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