entanglement swapping and quantum teleportation talk at: institute of applied physics johannes...
Post on 29-Mar-2015
228 Views
Preview:
TRANSCRIPT
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
top related