applying the Hadamard gate to the first qubit:

teleportation circuit54

Measurement devices

{

TeleportationCircuit

After Alice observes/measures her 2 qubits, she sends the resulting classical information to Bob:

teleportation circuit55

{

TeleportationCircuit

Bob applies the appropriate quantum gate to his qubit based on the classical information from Alice:

teleportation56
Teleportation

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

conclusion
Conclusion
  • Brief Review of Quantum Mechanics
  • Quantum Circuits/Gates
    • Classical Gates vs. Quantum Gates
    • Hadamard Gates
    • C-not Gates
    • Bell States
conclusion cont
Conclusion, cont.
  • No-Cloning
  • Distinguishability of Quantum States
  • Superdense Coding
    • Pauli Matrices
    • The Conditions
    • How it Works
conclusion cont59
Conclusion, cont.
  • Quantum Teleportation
    • What is it used for?
    • Teleportation Circuit
slide60

Bibliography

http://en.wikipedia.org/wiki/Inner_product_space

http://vergil.chemistry.gatech.edu/notes/quantrev/node14.html

http://en2.wikipedia.org/wiki/Linear_operator

http://vergil.chemistry.gatech.edu/notes/quantrev/node17.html

http://www.doc.ic.ad.uk/~nd/surprise_97/journal/vol4/spb3/

http://www-theory.chem.washington.edu/~trstedl/quantum/quantum.html

Gudder, S. (2003-March). Quantum Computation. American Mathmatical Monthly. 110, no. 3,181-188.

Special Thanks to:

Dr. Steve Deckelman