Quantum information with linear optics

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Quantum information with linear optics
A tutorial
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Outline
-What is quantum information? Qubits quantum
gates, realisations,what is it good for?
-Quantum information with linear optics
Examples Beam splitter Quantum data
compression Knill-Laflamme-Milburn
scheme Quantum scissors/teleportation
Generalised quantum measurements
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What is quantum information?
Classical bit
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What is it good for?
Quantum computers are good at -factoring large
numbers (cracking crypto) -database search What
lies behind many quantum algorithms is the
quantum Fourier transform.
QUANTUM- PUTER
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What more is it good for?
Quantum communication -quantum
teleportation -quantum key distribution (quantum
cryptography) -quantum data compression Requires
a small quantum computer, therefore more
realistic. Typically use photons.
Beate
Eavesdropping
Angus
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Qubit realisations with photons
Discrete Polarisation
Hgt
Vgt
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What is linear optics?
Eout is a linear function of Ein
U al,in0gt Sm Uml am,out0gt
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What is not linear optics?
Eout is a non-linear function of Ein
Squeezers, amplifiers
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What is the problem with linear optics?
Photons do not interact, so quantum version of
controlled-NOT gate (CNOT) is hard to make.
Classical CNOT
C T C T 0 0 0 0 0 1 0
1 1 0 1 1 1 1 1 0
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A way out One photon, many paths
00gt
01gt
10gt
11gt
CNOT is easy!
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Quantum data compression experiment
2 path qubits, 1 polarisation qubit
0
1
2
3
4
D4
D5
A
A
A
D6
A
C
B
D1
A
B
B
C
D
D
D
D2
D3
Y. Mitsumori, J. A. Vaccaro, S. M. Barnett, E.
Andersson, A. Hasegawa, M. Takeoka, M. Sasaki,
PRL 91, 217902 (2003).
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Another way out Particle statistics
Classical beam splitter
50
50
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Particle statistics Quantum beam splitter
Two photons incident on a beam splitter
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Beam splitter quantum maths
Linear transformation
ain (aoutibout) /v2 bin (iaoutbout)/v2
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Quantum state comparison
Q How do we tell if two quantum states are the
same or different without knowing anything about
the states?
agtcosqHgt sinqVgt
Ex. The polarisation states of photons
Vgt
q
Hgt
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Quantum state comparison
Solution Use a beam splitter! If the states of
the two photons are the same, they will always
exit together. Only if they are different can
they exit in different ports.
Barnett et al., Phys. Lett. A 307, 189 (2003),
quant-ph/0202087 Andersson et al., J Phys A
Math. Gen. 36, 2325 (2003), quant-ph/0208153 Jex
et al., J Mod. Opt. 51, Number 4/10 March 2004,
quant-ph/0305120.
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Quantum computation Knill-Laflamme-Milburn
scheme
Use quantum statistics to fake interaction
Two photons Mode a, b for the first, c, d for
the second
Knill et al. Nature 409, 46 (2001) theory Sanaka
et al. PRL 92, 017902 (2004) experiment.
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Non-deterministic QCQuantum scissors/teleportati
on
Dc
outgt
Db
0gt
b
a
c
1gt
ingt a0gtb1gt
If Db detects 1 and Dc detects 0 photons, then
outgta0gtb1gt
Probability of success 1/4 (or even 1/2)
Pegg, Phillips and Barnett, PRL 81, 1604 (1998)
theory Babichev, Ries, Lvovsky, Europhys. Lett.
64, 1 (2003) experiment
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Quantum computation Knill-Laflamme-Milburn
scheme
Use quantum statistics probabilistic
operation to fake interaction.
Have to try many times
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Generalised quantum measurements
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How do we distinguish between non-orthogonal
polarisation states?
Vgt
agt
-important in quantum communication and in
quantum cryptography
Hgt
bgt
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Optimal measurement has three outcomes
agt
Vgt
agt
?gt
agt bgt, bgt agt
q
Hgt
bgt
bgt
Minimum p?cos 2q ltabgt
(Ivanovic 1987, Dieks, Peres 1988, Jaeger,
Shimony 1995)
but how do we realise the measurement?
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Generalised quantum measurement with linear
optics
a
b
PBS
l/2
PBS
Clarke et al. PRA 63, 040305(R) (2001).
?
PBS
agt,bgt
0gt (ancilla)
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Ideal von Neumann measurements
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Photodetection
light pulse
detector efficiency h
Each photon detected with probability h
undetected with probability 1-h
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Generalised quantum measurements (POMs or POVMs)
Carl W. Helstrom
Alexander S. Holevo
Karl Kraus
E Davies
and others
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Distinguish between non-orthogonal states
agt
2gt
bgt
?gt
1gt
agt cosq0gtsinq1gt
yagt
bgt cosq0gt-sinq1gt
y?gt
0gt
agt bgt, bgt agt
ybgt
(Ivanovic 1987, Dieks, Peres 1988, Jaeger,
Shimony 1995)
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Summary

• Qubits linear optics intro

• Problem
• Interaction difficult with linear optics
• -Solutions
• One photon, many paths (quantum coding)
• Particle statistics (beam splitter, comparison)
• Non-deterministic operations (KLM scheme,
• quantum scissors/teleportation)

• Measurement Distinguish non-orthogonal states