Transcript ppt - Max-Planck
Max Planck Institute of Quantum Optics (MPQ) Garching / Munich, Germany
Entanglement swapping and quantum teleportation
Johannes Kofler Talk at: Institute of Applied Physics Johannes Kepler University Linz 10 Dec. 2012
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) …waves Isaac Newton (1643 –1727) ….particles
James Clerk Maxwell (1831 –1879) …electromagnetic waves Albert Einstein (1879 –1955) …quanta
The double slit experiment
Particles Waves Quanta
|
Superposition
: = |left + |right Picture: http://www.blacklightpower.com/theory/DoubleSlit.shtml
Superposition and entanglement
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 | = (| AB + | AB ) / 2 = (| AB + i| AB – 3| AB ) /
n
B entangled states, i.e.
not of form | A | B Example: | AB = (| AB + | AB + | AB + | AB ) / 2 = | AB
Quantum entanglement
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
Entanglement
“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?
Local Realism
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: measurement 2: color result: parity result:
A A
1 2 (Alice), (Alice),
B 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
A
1
B
1 +
A
1
B
2 +
A
2
B
1 –
A
2
B
2 = ±2 ≤ 2 Alice for all local realistic (= classical) theories CHSH version (1969) of Bell’s inequality (1964) Bob
Quantum violation of Bell’s inequality
With the entangled quantum state | AB = (| AB + | AB ) / 2 and for certain measurement directions
a
1 ,
a
2 the left hand side of Bell’s inequality 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)
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)
What is nature?
Niels Bohr (1885 –1962)
What can be said about nature?
Cryptography
Symmetric encryption techniques
plain text encryption cipher text decryption plain text 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“) 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
Quantum key distribution (QKD)
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 experimental realization (2000)
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)
8 km free space above Vienna (2005)
Millennium Tower Twin Tower Kuffner Sternwarte K. Resch
et al
.,
Opt. Express
13, 202 (2005)
Tokyo QKD network (2010)
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
The next step
ISS (350 km Höhe)
Moore’s law (1965)
Gordon Moore
2000 2010 2020 Transistor size
200 nm 20 nm 2 nm (?)
Computer and quantum mechanics
1981: Nature can be simulated best by quantum mechanics Richard Feynman 1985: Formulation of the concept of a quantum Turing machine David Deutsch
Quantum computer 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…
Qubits
General qubit state:
Bloch sphere:
P
(„0“) = cos 2 /2,
P
(„1“) = sin 2 /2 … phase (interference)
Physical realizations:
photon polarization: electron/atom/nuclear spin: atomic energy levels: superconducting flux: etc… |0 = | |0 = |up |0 = |ground |0 = |left |1 = | |1 = |down |1 = |excited |1 = |right | |R = |0 = |0 + |1 + i |1
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[(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“]
Possible implementations
NMR Trapped ions Photons SQUIDs NV centers Quantum dots Spintronics
Quantum teleportation
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
Quantum teleportation
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 trans formation has to be performed
Bell-state measurement
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
Entanglement swapping
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)
Delayed-choice entanglement swapping
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)
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
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)
Acknowledgments
A. Zeilinger X. Ma R. Ursin B. Wittmann T. Herbst S. Kropascheck