9 Algorithmic Cooling (20 points) The idea of this problem is to find out about how we can cool a qubit by doing a unitary transformation of a number of them. Here we will use 3 qubits and cool one of them. A qubit in contact with a bath has the state 1 (1 + Eb 0 1 pb = 3 0 1-EB and polarization P=tr[P2] = 5. If we have 3 of these qubits then we have the state p12a = pt pt . a) Ify > 0, show that the probability of being in the state (011) is smaller than the probability of being in (100). b) It is possible to reduce the entropy of the 1st spin by swapping the state (011) and (100) with the unitary transformation: [1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0] 0 0 0 0 0 0 1 =1-100100-10117011| + |1007011| +011/100 (2) Show that the new density matrix is diagonal with the following diagonal elements (1 + ) 1 (1 +)'(1 - ) (1 +)'(1 - 1) pifter (1 +)'(1-) * (1 + E) (1 - ) (1 + c)(1 - ) (1+) (1 - ) (1 - ) (4) e) Find the reduce density matrix of the 1st qubit for small value of pafter = trga pilter Show that the polarization (P = tr[poz]) is bigger after the swapping of states operation (as long as > 0) d) Calculate the entropy of the first qubit before and after the swap operation (remember S = -tr|p In pl -EipyIn pi where In is base 2). And show that the operation has reduced the entropy of the first qubit. e) Bonus question. Show that the entropy of qubit 2 and qubit 3 has increased for some E) 9 Algorithmic Cooling (20 points) The idea of this problem is to find out about how we can cool a qubit by doing a unitary transformation of a number of them. Here we will use 3 qubits and cool one of them. A qubit in contact with a bath has the state 1 (1 + Eb 0 1 pb = 3 0 1-EB and polarization P=tr[P2] = 5. If we have 3 of these qubits then we have the state p12a = pt pt . a) Ify > 0, show that the probability of being in the state (011) is smaller than the probability of being in (100). b) It is possible to reduce the entropy of the 1st spin by swapping the state (011) and (100) with the unitary transformation: [1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0] 0 0 0 0 0 0 1 =1-100100-10117011| + |1007011| +011/100 (2) Show that the new density matrix is diagonal with the following diagonal elements (1 + ) 1 (1 +)'(1 - ) (1 +)'(1 - 1) pifter (1 +)'(1-) * (1 + E) (1 - ) (1 + c)(1 - ) (1+) (1 - ) (1 - ) (4) e) Find the reduce density matrix of the 1st qubit for small value of pafter = trga pilter Show that the polarization (P = tr[poz]) is bigger after the swapping of states operation (as long as > 0) d) Calculate the entropy of the first qubit before and after the swap operation (remember S = -tr|p In pl -EipyIn pi where In is base 2). And show that the operation has reduced the entropy of the first qubit. e) Bonus question. Show that the entropy of qubit 2 and qubit 3 has increased for some E)