Cloning of quantum states

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

2. |Y |Y QCM | |0 |Y |0 |Y |Y |A U |AY The concept of cloning • Perfect Quantum Cloning Machine • produces perfect copies of input state • works for arbitrary input state • Description in Hilbert space H1H2HA • Is such machine allowed by laws of quantum mechanics?

3. Proof (Ad absurdum): |Y|0|A |Y|Y|AY U |F|0|A |F|F|AF thanks to unitarity: F|Y0|0A|A = F|YF|YAF|AY contradiction Perfect cloning is imposible • Two non-orthogonal quanum states cannot be cloned Assumptions: we have two states |Y, |F such that 0 < |F|Y| < 1 F|Y(1- F|YAF|AY)=0 • We have to loosen our requirements

4. U |Y|0|A 12A = |Yout Yout| |Yout H1H2HA 1 = Tr2A(12A) – reduced density matrix for clone 1 1 = 2 -symetric cloning 2 = Tr1A(12A) – reduced density matrix for clone 2 F = Y| 1|Y = Tr(|YY| 1) Imperfect cloning machines • Different kinds of imperfect cloning machines: - faithful but not universal (limited set of states) - universal but not faithful (fidelity less than 100%) - not faithful and not universal • Fidelity

5. |1 • Qubit |Y = a|0+b|1 |a|2+|b|2=1 Bloch sphere |YY| =1/2 ( + n) |0 Input state |Y = Clone 1 1= 2=1/2 (  +2/3 n) = 2/3|YY|+1/6  Blank state |0 F=5/6 - fidelity Clone 2 Optimal cloning machines for qubits |Y = cos(q/2) |1+sin(q/2)·exp(if) |0 • Optimal, universal cloning machine for qubits (Buzek,Hillery 1996)

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

7. Basis of Fock states: |0, |1, |2, … Infinite dimensional space. a, a†- anihilation, creation operators a |n = n |n-1 a† |n = n+1 |n+1 | - coherent state a | = | blank state In the Heisenberg picture: a1new = 1/2(a1+a2) a2new= 1/2(a1-a2 ) |0 |Y For initial state: |a|0 input state clone 1 Expectation value: a|0|a1new|a|0 = a/2 clone 2 Cloninig the states of light • Single mode of electromagnetic field • Beam splitter Single beam splitter is a very bad cloning machine

8. ancilla |0 a1new = a1+1/2(aA†+ a2) a2new = a1+ 1/2(aA†- a2) aAnew = a1†+ 2aA a1new= 2a1+ aA† aAnew = a1†+ 2aA blank state |0 - adds noise to copied state: (initial state = |Y1|02|0A ) |Y x2new- xnew2 = x2- x2 + 1/2 x = (a + a†)/2 input state clone 1 clone 2 Optimal cloning of coherent states • Optimal, universal cloning machine for coherent states Amplifier - preserves mean values of quadratures - does not distinguish any direction in phase space

9. † • Wigner function of clones  = |YY||00||00| - initial density matrix Wigner function of input state † † Wigner function of either of clones • Fidelity Wigner function picture of cloning • Wigner function

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

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

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