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Quantum teleportation in a nutshell Fabian Kössel
Quantum teleportation in a nutshell
Fabian Kössel
Technische Universität München and MPQ
June 12th, 2013
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Why Quantum Teleportation?
Transfer of a quantum state or QuBit from one position (Alice) to another
(Bob).
I Useful in Quantum information, Quantum cryptography,. . .
I It’s not about transportation or dis- and reassembling of matter.
Sorry, Trekkies!
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Recap: The QuBit
|ψ〉 = cos(θ2
)|0〉+ eiϕ sin
(θ2
)|1〉
QuBit on bloch sphere
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
How to transport a QuBit?
1.
Classically: Copy bit and transfer it, hence clone it.
I Doesn’t work for QuBit. Can’t retreive complete
information of state.
2.Physically transport QuBit, i.e. carry it from A to B.
I Lossy. Short coherence times of quantum state.
3. Quantum Teleportation
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
How to transport a QuBit?
1.
Classically: Copy bit and transfer it, hence clone it.
I Doesn’t work for QuBit. Can’t retreive complete
information of state.
2.Physically transport QuBit, i.e. carry it from A to B.
I Lossy. Short coherence times of quantum state.
3. Quantum Teleportation
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
How to transport a QuBit?
1.
Classically: Copy bit and transfer it, hence clone it.
I Doesn’t work for QuBit. Can’t retreive complete
information of state.
2.Physically transport QuBit, i.e. carry it from A to B.
I Lossy. Short coherence times of quantum state.
3. Quantum Teleportation
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Basic scheme of quantum teleportation
EPR
BSM
2 3
entangled1
|ψin〉1
classical information
Alice
Bob
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Recap: Entanglement and Bell basis
EPR
BSM
2 3
entangled1
|ψin〉1
classical information
Alice
Bob
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Recap: Entanglement and Bell basis
I Strong correlation between quanta that is unique in quantum
mechanics
I A special entangled basis of two-QuBit-system: Bell basis
|ψ±〉 = 1√2(|0〉 |1〉 ± |1〉 |0〉)
|φ±〉 = 1√2(|0〉 |0〉 ± |1〉 |1〉)
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Bell State Measurement (BSM)
EPR
BSM
2 3
entangled1
|ψin〉1
classical information
Alice
Bob
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Bell State Measurement (BSM)
Joined measurement of two incoming particles.
1
2
BSM2 bits
|ψ−〉 , |ψ+〉 , |φ−〉 or |φ+〉
I projection onto one of four Bell states
I thus 2 bits are needed to express outcome
I destroys incoming QuBits
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
EPR
BSM
2 3
entangled1
|ψin〉1
classical information
Alice
Bob
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
1. Alice shares with Bob an entangled pair
of QuBits |ψ−〉2,3.
2. Alice performs a joined BSM on her own
initial state and her own QuBit of the EPR
pair and detects on which of the four Bell
state the incoming QuBits were
projected.
3. Alice sends this information from the
BSM to Bob.
EPR
BSM
23
entangled1
|ψin〉1
classical information
Alice
Bob
1.
2.3.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
1. Alice shares with Bob an entangled pair
of QuBits |ψ−〉2,3.
2. Alice performs a joined BSM on her own
initial state and her own QuBit of the EPR
pair and detects on which of the four Bell
state the incoming QuBits were
projected.
3. Alice sends this information from the
BSM to Bob.
EPR
BSM
23
entangled1
|ψin〉1
classical information
Alice
Bob
1.
2.
3.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
1. Alice shares with Bob an entangled pair
of QuBits |ψ−〉2,3.
2. Alice performs a joined BSM on her own
initial state and her own QuBit of the EPR
pair and detects on which of the four Bell
state the incoming QuBits were
projected.
3. Alice sends this information from the
BSM to Bob.EPR
BSM
23
entangled1
|ψin〉1
classical information
Alice
Bob
1.
2.3.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
Three QuBit-System of incoming state |ψin〉1 and entangled |ψ−〉2,3 EPR pair.
First rewrite it (ignoring normalizations). . .
|ψ〉1,2,3 = |ψin〉1 |ψ−〉2,3
= |ψin〉1 |0〉2 |1〉3 − |ψin〉1 |1〉2 |0〉3
... → change to Bell basis of1
and2
= |ψ−〉1,2 U1 |ψin〉3
+ |ψ+〉1,2 U2 |ψin〉3
+ |φ−〉1,2 U3 |ψin〉3
+ |φ+〉1,2 U4 |ψin〉3
〈ψ+|1,2 U2 |ψin〉3
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
Three QuBit-System of incoming state |ψin〉1 and entangled |ψ−〉2,3 EPR pair.
First rewrite it (ignoring normalizations). . .
|ψ〉1,2,3 = |ψin〉1 |ψ−〉2,3
= |ψin〉1 |0〉2 |1〉3 − |ψin〉1 |1〉2 |0〉3
... → change to Bell basis of1
and2
= |ψ−〉1,2 U1 |ψin〉3
+ |ψ+〉1,2 U2 |ψin〉3
+ |φ−〉1,2 U3 |ψin〉3
+ |φ+〉1,2 U4 |ψin〉3
〈ψ+|1,2 U2 |ψin〉3
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
1. Alice shares with Bob an entangled pair
of QuBits |ψ−〉2,3.
2. Alice performs a joined BSM on her own
initial state and her own QuBit of the EPR
pair and detects on which of the four Bell
state the incoming QuBits were
projected.
3. Alice sends this information from the
BSM to Bob.
4. Bob applies one of four unitary
transformations Ui on his now collapsed
QuBit from the EPR pair and receives
Alice’s initial state.
EPR
BSM
23
entangled1
|ψin〉1
classical information
Alice
Bob
1.
2.3.
4.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Quantum Teleportation Protocol
1. Alice shares with Bob an entangled pair
of QuBits |ψ−〉2,3.
2. Alice performs a joined BSM on her own
initial state and her own QuBit of the EPR
pair and detects on which of the four Bell
state the incoming QuBits were
projected.
3. Alice sends this information from the
BSM to Bob.
4. Bob applies one of four unitary
transformations Ui on his now collapsed
QuBit from the EPR pair and receives
Alice’s initial state.
EPR
BSM
23
entangled1
|ψin〉1
classical information
Alice
Bob
1.
2.3. 4.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
global phase shift
|ψ−〉 :
U1 =
−
1 0
0 1
phase shift
|ψ+〉 :
U2 =−1 0
0 1
spin flip
|φ−〉 :
U3 =0 1
1 0
phase shift + spin flip
|φ+〉 :
U4 =0 −1
1 0
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Information
Both information channels are needed to reconstruct state!
unita
rytr
ansf
or
matio
nUi , i = 1, . . . ,4
Alice
BobI Classical information is not enough!
I You need a state on which you can
apply the unitary transformations.
I Otherwise one measurement would
be enough to fully characterize
state.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Information
Both information channels are needed to reconstruct state!
I Quantum channel is not enough!
I Ensemble is in a perfectly mixed
state. Need information about right
transformation.
I Otherwise causality would be
violated and instant transfer of
information would be possible.
∑i Ui |ψin〉
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Information
Both information channels are needed to reconstruct state!un
itary
tran
sfor
matio
nUi , i = 1, . . . ,4
Alice
Bob
∑i Ui |ψin〉
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
First experimental realisation – Setup
EPR
BSM
2 3
entangled1
|ψin〉1
classical information
Alice
Bob
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
First experimental realisation – Setup
EPR
Alice Bob
trigger
f1
f2
d1
d2
UV pulse
beam splitter
polarizing beam splitter
inital state preparation
EPR-Source
1 2 3
entangled
Weinfurter, Zeilinger 1997, Nature
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
EPR-Source
EPR
trigger
f1
f2
d1
d2
UV pulse
beam splitter
polarizing beam splitter
inital state preparation
EPR-Source
1 2 3
entangled
Weinfurter, Zeilinger 1997, Nature
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
EPR-Source
Spontaneous parametric down-conversion, Wikipedia
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
BSM with a beam splitter
Alice
trigger
f1
f2
d1
d2
UV pulse
beam splitter
polarizing beam splitter
inital state preparation
EPR-Source
1 2 3
entangled
Weinfurter, Zeilinger 1997, Nature
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
BSM with a beam splitter
a
b
|Ψ 〉 = |ψ〉inner · |χ〉spatial
With photons (bosons) the state has to be symmetric under
exchange of particles.
|ψ〉inner |χ〉spatial
|ψ−〉 = 1√2(|0〉1 |1〉2 − |1〉1 |0〉2) · 1√
2(|a〉1 |b〉2 − |b〉1 |a〉2)
|ψ+〉 = 1√2(|0〉1 |1〉2 + |1〉1 |0〉2) · 1√
2(|a〉1 |b〉2 + |b〉1 |a〉2)
|φ−〉 = 1√2(|0〉1 |0〉2 − |1〉1 |1〉2) · 1√
2(|a〉1 |b〉2 + |b〉1 |a〉2)
|φ+〉 = 1√2(|0〉1 |0〉2 + |1〉1 |1〉2) · 1√
2(|a〉1 |b〉2 + |b〉1 |a〉2)
spatialantisymmetric
spatial symmetric
Hong-Ou-Mandel
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Results
EPR
Alice Bob
trigger
f1
f2
d1
d2
UV pulse
beam splitter
polarizing beam splitter
inital state preparation
EPR-Source
1 2 3
entangled
Weinfurter, Zeilinger 1997, Nature
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Results
I Three-fold coincidence (click) at f1, f2 (→ |ψ−〉1,2) and output port d1 or
d2 corresponding to the input polarisation.
For example:
|ψ〉in = |1〉 Telep.PBS
8|0〉 no click!
3|1〉 click!
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Results
Checking teleporation for a complete basis and linear combinations.
Polarization Visibility
+45° 0.63± 0.02
-45° 0.64± 0.02
0° 0.66± 0.02
90° 0.61± 0.02
circular 0.57± 0.02
After substracting background noise a visibility of approximately 70%± 3%
was achieved.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Results
BUT protocoll isn’t fully implemented. Only discriminates |ψ−〉 from the rest at BSM.
→ By now there are more sophisticated experiments.
Experiment What? Remark
Photons 1997 1 photon via photon to photonworks at best in 25% of cases, very
long distances
Cavity 2013 2 atom via photon to atomhigher efficiency due to cavity, long
distance
Ions 2004 3 trapped ions full protocol, short distances
1Weinfurter, Zeilinger, et al. , 1997, Nature2Rempe, Ritter, et al., 2013, APS3Ozeri, Wineland, et al., 2004, Letters to Nature
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Summary
EPR
BSM
2 3
entangled1
|ψin〉1
classical information
Alice
Bob
Quantum teleportation makes it possible to reliably transfer a QuBit from on place to another.
By taking advantage of entanglement, first only expandable messenger particles have to be
exchanged through a (lossy) channel. And only, when this has worked, the relevant
information is transferred. Thus minimizing transfer losses of QuBit.
Motivation Theory Experiment Summary References
Quantum teleportation in a nutshell Fabian Kössel
Literature
M. D. Barret et al. “Deterministic quantum teleportation of atomic qubits”. In: Letters to
Nature (2004).
Charles H. Bennet et al. “Teleporting an Unkown Quantum State via Dual Classical and
Einstein-Podolsky-Rosen Channels”. In: Physical Review Letters (1993).
Dik Bouwmeester et al. “Experimental quantum teleportation”. In: Nature 390 (1997).
Samuel L. Braunstein and A. Mann. “Measurement of the Bell operator and quantum
teleportation”. In: The American Physical Society (1995).
Richard A. Campos, Bahaa E. A. Saleh, and Malvin C. Teich. “Teleporting an Unkown
Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels”. In: The
American Physical Society (1989).
Christian Noelleke et al. “Efficient Teleportation Between Remote Single-Atom
Quantum Memories”. In: The American Physical Society (2013).
Motivation Theory Experiment Summary References