1. Cloning of quantum states Rafał Demkowicz-Dobrzański IFT UW

2. 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 H 1  H 2  H A |0 || |A|A || || |A|A U

3. 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

4. Imperfect cloning machines Different kinds of imperfect cloning machines: Fidelity |   out    H 1  H 2  H A  12A =  out   out | U  1 = Tr 2A (  12A ) – reduced density matrix for clone 1  2 = Tr 1A (  12A ) – reduced density matrix for clone 2  1 =  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

5. Optimal cloning machines for qubits Qubit   a  b|1  |a| 2 +|b| 2 =1 |  cos  sin  )·exp(i  |0  Bloch 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

6. Optimal cloning machines for qubits N  M cloning of qubits(Gisin,Massar 1997) QCM || || |0    N M-N clone 1  clone M Cloning is strictly related to estimation theory Optimal cloning of two non-orthogonal states Telecloning = teleportation + cloning Optimal cloning (N  M) in d-dimensional space (Werner 1998)

7. 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 |  =  |  Beam splitter || input state |0  blank state clone 1 clone 2 For initial state: |  Expectation value:  a 1 new |  =  Single beam splitter is a very bad cloning machine In the Heisenberg picture: a 1 new = 1/  2(a 1 +a 2 ) a 2 new = 1/  2(a 1 -a 2 )

8. Optimal cloning of coherent states || input state |0  blank state clone 1 clone 2 Optimal, universal cloning machine for coherent states Amplifier |0  ancilla a 1 new =  2a 1 + a A † a A new = a 1 † +  2a A a 1 new = a 1 +1/  2(a A † + a 2 ) a 2 new = a 1 + 1/  2(a A † - a 2 ) a A new = a 1 † +  2a A - preserves mean values of quadratures - does not distinguish any direction in phase space  x 2 new  -  x new  2 =  x 2  -  x  2 + 1/2 x = (a + a † )/  2 - adds noise to copied state: (initial state = |   |   |  A )

9. Wigner function picture of cloning Wigner function † Wigner function of clones  = |  |  |  - initial density matrix † Wigner function of input state † Wigner function of either of clones Fidelity

10. Examples of cloned states Coherent state |  0  W input (  )= 2/  exp(-2|  -  0 | 2 ) W clone (  )= 1/  exp(-|  -  0 | 2 ) F=2/3 – optimal cloning of coherent states (Cerf, Iblisdir 2000) Fock state |1  W input (  )= -2/  (1-4|  | 2 ) exp(-2|  | 2 ) W clone (  )= 1/  |  | 2 exp(-|  | 2 ) F= 10/27

11. Wigner functions of clones are positive Direct relation between Wigner functions of input and clone states Q quasi probability distribution Q(  ) =  |  |  - positive W clone (  ) = Q input (  ) In this cloning process close relation to joint-meassurement

12. Final remarks N  M optimal cloning of coherent states (Cerf, Iblisdir 2000) 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

13. Cloning of quantum states Rafał Demkowicz-Dobrzański IFT UW