Entanglement swapping andquantum teleportation

Talk at: Institute of Applied Physics

Johannes Kepler University Linz

10 Dec. 2012

Johannes Kofler

Max Planck Institute of Quantum Optics (MPQ)Garching / Munich, Germany

Outlook

• Quantum entanglement

• Foundations: Bell’s inequality

• Application: “quantum information”

(quantum cryptography & quantum computation)

• Entanglement swapping

• Quantum teleportation

Light consists of…

Christiaan Huygens(1629–1695)

Isaac Newton(1643–1727)

James Clerk Maxwell(1831–1879)

Albert Einstein(1879–1955)

…waves ….particles …electromagnetic waves

…quanta

The double slit experiment

Picture: http://www.blacklightpower.com/theory/DoubleSlit.shtml

Particles Waves Quanta

Superposition:

| = |left + |right

Superposition and entanglement

1 photon in (pure) polarization quantum state:

superposition states

(in chosen basis)

| = |

Pick a basis, say: horizontal | and vertical |

Examples:

| = (| + |) / 2

| = |

| = (| + i|) / 2

2 photons (A and B):

Examples:

|AB = |A|B |AB

|AB = | AB

product (separable)

states: | A| B

|AB = (|AB + |AB) / 2

|AB = (|AB + i|AB – 3|AB) / n

entangled states, i.e.

not of form | A| B

Example: |AB = (|AB + |AB + |AB + |AB) / 2 = |AB

= |

= |

Quantum entanglement

Entanglement:

|AB = (|AB + |AB) / 2

= (|AB + |AB) / 2

BobAlice

locally: random

/: /: /: /: /: /: /: /:

/: /: /: /: /: /: /: /:

globally: perfect correlation

basis: result basis: result

Picture: http://en.wikipedia.org/wiki/File:SPDC_figure.png

Entanglement

Erwin Schrödinger

“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

which is also locally random and globally perfectly correlated?

= (|AB| + |AB|) / 2

Local Realism

Realism: objects possess definite properties prior to and independent of measurement

Locality: 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: A1 (Alice), B1 (Bob)measurement 2: parity result: A2 (Alice), B2 (Bob)

possible values: +1 (even / red)–1 (odd / black)

A1 (B1 + B2) + A2 (B1 – B2) = ±2

A1B1 + A1B2 + A2B1 – A2B2 ≤ 2

A1B1 + A1B2 + A2B1 – A2B2 = ±2

for all local realistic (= classical) theories

Alice

Bob

CHSH version (1969) of Bell’s inequality (1964)

Quantum violation of Bell’s inequality

John S. Bell

A1B1 + A1B2 + A2B1 – A2B2 ≤ 2

With the entangled quantum state

|AB = (|AB + |AB) / 2

and for certain measurement directions a1,a2 and b1,b2, the left hand side of Bell’s inequality

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)

becomes 22 2.83.

A1

A2

B1

B2

Interpretations

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

Einstein vs. Bohr

Albert Einstein

(1879–1955)

Niels Bohr

(1885–1962)

What is nature?What can be said

about nature?

Cryptography

plain text encryption cipher text decryption plain text

Symmetric encryption techniques

Asymmetric („public key“) techniques: eg. RSA

Secure cryptography

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“)

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

Gilbert Vernam Claude Shannon

Quantum key distribution (QKD)

0

0 0

1111

0

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 experimental realization (2000)

O rig ina l:

X O R X O RB itw eises B itw e ises

Versch lüsse lt:

A licesS chlüsse l

B obsS chlüsse l

E ntsch lüsse lt:

S chlüsse l: 51840 B it, B it Fehler W ahrsch. 0 .4 %

First quantum cryptography with entangled photons

Key length: 51840 bitBit error rate: 0,4%

T. Jennewein et al., PRL 84, 4729 (2000)

8 km free space above Vienna (2005)

K. Resch et al., Opt. Express 13, 202 (2005)

Millennium Tower Twin Tower

Kuffner Sternwarte

Tokyo QKD network (2010)

http://www.uqcc2010.org/highlights/index.html

Partners:

Japan: NEC, Mitsubishi Electric, NTT NICTEurope: Toshiba Research Europe Ltd. (UK), ID Quantique (Switzerland) and “All Vienna” (Austria).

Toshiba-Link (BB84): 300 kbit/s over 45 km

The next step

ISS (350 km Höhe)

Moore’s law (1965)

Gordon Moore

Transistor size

2000 200 nm2010 20 nm2020 2 nm (?)

Computer and quantum mechanics

David Deutsch

1985: Formulation of the concept of a quantum Turing machine

Richard Feynman

1981: Nature can be simulated best by quantum mechanics

Quantum computer

Classical input 01101… preparation

of qubitsmeasurement

on qubits

Classical Output

00110…

evolution

1

0

|Q = (|0 + |1)2

1

Bit: 0 or 1 Qubit: 0 “and” 1

Qubits

General qubit state:

Physical realizations:

photon polarization: |0 = | |1 = |

electron/atom/nuclear spin: |0 = |up |1 = |down

atomic energy levels: |0 = |ground |1 = |excited

superconducting flux: |0 = |left |1 = |right

etc…

P(„0“) = cos2/2, P(„1“) = sin2/2

… phase (interference)

| = |0 + |1|R = |0 + i |1

Bloch sphere:

Gates: Operations on one ore more qubits

Quantum algorithms

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[(64b/9)1/3(logb)2/3]}, qm: sub-poly. O(b3) [“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“]

Possible implementations

NV centers Quantum dots Spintronics

Trapped ionsNMR Photons

SQUIDs

Quantum teleportation

CA B

initial state(Charlie) source

entangled pairAlice Bob

classical channel

teleported state

C

Idea: Bennett et al. (1992/1993)

First realization: Zeilinger group (1997)

Bell-state measurement

Quantum teleportation

Entangled pair (AB):

|–AB = (|HVAB – |VHAB) / 2 |–AB = (|HVAB – |VHAB) / 2

|+AB = (|HVAB + |VHAB) / 2

|–AB = (|HHAB – |VVAB) / 2

|+AB = (|HHAB + |VVAB) / 2

Bell states:

Unknown input state (C):

| C = |HC + |VC

Total state (ABC):

|–AB | C = (1/2) (|HVAB – |VHAB) ( |HC + |VC)

= [ |–AC ( |HB + |VB)

+ |+AC (– |HB + |VB)

+ |–AC ( |HB + |VB)

+ |+AC (– |HB + |VB) ]

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 trans-formation has to be performed

Bell-state measurement

BS

PBS PBS

C A

H1 H2

V1 V2

|–AC = (|HVAC – |VHAC) / 2

|+AC = (|HVAC + |VHAC) / 2

singlet state, anti-bunching: H1V2 or V1H2

triplet state, bunching: H1V1 or H2V2

|–AC = (|HHAC – |VVAC) / 2

|+AC = (|HHAC + |VVAC) / 2cannot be distinguished with linear optics

Entanglement swapping

Idea: Zukowski et al. (1993)

First realization: Zeilinger group (1998)

Picture: PRL 80, 2891 (1998)

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

… … …

“quantum repeater”

Delayed-choice entanglement swapping

X. Ma et al., Nature Phys. 8, 479 (2012)

Bell-state measurement (BSM): Entanglement swapping

Mach-Zehnder interferometer and QRNG as tuneable beam splitter

Separable-state measurement (SSM): No entanglement swapping

Delayed-choice entanglement swapping

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)

Quantum teleportation over 143 km

Towards a world-wide “quantum internet”

X. Ma et al., Nature 489, 269 (2012)

Quantum teleportation over 143 km

X. Ma et al., Nature 489, 269 (2012)

605 teleportation events in 6.5 hours

State-of-the-art technology:

- frequency-uncorrelated polarization-entangled photon-pair source- ultra-low-noise single-photon detectors- entanglement-assisted clock synchronization

Acknowledgments

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