Max Planck Institute of Quantum Optics (MPQ)
Garching / Munich, Germany
Johannes Kofler
Talk at: Institute of Applied Physics
Johannes Kepler University Linz
10 Dec. 2012

• Quantum entanglement
• Foundations: Bell’s inequality
• Application: “quantum information”
(quantum cryptography & quantum computation)
• Entanglement swapping
• Quantum teleportation

Christiaan Huygens
(1629 –1695)
…waves
Isaac Newton
(1643 –1727)
….particles
James Clerk Maxwell
(1831 –1879)
…electromagnetic waves
Albert Einstein
(1879 –1955)
…quanta

Particles Waves Quanta
Superposition :
|
 
= |left

+ |right

Picture: http://www.blacklightpower.com/theory/DoubleSlit.shtml

1 photon in (pure) polarization quantum state:
Pick a basis, say: horizontal |
  and vertical |
 
Examples: |
 
= |
 
|
 
= |
 
|
 
= (|
 
+ |
 
) /

2 = |
 
|
 
= (|
 
+ i|
 
) /

2 = |
  superposition states
(in chosen basis)
2 photons (A and B):
Examples: |

AB
|

AB
|

AB
|

AB
= |
 
A
|
 
B
= |
 
AB

|
 
AB product (separable) states: |
 
A
|
 
B
= (|
 
AB
+ |
 
AB
) /

2
= (|
 
AB
+ i|
 
AB
– 3|  
AB
) / n entangled states, i.e.
not of form |
 
A
|
 
B
Example: |

AB
= (|
 
AB
+ |
 
AB
+ |
 
AB
+ |
 
AB
) / 2 = |
 
AB

Entanglement:
|

AB
= (|
 
AB
= (|
 
AB
+ |
 
AB
) /

2
+ |
 
AB
) /

2
Alice basis: result
 /  : 
 /  : 
 /  : 
 /  : 
 /  : 
 /  : 

/

:


/

:

Bob basis: result
 /  : 
 /  : 
 /  : 
 /  : 
 /  : 
 /  : 

/

:


/

:
 locally: random globally: perfect correlation
Picture: http://en.wikipedia.org/wiki/File:SPDC_figure.png

“Total knowledge of a composite system does not necessarily include maximal knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all.
” (1935)
What is the difference between the entangled state
|

AB
= (|  
AB
+ |  
AB
) /

2 and the (trivial, “classical”) fully mixed state probability ½: |  
AB probability ½: |  
AB

= (|
 
AB
 
| + |
 
AB
 
|) / 2
Erwin Schrödinger which is also locally random and globally perfectly correlated?

Realism:
Locality: objects possess definite properties prior to and independent of measurement a measurement at one location does not influence a
(simultaneous) measurement at a different location
Alice und Bob are in two separated labs
A source prepares particle pairs, say dice. They each get one die per pair and measure one of two properties, say color and parity measurement 1: color result: measurement 2: parity result:
A
A
1
2
(Alice), B
(Alice), B
1
2
(Bob)
(Bob) possible values: +1 (even / red)
–1 (odd / black)
A
1
( B
1
+ B
2
) + A
2
( B
1
– B
2
) = ±2
A
1
B
1
+ A
1
B
2
+ A
2
B
1
– A
2
B
2
= ±2

A
1
B
1

+

A
1
B
2

+

A
2
B
1
 – 
A
2
B
2
 ≤ 2
Alice for all local realistic
(= classical) theories
CHSH version (1969) of
Bell’s inequality (1964)
Bob

With the entangled quantum state
|

AB
= (|  
AB
+ |  
AB
) /

2 and for certain measurement directions a
1 the left hand side of Bell’s inequality
, a
2 and b
1
, b
2
,

A
1
B
1

+

A
1
B
2

+

A
2
B
1
 – 
A
2
B
2
 ≤ 2 becomes 2

2

2.83.
John S. Bell
A
2
A
1
B
1
B
2
Conclusion: entangled states violate Bell’s inequality (fully mixed states cannot do that) they cannot be described by local realism (Einstein: „Spooky action at a distance“) experimentally demonstrated for photons, atoms, etc. (first experiment: 1978)

Copenhagen interpretation quantum state (wave function) only describes probabilities objects do not possess all properties prior to and independent of measurements (violating realism) individual events are irreducibly random
Bohmian mechanics quantum state is a real physical object and leads to an additional “force” particles move deterministically on trajectories position is a hidden variable & there is a non-local influence (violating locality) individual events are only subjectively random
Many-worlds interpretation all possibilities are realized parallel worlds

Albert Einstein
(1879
–1955)
What is nature?
Niels Bohr
(1885
–1962)
What can be said about nature?

Symmetric encryption techniques plain text encryption cipher text decryption plain text
Asymmetric („public key“) techniques: eg. RSA

One-time pad
Idea: Gilbert Vernam (1917)
Security proof: Claude Shannon (1949)
[only known secure scheme]
Criteria for the key:
random and secret
(at least) of length of the plain text
is used only once („one-time pad“)
Gilbert Vernam Claude Shannon
Quantum physics can precisely achieve that:

Quantum Key Distribution (QKD)
Idea: Wiesner 1969 & Bennett et al . 1984, first experiment 1991
With entanglement: Idea: Ekert 1991, first experiment 2000

 0
 1 
1

0

0

1
 0
 1
Basis:
Result:

/
 
/
 
/
 
/
 
/
 
/
 
/
 …
0 1 1 0 1 0 1 …
Basis:
Result:

/
 
/
 
/
 
/
 
/
 
/
 
/
 …
0 0 1 0 1 0 0 …
Alice and Bob announce their basis choices (not the results)
if basis was the same, they use the (locally random) result
the rest is discarded
perfect correlation yields secret key: 0110…
in intermediate measurements, Bob chooses also other bases (22.5
°,67.5°) and they test Bell’s inequality
violation of Bell’s inequality guarantees that there is no eavesdropping
security guaranteed by quantum mechanics

First quantum cryptography with entangled photons
Alices
Schlüssel
Bobs
Schlüssel
Original: Verschlüsselt: Entschlüsselt:
Key length: 51840 bit
Bit error rate: 0,4%
Bitweises
XOR
Bitweises
XOR
Schlüssel: 51840 Bit, Bit Fehler Wahrsch. 0.4 %
T. Jennewein et al., PRL 84, 4729 (2000)

Millennium Tower Twin Tower
Kuffner Sternwarte
K. Resch et al ., Opt. Express 13, 202 (2005)

Partners:
Japan: NEC, Mitsubishi Electric, NTT NICT
Europe: Toshiba Research Europe Ltd. (UK),
ID Quantique (Switzerland) and “All Vienna”
(Austria).
Toshiba-Link (BB84): 300 kbit/s over 45 km http://www.uqcc2010.org/highlights/index.html

ISS (350 km Höhe)

Gordon Moore
Transistor size
2000

200 nm
2010

20 nm
2020

2 nm (?)

1981: Nature can be simulated best by quantum mechanics
Richard Feynman
1985: Formulation of the concept of a quantum Turing machine
David Deutsch

0
|Q
 1
= (|0

+ |1

)
Bit: 0 or 1
Classical input
01101… preparation of qubits
1 evolution
Qubit: 0 “and” 1 measurement on qubits
Classical
Output
00110…

General qubit state: Bloch sphere:
P („0“) = cos 2

/2, P („1“) = sin 2

/2
 … phase (interference)
Physical realizations:
 photon polarization: |0

= |
 
 electron/atom/nuclear spin: |0

= |up

 atomic energy levels:
 superconducting flux:
 etc…
|0
|0


= |ground
= |left


|1
|1
|1
|1




= |
 
= |down

= |excited
= |right


|  
= |0

+ |1

|R

= |0

+ i |1

Gates: Operations on one ore more qubits


Deutsch algorithm (1985) checks whether a bit-to-bit function is constant, i.e. f (0) = f (1), or balanced, i.e. f (0)
 f (1) cl: 2 evaluations, qm: 1 evaluation

Shor algorithm (1994) factorization of a b -bit integer cl: super-poly. O {exp[(64 b /9) 1/3 (log b ) 2/3 ]}, qm: sub-poly. O ( b 3 ) [“exp. speed-up”] b = 1000 (301 digits) on THz speed: cl: 100000 years, qm: 1 second

Grover algorithm (1996) search in unsorted database with N elements cl: O ( N ), qm: O (

N ) [„quadratic speed-up“]

NMR Trapped ions Photons
SQUIDs NV centers Quantum dots Spintronics

Idea: Bennett et al . (1992/1993)
First realization: Zeilinger group (1997)
Bell-state measurement classical channel teleported state
C
C initial state
(Charlie)
Alice
A entangled pair
B source
Bob

Entangled pair (AB):
|
 – 
AB
= (|HV

AB
– |VH 
AB
) /

2
Unknown input state (C):
|
 
C
=

|H

C
+

|V

C
Bell states:
|
 – 
AB
|

+

AB
|
 – 
AB
|

+

AB
= (|HV

AB
– |VH 
AB
) /

2
= (|HV

AB
+ |VH

AB
) /

2
= (|HH

AB
– |VV 
AB
) /

2
= (|HH

AB
+ |VV

AB
) /

2
Total state (ABC):
|
 – 
AB
|
 
C
= (1/

2) (|HV

AB
– |VH 
AB
) (

|H

C
+

|V

C
)
= [ |
 – 
AC
(

|H

B
+

|V

B
)
+ |

+

AC
( – 
|H

B
+

|V

B
)
+ |
 – 
AC
(

|H

B
+

|V

B
)
+ |

+

AC
( – 
|H

B
+

|V

B
) ] if A and C are found in |
 – 
AC then B is in input state if A and C are found in another
Bell state, then a simple transformation has to be performed

H
1
H
2
PBS PBS
BS
V
1
V
2
C A
|
 – 
AC
= (|HV

AC
– |VH 
AC
) /

2
|

+

AC
= (|HV

AC
+ |VH

AC
) /

2
|
 – 
AC
= (|HH

AC
– |VV 
AC
) /

2
|

+

AC
= (|HH

AC
+ |VV

AC
) /

2 singlet state, anti-bunching: H
1
V
2 or V
1
H
2 triplet state, bunching: H
1
V
1 or H
2
V
2 cannot be distinguished with linear optics

Idea: Zukowski et al . (1993)
First realization: Zeilinger group (1998)
… … …
“quantum repeater” initial state factorizes into 1,2 x 3,4 if 2,3 are projected onto a Bell state, then 1,4 are left in a Bell state
Picture: PRL 80 , 2891 (1998)

Bell-state measurement (BSM):
Entanglement swapping
Mach-Zehnder interferometer and
QRNG as tuneable beam splitter
Separable-state measurement (SSM):
No entanglement swapping
X. Ma et al ., Nature Phys. 8 , 479 (2012)

A later measurement on photons 2 & 3 decides whether photons 1 & 4 were in a separable or an entangled state
If one viewed the quantum state as a real physical object, one would get the seemingly paradoxical situation that future actions appear as having an influence on past events
X. Ma et al ., Nature Phys. 8 , 479 (2012)

Towards a worldwide “quantum internet”
X. Ma et al ., Nature 489 , 269 (2012)

State-of-the-art technology:
- frequency-uncorrelated polarization-entangled photon-pair source
- ultra-low-noise single-photon detectors
- entanglement-assisted clock synchronization
605 teleportation events in 6.5 hours
X. Ma et al ., Nature 489 , 269 (2012)

A. Zeilinger X. Ma R. Ursin B. Wittmann T. Herbst S. Kropascheck