1

SpinMany quantum experiments are done with photon polarization instead of electron spinHere is the correspondence between the two

And the measurement probabilities are the same

arizer Pol SG

D- 0

D 1

V 0

H 1oton PhElectron

x

x

z

z

+

2

SpinFrom our study of spin and Stern-Gerlachexperiments we know we must choose a direction (basis) to measure Sz for electrons or linear polarization for photons

50% D- x H - 1

50% D x H 1

0 V H z- 1

1 H H z 1detection ofbility Proba Filter n PhotoSG Electron

z

z

z

z

++

+

3

Spin or Polarization

4

Quantum Cryptography

How does the left side of the room pass a note to the right side of the room without me learning the contents of the message?The only mathematically proven way to transmit a message is to use a one-time pad

This requires a key the same length as the message and can be used only once

5

Quantum Cryptography

Note, the figures on the next few slides were take from a talk on quantum cryptography by Vadim Makarov from NTNU in Norway

6

Quantum CryptographyClassical key cryptography requires a secure channel for key distribution

Vulnerable, authentication not assured, evesdropping

Quantum cryptography can distribute the key using an open channel

7

Quantum CyptographyIn addition, a message transmitted classically can be passively monitoredA message transmitted quantum mechanically

8

Quantum Cryptography

9

Quantum CryptographyAlice generates a random keyAlice generates a random set of analyzersAlice sends the results to BobBob generates a random set of analyzersAlice and Bob publicly exchange what analyzers were used (sifting)Alice and Bob check whether randomly selected entries agreeThe remaining results are kept as a key

10

Quantum Cryptography

11

Quantum CryptographyHere is Alice

12

Quantum CryptographyHere is Bob

13

QubitA Qubit (or Qbit) is a quantum system with exactly two degrees of freedom

ExamplesElectron with spin up and spin downPhoton with horizontal and vertical polarization

( )

( ) ( ) 22

22

0 Pand 1Ppreviously learned weas and

1 where

01 states of ionsuperposit isqubit a

0or 1 0or 1 be canbit a

βα

βα

βαψ

==

=+

+=

14

EntanglementRecall the EPR paradox

15

EntanglementDescribes quantum states that have to be referenced to each other even though they are spatially separatedSay I can generate electron (or photon) pairs such that they always have opposite spins

What is the probability of A measuring up?What is the probability of A measuring down?What is the probability of B measuring up when A measures down?

We say the two spins are entangled

( )↑↓−↓↑= ,,2

1ψ

16

Quantum Teleportation

Quantum teleportation transfers a quantum state to an arbitrarily far location using a distributed entangled state and the transmission of classical informationThe state at A is physically not transferred but an identical copy appears at B

17

Quantum TeleportationRecall that spin can be measured with different SG analyzers (bases)We’ll define

( )

( )←−→=↓

←+→=↑

←=↓→=↑

↓=↓↑=↑

21 and

21 and

and

and

z

z

xx

zz

18

Quantum TeleportationLet’s define some operationsThese don’t destroy the quantum state

( )

( )

( )↑↓+↓↓+↓↑+↑↑

=↓↓+↑↓+↓↑+↑↑−

−−

↓−↑=↓+↑

↑+↓=↓+↑

,,,,

,,,,

up isfirst theif spin second thekeeps down isfirst theif spin second theflips

axis-z about the rotationa is

axis-xabout therotationa is

dcba

dcbaXC

XCXC

babaZ

Z

babaX

X

19

Quantum TeleportationWe want to transfer an arbitrary state from Lab A to Lab B

We also start with an entangled state

Then the overall state is

↓+↑ ba

( )↑↓−↓↑ ,,2

1

↑↓↓+↑↓↑−↓↑↓−↓↑↑ ,,2

,,2

,,2

,,2

baba

20

Quantum Teleportation

Spin 1 is the unknown spin to be transportedSpin 2 is the first spin of the entangled pair and is located Lab ASpin 3 is the second spin of the entangled pair and is located in Lab B

21

Quantum TeleportationApply the C-X operation to spins 1 and 2 in Lab A

↑↑↓+↑↓↑−↓↓↓−↓↑↑

↑↓↓+↑↓↑−↓↑↓−↓↑↑

,,2

,,2

,,2

,,2

becomes

,,2

,,2

,,2

,,2

baba

baba

22

Quantum TeleportationMeasure spin 2 in the z basis

The probability of getting spin up is

And also 1/2 for getting spin down

Let’s say we measure up, then the new state will be

↑↑↓+↑↓↑−↓↓↓−↓↑↑ ,,2

,,2

,,2

,,2

baba

21

222

2222

=+

=+baba

↑↑↓+↓↑↑ ,,,, ba

23

Quantum TeleportationMeasure spin 1 in the x basis

Then the probability of measuring spin right is

And 1/2 for getting spin left

Let’s say we measure right, then the new state will be

↑↑←+↑↑→+↓↑←+↓↑→

↑↑↓+↓↑↑

,,2

,,2

,,2

,,2

basisx thein ,,,, efirst writ

bbaa

ba

21

222

2222

=+

=+baba

↑↑→+↓↑→ ,,,, ba

24

Quantum TeleportationNow call Lab B and tell them the resultsLab B responds by

Our example is case 3

3 spin toX thenZapply :1 spinfor and 2 spinfor If3 spin toXapply :1 spinfor and 2 spinfor If3 spin toZapply :1 spinfor and 2 spinfor If

nothing do :1 spinfor and 2 spinfor If

←↑

→↑

←↓

→↓

↓↑→+↑↑→

↑↑→+↓↑→

,,,,

becomes ,,,,

ba

ba

25

Quantum TeleportationThus the state of spin 3 in Lab B is

Exactly the state we were trying to transmit

↓+↑

↓↑→+↑↑→

ba

ba ,,,,

26

Quantum Teleportation