cloning of quantum states rafał demkowicz-dobrzański ift uw
TRANSCRIPT
Cloning of quantum statesRafał Demkowicz-Dobrzański IFT UW
The concept of cloning
• Perfect Quantum Cloning Machine
- produces perfect copies of input state
- works for arbitrary input state
|0
|QCM
|
|
• Is such machine allowed by laws of quantum mechanics?
• Description in Hilbert space 12A
|0| |A || |AU
Perfect cloning is imposible
• Two non-orthogonal quanum states cannot be cloned
thanks to unitarity:
Assumptions: we have two states such that |
contradiction
|U
Proof (Ad absurdum):
|
|
|
• We have to loosen our requirements
Imperfect cloning machines
• Different kinds of imperfect cloning machines:
• Fidelity
| out12A12A = out out|
U
1 = Tr2A(12A) – reduced density matrix for clone 1
2 = Tr1A(12A) – reduced density matrix for clone 21 = 2 -symetric cloning
F = 1|= Tr(1)
- faithful but not universal (limited set of states)
- universal but not faithful (fidelity less than 100%)
- not faithful and not universal
Optimal cloning machines for qubits
• Qubit
ab|1 |a|2+|b|2=1
|cossin)·exp(i|0Bloch sphere
|( + n)
• Optimal, universal cloning machine for qubits (Buzek,Hillery 1996)
Blank state|0
Input state| Clone 1
Clone 2
1= 2=( +2/3 n) =
= 2/3|
F=5/6 - fidelity
Optimal cloning machines for qubits• NM cloning of qubits(Gisin,Massar 1997)
QCM
|
|
|0
|0
N
M-N
clone 1
clone M
2)M(N
N1)M(NF
Cloning is strictly related to estimation theory
• Optimal cloning of two non-orthogonal states
• Telecloning = teleportation + cloning
• Optimal cloning (NM) in d-dimensional space (Werner 1998)
Cloninig the states of light
• Single mode of electromagnetic field
Infinite dimensional space. Basis of Fock states: |0, |1, |2, …
a, a† - anihilation, creation operators a |n = n |n-1 a† |n = n+1 |n+1
| - coherent state a | = | n
n2
n|n!
α )
2
|α|exp(-α|
• Beam splitter
|input state
|0blank state
clone 1
clone 2
For initial state: |
Expectation value: a1new|=
Single beam splitter is a very bad cloning machine
In the Heisenberg picture: a1new = 1/2(a1+a2)
a2new = 1/2(a1-a2 )
Optimal cloning of coherent states
|input state
|0blank state
clone 1
clone 2
• Optimal, universal cloning machine for coherent states
Amplifier
|0ancilla
a1new = 2a1+ aA
†
aAnew = a1
† + 2aA
a1new = a1+1/2(aA
† + a2)
a2new = a1+ 1/2(aA
† - a2)
aAnew = a1
† + 2aA
- preserves mean values of quadratures
- does not distinguish any direction in phase space
x2new- xnew2 = x2- x2 + 1/2 x = (a + a†)/2
- adds noise to copied state: (initial state = |||A )
Wigner function picture of cloning• Wigner function
)eeTr(ρeπ
βdW β*aβa/|βα*βαβ* 2|
2
22
)(†
• Wigner function of clones
= ||| - initial density matrix
)ee(ρTreπ
βdW β*aβa
A/|βα*βαβ*
input11
2
122|
2
2
)( † Wigner function
of input state
)ee(ρTreπ
βdW
newnew β*aβaA
/|βα*βαβ*clone
112
122|
2
2
)( †
Wigner function of either of clones
• Fidelity
)()()( 2 cloneinputcloneinput WWdTrF
Examples of cloned states• Coherent state |0
Winput()= 2/ exp(-2|-0|2) Wclone()= 1/ exp(-|-0|2)
F=2/3 – optimal cloning of coherent states (Cerf, Iblisdir 2000)
• Fock state |1
Winput()= -2/ (1-4||2) exp(-2||2) Wclone()= 1/ ||2 exp(-||2)
F= 10/27
Wigner functions of clones are positive
• Direct relation between Wigner functions of input and clone states
)2)22 2
(We (αW inputβ||αd
clone
• Q quasi probability distribution
Q() = || - positive
)2)22 2
W(e Q(α β||αd
Wclone() = Qinput()
• In this cloning process
close relation to joint-meassurement
Final remarks• NM optimal cloning of coherent states (Cerf, Iblisdir 2000)
NMMN
MNF
• Superluminal communication via cloning (Dieks 1982)
- If perfect cloning was possible superluminal communication would be possible
- Alice and Bob share entagled qubit pair
- Alice can make two kinds of meassurements (projecting on two different basis)
- If cloning was possible Bob would know what basis Alice had chosen
Cloning of quantum statesRafał Demkowicz-Dobrzański IFT UW