1542 inner products

Presented by:Lacie Zimmerman

Adam SerdarJacquie OttoPaul Weiss

Alice and Bob’s Alice and Bob’s Excellent AdventureExcellent Adventure

• Brief Review of Quantum Mechanics

• Quantum Circuits/Gates

• No-Cloning

• Distinguishability of Quantum States

• Superdense Coding

• Quantum Teleportation

What’s to Come…What’s to Come…

Dirac Bra-Ket NotationDirac Bra-Ket NotationNotation

Inner ProductsOuter Products

Bra-Ket Notation InvolvesVector Xn can be represented two

waysKet |n>

zyxwv

Bra <n| = |n>t

***** zyxwv

*m is the complex conjugate of m.

Inner ProductsAn Inner Product is a Bra multiplied by a Ket

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

= <a|b> =

ponml

***** zyxwv ***** pzoynxmwlv

Outer ProductsAn Outer Product is a Ket multiplied by a Bra

|a><b| =

ponml

=

*****

*****

*****

*****

*****

pzpypxpwpvozoyoxowovnznynxnwnvmzmymxmwmvlzlylxlwlv

***** zyxwv

By Definition acbcba

• 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.

nCV

• Simplest state space - (Qubit) If and form a basis for , then an arbitrary qubit state has the form

, where a and b inhave .

• Qubit state differs from a bit because “superpositions” of a qubit state are possible.

2CV 0| 1| V

1|0|| bax C1|||| 22 ba

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.,

)(| 2t)(| 1t

2,1 ttU

)(|)(| 1,2 21tUt tt

Quantum measurements are described by a

finite set of projections, {Pm}, acting on the

state space of the system being measured.

• If is the state of the system immediately before the measurement.

•Then the probability that the result m occurs is given by .

|

||)( mPmp

• If the result m occurs, then the state of the

system immediately after the measurement is

)(|

|||

2/1 mpP

PP m

m

m

• 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

.

)(| it

n || 1

Quantum Uncertainty and Quantum Circuits

Classical Circuits vs. Quantum CircuitsHadamard Gates

C-not GatesBell States

Other Important Quantum Circuit Items

Classical Circuits vs.

Quantum CircuitsClassical 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…

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 GatesHadamard 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.

11

112

1H

Circuit Diagram of aHadamard Gate

Hx y

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 GatesC-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.

0100100000100001

CNU

Circuit Diagram of a C-Not Gate

x

y

x

yx

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 StatesBell 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.

Hx

yxy

Bell State EquationsThe following equations map the previous Bell State Generator:

0011002

100002

100

0110012

111012

101

1011002

110002

110

1110012

111012

111

So we can write: 2

110 yy x

xy

Other Important Quantum Circuit Items

• Controlled U-Gates• Measurement Devices

Controlled U-GateA 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:

Un n

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

Measurement DevicesThese 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.

Cloning of a Quantum

State

CloningCan copying of an unknown qubit

state really happen?

By copy we mean:

1. Take a quantum state

2. Perform an operation

3. End with an exact copy of

Z

Z

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

• 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

Z 1b 0aZ Z ZZ0

10b 00a 0)1b 0(a Z

ZZ

11b² 10ab 01ab 00a²

Limits on Copying

Note that:

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

11b² 10ab 01ab 00a²10b00a

0 1

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 SZ

ZZSZU

ZS

…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

yy )sy U(& zz )szU(

²yz yz yz

y z y zz

zy y

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• 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

n21 y,...,y,y

n21 y...yy

mmm yy P my mP

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.

mm y|P|y P(m) m

mmmm yyyy P(m)

1 11P(m)

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.

• Pauli Matrices

• Alice & Bob

• The Conditions

• How it Works

0110

X

00i

iY

10

01Z

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.

HOW IT WORKS…• Alice and Bob initially share a 2-qubit in the

entangled Bell state

which is just a pair of quantum particles.

2

1100

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.

HOW IT WORKS…1) Alice keeps the first qubit (particle).

2) Bob keeps the second qubit (particle).

3) Bob moves far away from Alice.

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.

The four resulting states are:

.)(:11

,:10

,)(:01

,:00

11

10

01

00

i

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.

TeleportationTeleportation

What is it used for? Teleportation Circuit

TeleportationTeleportation•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.

2/110000


•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 Teleportation CircuitCircuit

Top 2 wires represent Alice's system

Bottom wire represents Bob’s system

43210

2

1

Z

xyX

MMH

00 {

Single line denotes quantum information being transmitted

Double line denotes classical info being transmitted

TeleportationTeleportationCircuit Circuit

11001110002

1

10

000

ba

ba

Initial State

43210

2

1

Z

xyX

MMH

00 {

TeleportationTeleportationCircuitCircuit

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

01101110002

11 ba

43210

2

1

Z

xyX

MMH

00

C-Not gate

{


01111010

01011000

21

2 baba

baba

After applying the Hadamard gate to the first qubit:

43210

2

1

Z

xyX

MMH

00

Hadamard gate

{


.0111 ,1010

,0101 ,1000

33

33

baba

baba

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

43210

2

1

Z

xyX

MMH

00 {

Measurement devices


43210

2

1

Z

xyX

MMH

00 {

.101001:11

,1010:10

,1001:01

,:00

4311

4301

4310

43300

babZaZbXaXZXZ

babZaZXZ

babXaXXZ

IXZ

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


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

4

ConclusionConclusion• 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, cont.Conclusion, cont.• Quantum Teleportation

- What is it used for?

- Teleportation Circuit

Bibliographyhttp://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

Education

1542 inner products