Download Quantum computation communication theory

October 23, 2002
Theoretical Aspects of
Experimental Entanglement and
Teleportation
Horace P. Yuen, Max Raginsky, Mauro
D’Ariano, Masanao Ozawa
Entanglement and Teleportation
•
•
•
•
Qubit Entanglement
Qumode Entanglement
Entanglement Concentration
Quantum Repeaters
– Device & System Imperfections
– Possible Applications vs. Classical
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Entanglement
• Qubit:
Ψ
−
=
1
2
(0
1
1
2
− 1
0
1
2
)
maximally entangled
—
measure of degree of entanglement E
Bell states Ψ µ
(basis)
Φµ ∆
• Qumode:
1
2
(0
1
0
2
2-mode squeezed state
Ψ
AB
[(
= exp r a a − a A a B
λ = tanh r
E(Ψ )
†
A
†
B
)]
0 = 1− λ
2
µ 1
3
1
∞
2
λ
∑ n
n =0
can be increased by
QND measurement of NA+NB
1
A
2
n
)
B
Effect of Imperfection
• Error model of
– 1-qubit operations p1
– 2-qubit operations p2
– 1-qubit measurement η
1− p 1
2
1−η error propabilities
• Imperfect Bell-measurement described by
– 1 imperfect 2-qubit operation
– 2imperfect 1-qubit measurements
⇒
Fmax maximum attainable F, Fmax < 1
Fmin minimum F necessary for improvement
F~1-ε
ε linear is imperfection
4
General Problem
• If state is totally unknown, can(not) use teleportation
– Qubit Case:
↑1→ 2− →1↑ 2
0
Need to know relative polarization to make the appropriate Bell
measurement
– Qumode case:
• q0
q1
q2
p1
p2
Need to know relative phase to make the appropriate homodyne
measurement
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Coherence and Teleportation
•
•
Optical Coherence Requirement for Continuous Variable Quantum
Teleportation
– Rudolph and Sanders PRL 87, 2001
Optical Coherence: a convenient fiction
– Molmer
PRA 55, 1997
? quantum coherence
* uniform random phase in (laser) coherent state amplitude
⇒ diagonal in number-state representation
⇒ no coherent 2-mode squeezed state
* usual coherence a result of same source
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Coherence cont.
• The Quantum State of propagation laser field
– van Enk and Fuchs
q-ph Nov 2001 (Journal of quantum Information)
Field state inside cavity vs. outside
• The Partitum Ensemble Fallacy Fallacy
– Nemoto and Braunstein
q-ph Jul 2002
Phase unobservable in-principle
*privileged role of coherent state
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Space-time coherence
• Any single-mode excited field is fully space-time coherent
– So ρ diagonal in n O.K.
– “Coherent states” special in that a multimode CS field is
equivalent to a single-mode one
– Only fixed relative phases between modes essential for
homodyne detection
• Field expansion (quantum Karhunen-Loeve)
E( x, t ) = ∑ akΨ k (x, t )
k
Joint ρ for all k-modes, α k classically
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Difference between LED & LASER
• Multimode random variables {αk } vs single appropriate
mode (random phase or not)
• Phase Φ(t) fluctuates slowly for lasers − a fixed unknown
parameter Φ0 over observation time
•
Φ0 can be measured
– Split strong beam and measure the phase
– Laser field does have a definite phase
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Squeezed States Teleportation
c = µa + νb
d = µb + νa
†
†
[a,a ] = I
†
[a, b] = 0
etc.
ν → νe
jφ
φ pump laser phase
• Only need to look at appropriately phase-locked
• Phase of state to be teleported defined also by
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{ a,b} modes
{ a,b}
Fallacies:
• Confuse unknown parameter with a random variable
• Confuse ensemble average with time average
• Confuse relative phase with absolute phase
– cancellation of fluctuation
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Entanglement at distance:
Qubits vs. Qumodes
• Problem:
– Obtain maximally entangled states suitable for teleportation in the
presence of loss.
– Compare efficiency of the two following schemes for establishing N
shared ebits (entangled qubit pairs) in the presence of loss:
• (a) N singlets: N single ebits with qubits encoded on polarization;
• (b) 1 twin beam: 1 pair of entangled qumodes + LOCC to obtain a
maximally entangled state of dim 2N, with qubits encoded on photon
number.
• Result:
– Scheme (a) is exponentially more efficient than scheme (b).
Moreover:
– Scheme (a) allows knowingly successful teleportation with a fixed
protocol/machine.
G. M. D’Ariano and M. F. Sacchi, quant-ph/0205073 (to appear on Forscr. der Phys.)
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Anonymous Key Identification (AKI)
• If quantum memory is available, one can have
1. A identifies to B without on-line third party
2. B cannot pretend to be A
3. Can solve the N2 problem in key management as follows:
challenge
A
| θA + θB 〉, | θ’B 〉
B
| θA 〉 stored,
θA known only to A,
picks random θB, θ’B
response
A
B
| θB 〉
| θA + θ’B 〉 to replenish
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Checks | θ B 〉,
stores | θA 〉
Two Distinct Issues
• Whether there is, for US QBC:
– An impossibility theorem
– A protocol provably unconditionally secure
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A priori No Impossibility Theorem
without QBC Definition
• Similar to:
– No Church-Turing theorem (only Thesis) without
definition of “algorithm”
• Different Forms of algorithms:
– Turing, Post, Markov, etc.
• Different Types of QBC protocols:
– 1,2,3,4,… from cheating detection,
quantum games,
evidence questioning etc.
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Fundamental Stability Issues
for Large-Scale Quantum Computers
•
As we have shown, any physical channel can be modeled arbitrarily
closely by a Strictly Contractive Quantum Channel (SCC)
– SCC causes any initial state to evolve exponentially
to a unique final state
– Maximally-mixed-state (MMS) preserving SCC
strictly increases the entropy of any other state
•
•
Goal: Determine stability of large scale quantum computers under the
influence of MMS-preserving SCCs
Results:
– Derived lower bounds on increase of entropy due to such SCCs
– Derived limits on run-time and size of circuit-based quantum computers
– Cellular automata (interacting particle systems) provide a more robust
medium for quantum information processors
M. Raginsky, "Entropy production rates of bistochastic SCC on a matrix algebra,” mathph/0207041; to appear in J. Phys A.
“Entropy-energy balance in noisy quantum computers,” QCMC’02 Proceedings, to
appear.
“Almost any quantum spin system with short-range interactions can support toric
codes,” Phys. Lett. A 294, 153 (2002).
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Quantum Limits on Quantum Computing
Induced by Conservation Laws
• Assume:
– (i)Computational basis is represented by a component of spin
– (ii)Physical realizations obey the angular momentum conservation
laws
• Then, controlled NOT gates cannot be realized with error
probability less than 1/4S2, where S is the size of the ancilla.
– Fermionic ancilla: S = # of qubits included in the ancilla particles
– Bosonic ancilla: S2 = 4 x (the average photon # in the ancilla field)
• To prevent this limitation altogether, we need to code the
computational basis to commute with the angular momentum.
M. Ozawa, “Conservation laws, uncertainty relations, and quantum limits
of measurements,” Phys. Rev. Lett. 88, 050402 (2002);
“Conservative Quantum Computing,” Phys. Rev. Lett. 89, 0507902 (2002).
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Future Plan
I.
Study the possibility of robust teleportation of a qubit vs
qumode
II.
Develop appropriate direct criterion for imperfect
teleportation of quantum states with corresponding efficient
entanglement concentration schemes
III. Quantify AKI key management protocol performance
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